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14th International Geometry Symposium

25-28 May 2016

ABSTRACT BOOK

Pamukkale University

Denizli - TURKEY

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

14th International Geometry Symposium

ABSTRACT BOOK

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Proceedings of the 14th International Geometry Symposium

Edited By: Dr. Şevket CİVELEK Dr. Cansel YORMAZ

E-Published By:

Pamukkale University Department of Mathematics Denizli, TURKEY

All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means or whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder. Authors of papers in these proceedings are authorized to use their own material freely. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to:

Assoc. Prof. Dr. Şevket CİVELEK Pamukkale University Department of Mathematics Denizli, TURKEY Email: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Proceedings of the 14th International Geometry Symposium

May 25-28, 2016 Denizli, Turkey.

Jointly Organized by Pamukkale University Department of Mathematics Denizli, Turkey

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 PREFACE

This volume comprises the abstracts of contributed papers presented at the 14th International Geometry Symposium, 14IGS 2016 held on May 25-28, 2016, in Denizli, Turkey. 14IGS 2016 is jointly organized by Department of Mathematics, Pamukkale University, Denizli, Turkey. The sysposium is aimed to provide a platform for Geometry and its applications. The sysposium is proposed to offer a motivating environment to encourage discussion and exchange of ideas leading to endorsement of geometric subjects and structures.

This is a peer reviewed sysposium and all the papers included in the sysposium proceedings have been selected after a rigorous review process performed by the international Scientific committee. I would like to extend my appreciation to the International Advisory Committee, the International Scientific Committee and Local Organizing Committee for the devotion of their precious time, advice and hard work to prepare for this Sysposium. Appreciation is also due to our sponsors including Denizli Ticaret Odası, Ozan Tekstil, Denizli Sanayi Odası, Gamateks, Denizli Valiliği, Pamukklae Belediyesi, Denizli İhracatcılar Birliği, Denizli Ticaret Borsası, Murat Eğitim Kurumları, Pamukkale University and Colossae Thermal Hotel. I would like to acknowledge and give special appreciation to our invited speakers who are Prof. Dr. H. Hilmi HACISALİHOĞLU, Prof. Dr. Ali GÖRGÜLÜ, Prof. Dr. Osman GÜRSOY, Prof. Dr. Cengizhan MURATHAN, Prof. Dr. Gennadi SARDANASHVILY, Prof. Dr. Manuel De LEON, Prof. Dr. Mukut Mani TRIPATHI, Prof. Dr. Uday CHAND DE and Prof. Dr. Ioan BUCATARU for their valuable contribution, our delegates for being with us and sharing their experiences and our invitees for participating in 14IGS 2016, Denizli, Turkey.

Assoc. Prof. Dr. Şevket CİVELEK

Head of Organizing Committee

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

TABLE OF CONTENTS

CONTENTS PAGE HONARARY CHAIRMANS 12 Prof. Dr. Hüseyin BAĞCI Rector Prof. Dr. H. Hilmi HACISALİHOĞLU ORGANIZING COMMITTEE CHAIRMAN 12 Assoc. Prof. Dr. Şevket CİVELEK ORGANIZING COMMITTEE 12 SCIENTIFIC COMMITTEE 12 ADVISORY COMMITTEE 15 SECRETERIA Assoc. Prof. Dr. Cansel YORMAZ Assoc. Prof. Dr. Serpil HALICI 15 WEB DESINGNER Alper ÇAKIR 15 SPONSORS 17 INVITED SPEAKERS 20 “On The Fractals in Spider Network” Prof. Dr. H. Hilmi HACISALİHOĞLU 24 “On the Line Geometry” Prof. Dr. Osman GÜRSOY 25 “Mystery Behind the Contact Structure” Prof. Dr. Cengizhan MURATHAN 26 “The Differential Calculus on N-Graded Manifolds” Prof. Dr. Gennadi SARDANASHVILY 27 “Noether’s Theorems in a General Setting” Prof. Dr. Gennadi SARDANASHVILY 28 “The Geometry of the Hamilton-Jacobi Equation” Prof. Dr. Manuel de LEON 29 “Inequalities for Algebraic Casorati Curvatures and their Applications” 30 Prof. Dr. Mukut Mani TRIPATHI ABSTRACTS OF ORAL PRESENTATIONS 31 Lagrange Mechanical Systems on the Walker Manifolds and killing Magnetic Curves 32 Şevket CİVELEK Hamiltonian Energy Systems On Supermanifolds Cansel YORMAZ, Simge ŞİMŞEK 33 On Geometry of Quaternions whose Coefficients Fibonacci Numbers Serpil HALICI Şule ÇÜRÜK 34 Vector Matrix Representation of Octonions Serpil HALICI, Adnan KARATAŞ 35 On the Involute Supercurves Cumali EKİCİ, Cansel YORMAZ , Hatice TOZAK 36 On conharmonically flat Sasakian Finsler structures on tangent bundles Nesrin ÇALIŞKAN 37 Weyl-Euler-Lagrange Equations on Twistor Space for Tangent Structure Zeki KASAP 38 Spherical Circles Taxicab Süleyman YÜKSEL 39 Characterizations for new partner curves in the Euclidean 3-space Onur KAYA Mehmet ÖNDER 40 Some Notes on Almost Lorentzian r-Paracontact Structures on Tangent Bundle Haşim ÇAYIR 41 Some Relationships between Darboux and Typ-2 Bishop Frames Defined on Surface in Euclidean 3- 42 space Amine YILMAZ Emin ÖZYILMAZ A new Type of Almost Contact Manifolds Gülhan AYAR Alfonso CARRIAZO Nesip AKTAN 43 Geodesics on the Tangent Sphere Bundle of Pseudo Riemannian 3-Sphere İsmet AYHAN 44 Semi-Slant Riemannian Submersions From Locally Product Riemannian Manifolds 45 Hakan Mete TAŞTAN, Fatma ÖZDEMİR , Cem SAYAR On A New Type Of Framed Manifolds Nesip AKTAN Mustafa YILDIRIM Yavuz Selim BALKAN 46 Euler-Lagrange and Hamilton-Jacobi Equations on a Riemann Almost Contact Model of a Cartan 47 Space of order k Ahmet MOLLAOĞULLARI, Mehmet TEKKOYUN On Isotropic Leaves of Lightlike Hypersurfaces Mehmet GÜLBAHAR 48 Some Characterizations For Complex Lightlike Hypersurfaces 49 Erol KILIÇ Mehmet GÜLBAHAR Sadık KELEŞ

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A compactness theorem by use of m-Bakry-Emery Ricci tensor 50 Yasemin SOYLU Murat LİMONCU A Special Connection On 3-Dimensional Quasi-Sasakian Manifolds 51 Azime ÇETİNKAYA AHMET YILDIZ Getting an Hyperbolical Rotation Matrix by Using Householder’s Method in 3-Dimensional Space 52 Hakan ŞİMŞEK Mustafa ÖZDEMİR Timelike Translation Surfaces According To Bishop Frame In Minkowski 3-Space 53 Zehra EKİNCİ Melike YAĞCI Hasimoto Surfaces in Minkowski 3-Space with Parallel Frame 54 Melek ERDOĞDUİ Mustafa ÖZDEMİRI On the Line Congruences Ferhat TAŞ 55 Minimal Surfaces and Harmonic Mappings Hakan Mete TAŞTAN, Sibel GERDAN 56 Cubical Cohomology Groups of Digital Images Özgür EGE 57 Ruled Surface Reconstruction in Euclidean Space Mustafa DEDE, Cumali EKİCİ 58 On the spacelike parallel ruled surfaces with Darboux frame Muradiye ÇİMDİKER Cumali EKİCİ 59 On Triakis Octahedron Metric and Its Isometry Group Gürol BOZKURT Temel ERMİŞ 60 Umbilic Surfaces in Lorentz 3-Space Esma DEMİR ÇETİN Yusuf YAYLI 61 On the Mannheim curves in the three-dimensional sphere Tanju KAHRAMAN Mehmet ÖNDER 62 Complete lift s of tensor Fields of Type (1,1) on Cross-Section in a Special Class of Semi Cotangent 63 Bundles Furkan YILDIRIM, Kürşat AKBULUT Notes On The Curves According To Type-I Bishop Frame in Euclidean Plane 64 Süha YILMAZ Yasin ÜNLÜTÜRK Semi-invariant semi-Riemannian submersions from para-Kahler manifolds 65 Yılmaz GÜNDÜZALP Mehmet Akif AKYOL Lagrangian Dynamics on Matched Pairs Oğul ESEN Serkan SÜTLÜ 66 Reduction of Tulczyjew’s Triplet Oğul ESEN Hasan GÜMRAL 67 Spherical Motions And Dual Frenet Formulas Aydın ALTUN 68 The Timelike Bezier Spline in Minkowski 3 Space 69 Hatice KUŞAK SAMANCI Özgür BOYACIOĞLU KALKAN Serkan ÇELİK The Geometric Approach of Yarn Surface and Weft Knitted Fabric 70 Hatice KUŞAK SAMANCI Filiz YAĞCI Ali ÇALIŞKAN Some Solutions of the Non-minimally coupled electromagnetic fields to gravity Özcan SERT 71 Differantial Equations of Motion Objects with An Almost Paracontact Metric Structure 72 Oğuzhan ÇELİK Zeki KASAP Characterizations of Some Special Time-like Curves In Lorentzian Plane 73 Abdullah MAĞDEN Süha YILMAZ Yasin ÜNLÜTÜRK Contributions to Differential Geometry of Space-like Curves In Lorentzian Plane 74 Yasin ÜNLÜTÜRK Süha YILMAZ n1 75 On The Massey Theorem in E Cumali EKİCİ and Ali GÖRGÜLÜ Statistical Manifolds: New Approaches and Results Muhittin Evren AYDIN Mahmut ERGUT 76 Similarity and Semi-similarity Relations on Generalized Quaternions Abdullah İNALCIK 77 Examples of Curves which Spherical Indicatrices are Spherical Conics 78 Mesut ALTINOK Levent KULA On The Special Smarandache Curves Pelin POŞPOŞ TEKİN Erdal ÖZÜSAĞLAM 79 On Generalized Beltrami Surfaces in Euclidean Spaces 80 Didem KOSOVA Kadri ARSLAN Betül BULCA On the second order involute curves in 퐸3 Şeyda KILIÇOĞLU Süleyman ŞENYURT 81 4 Rational Surfaces Generated From The Split Quaternion Product of Two Rational Space Curves in 퐸2 82 Veysel Kıvanç KARAKAŞ Levent KULA Mesut ALTINOK Contact Pseudo-Slant Submanifolds of a Kenmotsu Manifold 83 Süleyman DİRİK Mehmet ATÇEKEN Ümit YILDIRIM

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 f-Biharmonicity Conditions for Curves Fatma KARACA Cihan ÖZGÜR 84 Rotational Surfaces in 3-Dimensional Isotropic Space Alper Osman ÖRENMİŞ 85 On the Generalization of Geometric Design and Analysis of a MMD Machine 86 Engin CAN Hellmuth STACHEL About The Generated Spacelike Bezier Spline with a Spacelike Principal Normal in Minkowski 87 3-Space Hatice KUŞAK SAMANCI Serkan ÇELİK 3 88 Constant Ratio Quaternionic Curves in Euclidean 3-Space E Günay ÖZTÜRK İlim KİŞİ Sezgin BÜYÜKKÜTÜK Tube Surfaces with Type-2 Bishop Frame 89 Ali ÇAKMAK Sezai KIZILTUĞ 1 90 Some Characterizations of Curves in Pseudo-Galilean 3-Space G3 İlim KİŞİ Sezgin BÜYÜKKÜTÜK Günay ÖZTÜRK On Factorable Surfaces in Euclidean 4-Space E4 Sezgin BÜYÜKKÜTÜK Günay ÖZTÜRK 91 Some Notes on Tachibana and Vishnevskii Operators Seher ASLANCI Haşim ÇAYIR 92 Isometry Groups of CO and TO Spaces Zeynep CAN Özcan GELİŞGEN Rüstem KAYA 93 Some Ruled Surfaces Related To W-Direction Curves 94 İlkay ARSLAN GÜVEN Semra KAYA NURKAN Filiz ÖZSOY Screen Semi-invariant Half-lightlike Submanifolds of a Semi-Riemannian Product Manifold 95 Oğuzhan Bahadır Hamilton Equations of Frenet-Serret Frame on Minkowski Space 96 Zeki KASAP Emin OZYILMAZ

On a Novel Formula for Reidemeister Torsion of Orientable Σg,n,b Riemann Surfaces 97 Esma DİRİCAN Yaşar SÖZEN Prime Decomposition of 3-Manifolds and Reidemeister Torsion Yaşar SÖZEN Esma DİRİCAN 98 A Note On Reidemeister Torsion of G-Anosov Representations Hatice ZEYBEK Yaşar SÖZEN 99 Some Characterizations of a Timelike Curve in R^3_1 M. Aykut AKGÜN A. İhsan SVRİDAĞ 100 Semi-invariant submanifolds of almost α-cosymplectic f-manifolds 101 Selahattin BEYENDİ Nesip AKTAN Ali İhsan SİVRİDAĞ Nearly Trans-Sasakian Manifolds With Quarter- Symmetric Non-Metric Connection 102 Oğuzhan BAHADIR Ertuğrul AKKAYA On Generalized Spherical Surfaces in Euclidean Spaces 103 Bengü BAYRAM Kadri ARSLAN Betül BULCA H-curvature Tensors of IK-Normal Complex Contact Metric Manifold 104 Aysel TURGUT VANLI İnan ÜNAL Quaternionic Bertrand Direction Curves 105 Burak ŞAHİNER Mehmet ÖNDER Some Results About Harmonic Curves On Lorentzian Manifolds 106 Sibel SEVİNÇ Gülşah AYDIN ŞEKERCİ A. Ceylan ÇÖKEN Relations Among Lines of Complex Hyperbolic Space Ramazan ŞİMŞEK 107 Elastic Strips with Null Directrix Gözde ÖZKAN TÜKEL Ahmet YÜCESAN 108 Bézier Geodesic-like Curves on 2-dimensional Pseudo-hyperbolic Space 109 Ayşe AKINCI Ahmet YÜCESAN On the Kinematics of the Hyperbolic Spinors and Split Quaternions 110 Mustafa TARAKÇIOĞLU Tülay ERİŞİR Mehmet Ali GÜNGÖR Murat TOSUN A Survey on Rectifying Curves in Lorentz n-Space 111 Tunahan TURHAN Vildan ÖZDEMİR Nihat AYYILDIZ Applications of Complex Form of Instantaneous Invariants to Planar Path-Curvature Theory 112 Kemal EREN Soley ERSOY Some Solutions on the Flux Surfaces 113 Zehra ÖZDEMİR İsmail GÖK F. Nejat EKMEKCİ Yusuf YAYLI

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A Note on Warped Product Manifolds With Certain Curvature Conditions 114 Sinem GÜLER Sezgin ALTAY DEMİRBAĞ Submanifolds with Finite Type Spherical Gauss Map in Sphere Burcu BEKTAŞ Uğur DURSUN 115 Rectifying Salkowski Curves with Serial Approach in Minkowski 3-Space 116 Beyhan YILMAZ, İsmail GÖK Yusuf YAYLI Normal Section Curves on Semi-Riemannian Manifolds 117 Feyza Esra ERDOĞAN Selcen YÜKSEL PERKTAŞ On Generalized D-Conformal Deformations of Some Classes of Almost Contact Metric Manifolds 118 Nülifer ÖZDEMİR Semi-invariant   -Riemannian submersions from almost contact manifolds 119 Mehmet Akif AKYOL Ramazan SARI Elif AKSOY Lucas Collocation Method to Determination Spherical Curves in Euclidean 3-Space 120 Muhammed ÇETİN Hüseyin KOCAYİĞİT Mehmet SEZER Dual Euler-Rodrigues Formula Derya KAHVECİ Yusuf YAYLI İsmail GÖK 121 On The Isometry Group of Deltoidal Hexacontahedron Space 122 Zeynep ÇOLAK Özcan GELİŞGEN Seiberg-Witten Equations on 6-Dimensional manifold Without Duality 123 Serhan EKER Nedim Değirmenci On the Concircular Curvature Tensor of a Normal Paracontact Metric Manifold 124 Ümit YILDIRIM Mehmet ATÇEKEN Süleyman DİRİK Sierpinski-type Fractals in Galilean Plane 125 Elif Aybike BÜYÜKYILMAZ Yusuf YAYLI İsmail GÖK Some Characterizations for Bertrand and Mannheim offsets of null-scrolls 126 Pınar BALKI OKULLU Mehmet ÖNDER On Spatial Quaternionic Involute Curve A New View 128 Süleyman ŞENYURT Ceyda CEVAHİR Yasin ALTUN On the Darboux Vector Belonging to involute Curve a Different View 129 Süleyman ŞENYURT Yasin ALTUN Ceyda CEVAHİR Surface family with a common natural asymptotic lift 130 Ergin BAYRAM Evren ERGÜN Emin KASAP A new method for designing involute trajectory timelike ruled surfaces in Minkowski 3-space 131 Mustafa BİLİCİ On Archimedean Polyhedral Metric and Its Isometry Group 132 Özcan GELİŞGEN Temel ERMİŞ New Results for General Helices in Minkowski 3-space Kazım İLARSLAN 133 On The Semi-Parallel Tensor Product Surfaces In Semi-Euclidean Space E₂⁴ 134 Mehmet YILDIRIM Generalized Pseudo Null Bertrand Curves in semi-Euclidean 4 space 135 Osman KEÇİLİOĞLU Ali UÇUM A New Method To Obtain Special Curves In The Three-Dimensional Euclidean Space 136 Fırat YERLİKAYA Savaş KARAAHMETOĞLU İsmail AYDEMİR * 137 Some Notes on Integrability Conditions and Tachibana operators on Cotangent Bundle T (M n ) Haşim ÇAYIR On the parametric representation of the zero constant mean curvature surface family in Minkowski 138 space Sedat KAHYAOĞLU Emin KASAP Equivalence Problem for a Riccati Type Pde in Three Dimensions 139 Tuna BAYRAKDRA Abdullah Aziz ERGİN Meridian Surfaces of Weingarten Type in 4-dimensional Euclidean Space 피4 140 Betül BULCA Günay ÖZTÜRK Bengü Bayram Kadri ARSLAN The Fermi-Walker Derivative and Principal Normal Indicatrix in Minkowski 3-Space 141

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Fatma KARAKUŞ Yusuf YAYLI Split Semi-Quaternions and Semi-Euclidean Planar Motions 142 Murat BEKAR Yusuf YAYLI Suborbital Graphs For a Special Möbius Transformation on The Upper Half Plane ℍ 143 Murat BEŞENK Control invariants of non-directional Bezier curve İdris ÖREN 144 On the Generalization of Quaternions Muttalip ÖZAVŞAR E. Mehmet ÖZKAN 145 C-Curves in Minkowski Space Emre ÖZTÜRK Yusuf YAYLI 146 A Generalization of Cheeger-Gromoll Metric on Tangent Bundle 147 Murat ALTUNBAŞ Aydın GEZER On Osculating Curves in Semi-Euclidean 4 Space Nihal KILIÇ ASLAN Hatice ALTIN ERDEM 148 On The Complete Arcs in The Left Near Field Projective Plane Of Order 9 149 Elif ALTINTAŞ Ayşe BAYAR Ziya AKÇA Süheyla EKMEKÇİ Bi- f -harmonic immersions Selcen YÜKSEL PERKTAŞ Feyza Esra ERDOĞAN 150 Characterizations for Timelike Slant Ruled Surfaces in Dual Lorentzian Space 151 Seda ALTINGÜL Mustafa KAZAZ On Numerical Computation of Fibered Projective Planes 153 Mehmet Melik UZUN Ziya AKÇA Süheyla EKMEKÇİ Ayşe BAYAR On The Polar Taxicab Metric In Three Dimensional Space Temel ERMİŞ Özcan GELİŞGEN 154 Generalized Null Bertrand Curves in semi-Euclidean 4-space Ali UÇUM 155 On Freeness Conditions of Crossed Modules 156 Tufan Sait KUZPINARI Alper ODABAŞ Enver Önder USLU On The Fibered Projective Planes 157 Süheyla EKMEKÇİ Ziya AKÇA Ayşe BAYAR On Some Classical Theorems in Intuitionistic Fuzzy Projective Plane 158 Ayşe BAYAR Süheyla EKMEKÇİ Ziya AKÇA A Computer Search for some Subplanes of Projective Plane Coordinatized a Left Nearfield 159 Ziya AKÇA Ayşe BAYAR Süheyla EKMEKÇİ The Dual Euler Parameters in Dual Lorentzian Space 160 Buşra AKTAŞ Halit GÜNDOĞAN The Tangent Operator in Lorentzian Space Olgun DURMAZ Halit GÜNDOĞAN 161 Surfaces endowed with canonical principal direction in Minkowski 3-space 162 Alev KELLECİ Nurettin Cenk TURGAY Mahmut ERGÜT A Study On The Elastic Curves 163 Gülşah AYDIN ŞEKERCİ Sibel SEVİNÇ A. Ceylan ÇÖKEN Some Results About Harmonic Curves On Lorentzian Manifolds 164 Sibel SEVİNÇ Gülşah AYDIN ŞEKERCİ A. Ceylan ÇÖKEN 4 On Spherical Indicatries Of Partially Null Curves In R2 165 Ümit Ziya SAVCI Süha YILMAZ The New Frame Approach For Spatial Curves Çağla RAMİS Yusuf YAYLI 166 On Curvatures of Surfaces via Quaretnions in Minkowski Space 167 Muhammed Talat SARIAYDIN Talat KÖRPINAR Vedat ASİL On Weierstrass Representation Formula In Bianchi Type-I Spacetime 168 Talat KÖRPINAR Gülden ALTAY Handan ÖZTEKİN Mahmut ERGÜT Metric n-Hyperplanes of Euclidean and Hyperbolic Geometry Oğuzhan DEMİREL 169 On Golden Riemannian Tangent Bundles with C-G Metric 170 Ahmet KAZAN H. Bayram KARADAĞ Pseudosymmetric Lightlike Hypersurfaces in indefinite Sasakian Space Forms* 171 Sema KAZAN Bayram ŞAHİN Determination of the curves of constant breadth according to Bishop Frame in Euclidean 3-space by a 172 Galerkin-like method Şuayip YÜZBAŞI Murat KARAÇAYIR Mehmet SEZER

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A Laguerre method to determinate the curves of constant breadth according to 173 Bishop Frame in Euclidean 3-space Şuayip YÜZBAŞI Mehmet SEZER Esra SEZER Anet Parallel Surfaces in Heisenberg Group 174 Gülden ALTAY Talat KÖRPINAR Mahmut ERGÜT New Characterization of Involute Curves in Universal Covering Group 175 Handan ÖZTEKİN Talat KÖRPINAR Gülden ALTAY Mahmut ERGÜT Normal Section Curves on Semi-Riemannian Manifolds 176 Feyza Esra ERDOĞAN Selcen YÜKSEL PERKTAŞ LS(2,D)− Equivalence Conditions of Dual Control Points in D2 Muhsin İNCESU 177 Timelike Directional Tubular Surfaces Mustafa DEDE Hatice TOZAK Cumali EKİCİ 178 A new type of associated curves Evren ZIPLAR Yusuf YAYLI İsmail GÖK 180 Spherical Curves with Modified Orthogonal Frame Bahaddin BÜKCÜ Murat Kemal KARACAN 181 Normal Section Curves on Semi-Riemannian Manifolds 182 Feyza Esra ERDOĞAN Selcen YÜKSEL PERKTAŞ Spherical Indicatrices with Modified Orthogonal Frame 183 Bahaddin BÜKCÜ Murat Kemal KARACAN Statistical Evaluation of Relationship between Analytic Geometry Course Achievement and Student 184 Selection and Placement Exam Scores of In-service Elementary Mathematics Education Teachers at Faculty of Education Şüheda BİRBEN GÜRAY On the Quasi-Conformal Curvature Tensor of a Normal Paracontact Metric Manifold 185 Mehmet ATÇEKEN Ümit YILDIRIM Süleyman DİRİK ABSTRACTS OF POSTER PRESENTATIONS 186 A New Approach to Offsets of Ruled Surfaces Mehmet ÖNDER Tolga KASIRGA 187 On Fractional Geometric Calculus Nesip AKTAN Nusret TÜMKAYA 188 푁∗퐶∗- Smarandache Curve of Bertrand Curve Pair According to Frenet Frame 189 Süleyman ŞENYURT Abdussamet ÇALIŞKAN On The curves of AW(k)-type according to the Bishop Frame 190 Erdal ÖZÜSAĞLAM Pelin POŞPOŞ TEKİN Surfaces with a common isophote curve in Euclidean 3-space 191 O. Oğulcan TUNCER İsmail GÖK Yusuf YAYLI An Apollonius circle in the Taxicab Plane Geometry Aybüke EKİCİ Temel ERMİŞ 192 The Fermi-Walker Derivative On the Tangent Indicatrix 193 Yusuf DURSUN Fatma KARAKUŞ Yusuf YAYLI The Fermi-Walker Derivative On the Binormal Indicatrix 194 Ayşenur UÇAR Fatma KARAKUŞ Yusuf YAYLI On Intersection Curve of Two Surfaces* 195 Benen AKINCI Mesut ALTINOK Bülent ALTUNKAYA Levent KULA On Almost α-Kenmotsu Manifolds of Dimension 3 Hakan ÖZTÜRK 196 Rectifying curves in Minkowski n-space Osman ATEŞ İsmail GÖK Yusuf YAYLI 197 Tubular surfaces with a new idea in Minkowski 3-space 198 Erdem KOCAKUŞAKLI Fatma ATEŞ İsmail GÖK Nejat EKMEKCİ The Kinetic Energy Formula For The Closed Planar Homothetic Inverse Motions in Complex Plane 199 Önder ŞENER Ayhan TUTAR On the Horizontal Bundle of a Pseudo-Finsler Manifold 200 İsmet AYHAN Şevket CİVELEK A. Ceylan ÇÖKEN Surfaces with a common isophote curve in Euclidean 3-space 201 O. Oğulcan TUNCER İsmail GÖK Yusuf YAYLI Complete Lifts of Metallic Structures to Tangent Bundles Mustafa ÖZKAN Emre Ozan UZ 202 On fuzzy subgeometries of fuzzy n-dimensional projective space Ziya AKÇA 203 On the group of isometries of the generalized Taxicab plane Süheyla EKMEKÇİ 204 On Taxicab Circular Inversions Ayşe BAYAR 205

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Lie Group Analysis For Some Partial Differential Equations 206 Zeliha S. KÖRPINAR Gülden ALTAY On the Mechanical System on the Killing Curves Osman ULU Şevket CİVELEK 207 Spherical Indicatrix Curves of Spatial Quaternionic Curve Süleyman ŞENYURT Luca GRILLI 208 Mechanical Energy Of Particles On Minkowski 4-Space On Circle 209 Simge ŞİMŞEK Cansel YORMAZ A Physical Space-Modeled Approach To Energy Equations With Bundle Structure For Minkowski 4- 210 Space Simge ŞİMŞEK

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

HONARARY CHAIRMANS Prof. Dr. Hüseyin BAĞCI Rector Pamukkale University Prof. Dr. H. Hilmi HACISALİHOĞLU Bilecik Şeyh Edebali University

ORGANIZING COMMITTEE CHAIRMAN Assoc. Prof. Dr. Şevket CİVELEK Pamukkale University

ORGANIZING COMMITTEE Prof. Dr. M. Ali SARIGÖL Pamukkale University Assoc. Prof. Dr. Şevket CİVELEK Pamukkale University Assoc. Prof. Dr. Cansel YORMAZ Pamukkale University Assoc. Prof. Dr. Serpil HALICI Pamukkale University Assoc. Prof. Dr. Mustafa AŞÇI Pamukkale University Assoc. Prof. Dr. Özlem GİRGİN ATLIHAN Pamukkale University Assoc. Prof. Dr. İbrahim ÇELİK Pamukkale University Assoc. Prof. Dr. Alp Arslan KIRAÇ Pamukkale University Assoc. Prof. Dr. İsmail YASLAN Pamukkale University Assoc. Prof. Dr. Mehmet Ali ÇELİKEL Pamukkale University Assist. Prof. Dr. Erkan KAÇAN Pamukkale University Dr. Zeki KASAP Pamukkale University

SCIENTIFIC COMMITTEE Prof. Dr. H. Hilmi HACISALİHOĞLU Bilecik Şeyh Edebali University Prof. Dr. Mahmut ERGUT Namık Kemal University Prof. Dr. Salim YÜCE Yıldız Teknik University Prof. Dr. Murat TOSUN Sakarya University Prof. Dr. Ali ÇALIŞKAN

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Ege University Prof. Dr. Gennadi SARDANASHVILY Moscow State University, Russia Prof. Dr. Manuel de LEON Instituto de Ciencias Matematicas, Spain Prof. Dr. Ioan BUCATARU Alexandru Ioan Cuza University, Romania Prof. Dr. Cornelia-Livia BEJAN Alexandru Ioan Cuza University, Romania Prof. Dr. Mukut Mani TRIPATHI Banaras Hindu University, India Prof. Dr. Uday CHAND DE Calcutta University, India Prof. Dr. Wendy Goemans KU Leuven University, Belgium Assoc. Prof. Dr. Miguel Brozos VÁZQUEZ Titlar de Universidad, Spain Prof. Dr. Mustafa ÇALIŞKAN Gazi University Prof. Dr. H. Hüseyin UĞURLU Gazi University Prof. Dr. Baki KARLIĞA Gazi University Prof. Dr. Aysel TURGUT VANLI Gazi University Prof. Dr. Yusuf YAYLI Ankara University Prof. Dr. Nejat EKMEKCİ Ankara University Prof. Dr. Cengizhan MURATHAN Uludağ University Prof. Dr. Kadri ARSLAN Uludağ University Prof. Dr. Süleyman ÇİFTÇİ Uludağ University Prof. Dr. Abdullah Aziz ERGİN Akdeniz University Prof. Dr. Abdilkadir Ceylan ÇÖKEN Akdeniz University Prof. Dr. Mustafa Kemal SAĞEL Mehmet Akif Ersoy University Prof. Dr. Rıfat GÜNEŞ İnönü University Prof. Dr. Sadık KELEŞ İnönü University Prof. Dr. Bayram ŞAHİN İnönü University Prof. Dr. Ahmet YILDIZ İnönü University

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Prof. Dr. A. İhsan SİVRİDAĞ İnönü University Prof. Dr. Mehmet BEKTAŞ Fırat University Prof. Dr. Halit GÜNDOĞAN Kırıkkale University Prof. Dr. Osman GÜRSOY Maltepe University Prof. Dr. Kazım İLARSLAN Kırıkkale University Prof. Dr. Bülent KARAKAŞ Yüzüncü Yıl University Prof. Dr. Ali GÖRGÜLÜ Osmangazi University Prof. Dr. Rüstem KAYA Osmangazi University Prof. Dr. Nevin GÜRBÜZ Osmangazi University Prof. Dr. Mehmet TEKKOYUN Çanakkale Onsekiz Mart University Prof. Dr. Levent KULA Ahi Evran University Prof. Dr. Abdullah MAĞDEN Atatürk University Prof. Dr. Arif SALİMOV Atatürk University Prof. Dr. Nuri KURUOĞLU İstanbul Gelişim University Prof. Dr. Ertuğrul ÖZDAMAR Bahçeşehir University Prof. Dr. Cem TEZER Orta Doğu Teknik University Prof. Dr. Ersan AKYILDIZ Orta Doğu Teknik University Prof. Dr. Hurşit ÖNSİPER Orta Doğu Teknik University Prof. Dr. Abdülkadir ÖZDEĞER Kadir Has University Prof. Dr. Cihan ÖZGÜR Balıkesir University Prof. Dr. İsmail AYDEMİR Ondokuz Mayıs University Prof. Dr. Emin KASAP Ondokuz Mayıs University Prof. Dr. Ayhan SARIOĞLUGİL Ondokuz Mayıs University Prof. Dr. A. Sinan SERTÖZ Bilkent University Prof. Dr. Uğur DURSUN Işık University

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Prof. Dr. Nesip AKTAN Necmettin Erbakan University Prof. Dr. Mehmet BEKTAŞ Fırat University ADVISORY COMMITTEE Prof. Dr. Selahattin ÖZÇELİK Pamukkale University Prof. Dr. Ali YILMAZ Pamukkale University Prof. Dr. Alaattin ŞEN Pamukkale University Prof. Dr. Bilal SÖĞÜT Pamukkale University Prof. Dr. Ayşegül DAŞCIOĞLU Pamukkale University Assoc. Prof. Dr. Handan YASLAN Pamukkale University Assoc. Prof. Dr. İsmet AYHAN Pamukkale University Assoc. Prof. Dr. Özcan SERT Pamukkale University Assoc. Prof. Dr. Cumali EKİCİ Osmangazi University Assoc. Prof. Dr. Emin ÖZYILMAZ Ege University Assoc. Prof. Dr. Mustafa ÖZDEMİR Akdeniz University Assist. Prof. Dr. Şahin CERAN Pamukkale University Assist. Prof. Dr. Gülseli BURAK Pamukkale University Assist. Prof. Dr. Hüseyin KOCAYİĞİT Celal Bayar University Assist. Prof. Dr. Sibel PAŞALI Muğla Sıtkı Koçman University

SECRETERIA Assoc. Prof. Dr. Cansel YORMAZ Pamukkale University Assoc. Prof. Dr. Serpil HALICI Pamukkale University

WEB DESINGNER Alper ÇAKIR

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Sponsors

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 Platinum Sponsor Denizli Ticaret Odası

Gold Sponsor Silver Sponsor Ozan Tekstil Denizli Sanayi Odası

Bronze Sponsor

Gamateks Denizli Valiliği

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Pamukkale Belediyesi Denizli İhracatçılar Birliği

Denizli Ticaret Borsası Murat Eğitim Kurumları

Pamukkale University Colossae Thermal Hotel

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Invited Speakers

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

INVITED SPEAKERS

Prof. Dr. H. Hilmi HACISALİHOĞLU Bilecik Şeyh Edebali University, Turkey

Title of the Speaker’s Presentation “On The Fractals in Spider Network”

Prof. Dr. Ali GÖRGÜLÜ Eskişehir Osmangazi University, Turkey

Title of the Speaker’s Presentation “On Intrinsic Equations for an Elastic Line on an Oriented Surface ”

Prof. Dr. Osman GÜRSOY Maltepe University, Turkey

Title of the Speaker’s Presentation “On the Line Geometry”

Prof. Dr. Cengizhan MURATHAN Uludağ University, Turkey

Title of the Speaker’s Presentation “Mystery Behind the Contact Structure”

Prof. Dr. Gennadi A. SARDANASHVILY Moscow State University, Russia

Title of the Speaker’s Presentation “The Differential Calculus on N-Graded Manifolds”

Prof. Dr. Manuel de LEON Instituto de Ciencias Matematicas, Spain

Title of the Speaker’s Presentation “The Geometry of the Hamilton-Jacobi Equation”

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Prof. Dr. Ioan BUCATARU Alexandru Ioan Cuza University, Romania

Title of the Speaker’s Presentation “Projective Deformations in Finsler Geometry and Hilbert’s fourth Problem”

Prof. Dr. Mukut Mani TRIPATHI Banaras Hindu University, India

Title of the Speaker’s Presentation “Inequalities for Algebraic Casorati Curvatures and their Applications”

Prof. Dr. Uday CHAND DE Calcutta University, India

Title of the Speaker’s Presentation “On Generalized Robertson-Walker Space-times”

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Abstracts of Invited Speakers’ Presentations

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On the Fractals in Spider network

H. Hilmi HACISALİHOĞLU1

Abstract

In this study, the fractals in the spiders networks(cobwebs) have examined. There are mystery in building cobwebs how they can be built by the spiders. They are made in a particular geometric layout braiding the cobwebs. All cobwebs of spiders are excellent geometric design and there are fractal geometric structures on the cobwebs.

Key Words: Fractals, cobwebs, geometric structures

References

1 Bilecik Şeyh Edebali University, Faculty of Art and science, Department of Mathematics, Bilecik E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On The Line Surfaces

Osman GÜRSOY2

As known the geometry of a trajectory surfaces tracing by an oriented line (spear) is important in line geometry* and spatial kinematics. Until, early 1980s, although two real integral invariants, the pitch of angle xand the pitch ℓx of an x-trajectory surface were known, any dual invariant of the surface were not. Because of the deficiency, the line geometry wasn’t being sufficiently studied by using dual quantities. A global dual invariant, x , of an x-closed trajectory surface is introduced and shown that there is a magic relation between the real invariants, x=x-  ℓx [1]. It gives suitable relations, such as x  2  Ax   Gx or x  2  ax   gxds and  x  ax   (u  v )dudv which have the new geometric interpretations of an x -trajectory surface where ax is the measure of the spherical area on the unit sphere, described by the generator of x-closed trajectory surface and  u and  v are the distribution parameters of the principal surfaces of the X(u,v)-closed congruence. Therefore, all the relations between the global invariants, x ,ℓx, ax , a*, g x ,g*, K,T,  and s1 of x-c.t.s. are worth reconsidering in view of the new geometric explanations. Thus, some new results and new explanations are gained. Furthermore, as a limit position of the surface, some new theorems and comments related to space curves are obtained [2,3].

Key Words: Dual Angle of Pitch, Dual Integral Invariant, Line Surface. AMS 2010: 51K99, 53C22.

References

[1] Gursoy, O., On Integral Invariant of A Closed Ruled Surface, Journal of Geometry, vol.39, 80-91, 1990, S.W. [2] Gursoy O., Some Results on Closed Ruled Surfaces and Closed space Curves, Mech.Mach.Theory, 27, (1990), 323-330. [3] Gursoy O., Kucuk A., On the Invariants of Trajectory Surfaces, Mech.Mach. Theory, 34, (1999), 587- 597.

2 Maltepe University, Faculty of Education, Elementary Mathematics Education Department, Istanbul/ Turkey, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Mystery Behind the Contact Structure

Cengizhan Murathan3

Abstract

Christiaan Huygens (1678) stated that light is a wave that propagates through space much like ripples in water or sound in air. This theory is called wave theory of light. Using contact geometry (structure), one can explain wave theory of light. Then we will give some information contact structure.

Key Words: Wave theory, 1-jet, contact transformation, contact structure

References [1] Mclnerney Andrew, First Steps in differential geometry, Springer, New York 2003 [2] Geiges Hasjörg, An introduction to contact topology, Cambridge University press New York, 2008

3 Uludağ University, Faculty of Art and science, Department of Mathematics, Görükle Campus, 16059, Bursa E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

The Differential Calculus on N-Graded Manifolds

Gennadi SARDANASHVILY4

Abstract

The Chevalley–Eilenberg differential calculus and differential operators over N-graded commutative rings are constructed. This is the straightforward generalization of the differential calculus over commutative rings, and it is the most general case of the differential calculus over rings that is not the non-commutative geometry. Since any N-graded ring possesses the canonical Grassmann-graded structure, this also is the case of the graded differential calculus over Grassmann algebras and the supergeometry on graded manifolds.

Key Words: differential calculus, Chevalley–Eilenberg complex, graded algebra, graded manifold.

References

[1] Sardanashvily G., Lectures on Differential Geometry of Modules and Rings, Lambert Academic Publishing, Saarbrucken, 2012; arXiv: 0910.1515.

4 Moscow State University, Department of Theoretical Physics Russia E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Noether’s Theorems in a General Setting

Gennadi SARDANASHVILY5

Abstract

Noether theorems are formulated in a general case of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. A problem is that any Euler-Lagrange operator satisfies Noether identities, which therefore must be separated into the trivial and non-trivial ones. These Noether identities can obey first-stage Noether identities, which in turn are subject to the second-stage ones, and so on. Thus, there is a hierarchy of non-trivial Noether and higher-stage Noether identities. This hierarchy is described in homology terms. If a certain homology regularity conditions holds, one can associate to a reducible degenerate Lagrangian the exact Koszul-Tate chain complex possessing the boundary operator whose nilpotentness is equivalent to all complete non-trivial Noether and higher-stage Noether identities. Since this complex is necessarily Grassmann-graded, Lagrangian theory on graded bundles is considered from the beginning, and is formulated in terms of the Grassmann-graded variational bicomplex. Its cohomology defines a first variational formula whose straightforward corollary is the first Noether theorem. Second Noether theorems associate to the above mentioned Koszul-Tate complex a certain cochain sequence whose ascent operator consists of the gauge and higher-order gauge symmetries of a Lagrangian system. If gauge symmetries are algebraically closed, this ascent operator is prolonged to the ilpotent BRST operator which brings the gauge cochain sequence into a BRST complex, and thus provides a BRST extension of an original Lagrangian system.

Key Words: Noether identity, gauge symmetry, BRST theory.

References

[1] Sardanashvily G., Noether theorems in a general setting, arXiv: 1411.2910.

[2] Sardanashvily G., Higher-stage Noether identities and second Noether theorems, Adv. Math. Phys. Vol. 2015 (2015) 127481.

[3] Sardanashvily G., Noether's Theorems. Applications in Mechanics and Field Theory, Springer, 2016.

5 Moscow State University, Department of Theoretical Physics Russia E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Noether’s Theorems in a General Setting

Manuel Se LEON6

Abstract

Key Words: Noether identity, gauge symmetry, BRST theory.

References

[1] Sardanashvily G., Noether theorems in a general setting, arXiv: 1411.2910.

[2] Sardanashvily G., Higher-stage Noether identities and second Noether theorems, Adv. Math. Phys. Vol. 2015 (2015) 127481.

[3] Sardanashvily G., Noether's Theorems. Applications in Mechanics and Field Theory, Springer, 2016.

6 Moscow State University, Department of Theoretical Physics Russia E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Noether’s Theorems in a General Setting

Mukut Mani TRIPATHI7

Abstract

Key Words: Noether identity, gauge symmetry, BRST theory.

References

[1] Sardanashvily G., Noether theorems in a general setting, arXiv: 1411.2910.

[2] Sardanashvily G., Higher-stage Noether identities and second Noether theorems, Adv. Math. Phys. Vol. 2015 (2015) 127481.

[3] Sardanashvily G., Noether's Theorems. Applications in Mechanics and Field Theory, Springer, 2016.

7 Banaras Hindu University, India E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Abstracts of Oral Presentations

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Lagrange Mechanical Systems on the Walker Manifolds and killing Magnetic Curves

Şevket CİVELEK8

Abstract

In this study, the properties of Walker Manifolds and Killing Magnetic Curves are presented. Later, the Lagrangian energy systems have been set up on the Walker Manifolds by using the Killing Magnetic Curves and some physical and geometric comments are given about this study.

Key Words: Walker Manifolds, Killing Magnetic Curves, Lagrange Mechanical systems

References

[1] Civelek, Ş., Aycan, C., Dağlı, S., Improving Hamiltoınian Energy Equations On The Kahler Jet Bundles", Int. Jour. of Geo. Met. in Modern Phy. (ISI), Vol 10 No:3, 1-15 pp., 2013 , DOI: 10.1142/S0219887812500880

[2] Aycan, C., Civelek, Ş., Dağlı, S., Improving On Lagrangian Systems On Kahler Jet Bundles, Int. Jour. of Geo. Met. in Modern Phy. (ISI), Vol 10, no 7, 1-13 pp., 2013 , DOI: 10.1142/S0219887813500266

[3] Jleli, M., Munteanu, I. M. And Nistor, E. I., Magnetic Trajectories in an Almost Contact Metric Manifold, Results. Math. 67(2015),125–134. 2008.

[4] Özdemir, Z., Gök. İ., Yaylı, Y., Ekmekci, F.N., Notes on magnetic curves in 3D semi-Riemannian manifolds, Turkish Journal of Mathematics , (2015) 39, 412 - 426.

[5] Bejan, C. L., Romaniuc, S. L. D., Walker manifolds and Killing magnetic curves, Differential Geometry and its Applications 35 (2014) 106–116.

[6] Calvaruso, G., Munteanu M.,I., Perrone, A., Killing magnetic curves in three-dimensional almost paracontact manifolds, J. Math. Anal. Appl. 426 (2015) 423–439

8 Pamukkale University, Faculty of Art and science, Department of Mathematics, Kınıklı Campus, 20100, Kınıklı/Denizli, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Hamiltonian Energy Systems On Supermanifolds

Cansel YORMAZ9, Simge ŞİMŞEK10

Abstract

Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part of many classical and quantum field theories involving odd fields.

It is formulated in terms of 푍2 _graded modules and shaves over 푍2 _graded commutative algebras.(super commutative algebras)

Supermanifolds also phrased in terms of sheaves of graded commutative algebras. They are consructed by collecting of sheaves of supervector spaces anda re generalizations of the manifold concept based on ideas coming from supersymmetry. Supermanifold is a manifold with bosonic and fermionic coordinates.

On the other hand,Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics. It uses a different mathematical formalism, providing more abstract understanding of the theory.

In this study, the Hamiltonian energy systems has been proved on supermanifolds. At the same time, we have created an opportunity for this study to make physical and geometric comments by giving an example.

Key Words: Supergeometry, supermanifold, Hamiltonian energy systems

References

[1] Aycan, C., 2003, The Lifts of Euler-Lagrange and Hamiltonian Equations on the Extended Jet Bundles, D. Sc. Thesis, Osmangazi Univ. , Eskişehir.

[2] Dağlı, S., 2012, The Jet Bundles And Mechanic Systems On Minkowski 4-Space, PhD Thesis, Denizli.

[3] Supermetics On Supermaifolds, G. Sardanashvily, International Journal of Geometric Methods in Modern Physics, Vol. 5, No. 2 (2008) 271-286.

[4] Classical Field Theory and Supersymmetry, Daniel S. Freed, IAS/Park City Mathematics Series. Volume 11, 2001.

9 Pamukkale University, Faculty of Art and science, Department of Mathematics, Kınıklı Campus, 20100, Kınıklı/Denizli, E-mail: [email protected] 10 Pamukkale University, Acıpayam MYO, Acıpayam Campus, 20800, Acıpayam/Denizli, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Geometry of Quaternions whose Coefficients Fibonacci Numbers

Serpil HALICI11 Şule ÇÜRÜK12

Abstract

In this study, we investigate the quaternions. And we consider the quaternions whose coefficients Fibonacci numbers. We give some fundamental porperties of these quaternions. Moreover, we consider the geometric interpretation of quaternions and quaternionsmultiplications.

Keywords. Ouaternions, Fibonacci Numbers, Rotations.

References

[1] Horadam, A. F.: Complex Fibonacci Numbers and Fibonacci Quaternions, Amer. Math. Monthly, 70(1963), 289,291.

[2]Halici, S.: Halici, Serpil. "On Fibonacci quaternions." Advances in Applied Clifford Algebras 22.2 (2012): 321-327.

[3] Halici, S.: Halici, Serpil. "On complex Fibonacci quaternions." Advances in Applied Clifford Algebras 23.1 (2013): 105-112.

[4] Halici, S.: Halici, Serpil. "On Dual Fibonacci Octonions." Advances in Applied Clifford Algebras 25.4 (2015): 905-914.

[5] Hacisalihoglu, H. H.: Hareket Geometrisi ve Kuaternionlar Teorisi, Gazi Üni. Yayınları, No.30, 1983.

[6] Kuipers, J. B. : Quaternions and Rotation Sequences,Princeton Uni., Press, 2002.

[7] Girard, Patrick R. : Quaternions, Clifford Algebras and Relativistic Physics, Birkhauser Verlag AG., 2007.

11 Pamukkale University, Faculty of Art and science, Department of Mathematics, Kınıklı Campus, 20100, Kınıklı/Denizli, E-mail: [email protected] 12 Pamukkale University, Faculty of Art and science, Department of Mathematics, Kınıklı Campus, 20100, Kınıklı/Denizli, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Vector Matrix Representation of Octonions

Serpil HALICI13, Adnan KARATAŞ 14

Abstract

In this study, we consider vector matrix representation of quaternions and octonions, respectively. We investigate some matrix representations for them. In [5], the author defined a new matrix which includes vectors. He introduced a new multiplication which contains dot product and vectoral product. We study the vector matrix representation of quaternions and octonions and investigate them in the view of geometry.

Key Words: Quaternions, Octonions, Split Octonions, Matrix Representation.

References

[1] Baez, J. C. The Octonions. Bull. Amer. Math. Soc., 39, Pp 145-205, 2002. [2] Tıan, Y. Matrix Representation Of Octonions And Their Applications. Advances İn Applied Clifford Algebras, 10, Pp 61-90, 2000. [3] Ward, J. P. Quaternions And Cayley Numbers, Mathematics And Its Applications, 1997 Kluwer Academic Publishers. [4] Daboul, J., Delbourgo, R. Matrix Representation Of Octonions And Generalizations. Journal Of Mathematical Physics, 1999, 40.8: 4134-4150. [5] Zorn, M. Alternativkörper Und Quadratische Systeme. In: Abhandlungen Aus Dem Mathematischen Seminar Der Universität Hamburg. Springer Berlin/Heidelberg, 1933. P. 395-402. [6] Chanyal, B. C. Split Octonion Reformulation Of Generalized Linear Gravitational Field Equations. Journal Of Mathematical Physics, 2015, 56.5: 051702. [7] Halici, S., Karataş, A. Some Matrix Representations Of Fibonacci Quaternions And Octonions. Advances In Applied Clifford Algebras, 1-10.

13 Pamukkale University, Faculty of Art and science, Department of Mathematics, Kınıklı Campus, 20100, Kınıklı/Denizli, E-mail: [email protected] 14 Pamukkale University, Faculty of Art and science, Department of Mathematics, Kınıklı Campus, 20100, Kınıklı/Denizli, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On the Involute Supercurves

Cumali EKİCİ15, Cansel YORMAZ16 , Hatice TOZAK17

Abstract

Using the Banach Grassmann algebra BL , given by Rogers [1], a new scalar product, a new definition of the orthogonality and of the Frenet frame associated to supersmooth supercurve are introduced on the (m,n)-

mn+ dimensional total super-Euclidean space BL . In this study, definition of the involute supercurve in is

22+ given and also some theorems for the involute supercurve in BL are obtained.

Key Words: Supercurve, Super-Euclidean space, Frenet frame, Involute supercurve.

References [1] Rogers A., Graded Manifolds, Supermanifolds and Infinite-Dimensional Grassmann Algebras, Commun. Math. Phys. 105(1986), 375-384 [2] Rogers, A., Supermanifolds theory and applications, World Scientific Publishing Company, 2007. [3] Rogers A., Graded Manifolds, Supermanifolds and Infinite-Dimensional Grassmann Algebras, Commun, Math. Phys. 105(1986), 375-384. [4] Batchelor, M., Structure of Supermanifolds, Transactions of the American Mathematical Society, 1979. [5] Leites, D. A., Introduction to the theory of Supermanifolds, Russ. Math. Surv. 35 1(httpiopscience.iop.org0036-0279351R01), 1980. [6] Batchelor, M., Two approaches to Supermanifolds. Trans. Am. Math. Soc. 258(1980), 257-270. [7] Jadczyk, A. and Pilch, K., Superspaces and Supersymmetries, Communations in Mathematical Physics. 78(1981), 373-390. [8] Berezin, F. A., Leites, D. A., Supervarieties, Sov. Math. Dokl. 16 (1975), 1218-1222. [9] Bartocci, C., Bruzzo, U., Ruiperez, D.H., The Geometry of Supermanifolds (Mathematics and Its Applications), Springer, 1991. [10] DeWitt, B., Supermanifolds, Cambridge University press, 1992. [11] Cristea, V. G., Existence and uniqueness theorem for Frenet frame supercurves, Note di Matematica, 24(1) (2004/2005), 143-167.

15 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-Computer, Meşelik Campus, 26480, Eskişehir, E-mail: [email protected] 16 Pamukkale University, Faculty of Art and science, Department of Mathematics, Kınıklı Campus, 20100, Kınıklı/Denizli, E-mail: [email protected] 17 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-Computer, Meşelik Campus, 26480, Eskişehir, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On conharmonically flat Sasakian Finsler structures on tangent bundles

Nesrin ÇALIŞKAN18

Abstract In this paper, conharmonic curvature tensor 퐾 of Sasakian Finsler structures on tangent bundles is defined. In this manner, conharmonically flat Sasakian Finsler structures that are Einstein are discussed. Some structure theorems including such kind of structures are examined: It is shown that ‘If an 푚-dimensional conharmonically flat (퐻푇푀, 휙퐻, 휉퐻, 휂퐻, 퐺퐻) is Einstein then it is locally isometric to 푆푚(1). Additionally, the proof of the theorem: ‘If an 푚-dimensional conharmonically flat (퐻푇푀, 휙퐻, 휉퐻, 휂퐻, 퐺퐻) is an Einstein manifold and it satisfies 푅(푋퐻, 푌퐻). 퐾 = 0, then it is localy isometric to 푆푚(1)’ is given.

Mathematics Subject Classification (2010): 53D15; 53C05; 53C15; 53C60

Key Words: Conharmonic curvature tensor; Sasakian Finsler structure; Einstein manifold; tangent bundle References [1] Asanjarani A., Bidabad B. Classification of complete Finsler manifolds through a second order differential equation, Differential Geometry and its Applications, 2008, 26, 434-444 [2] Bejancu A., Finsler Geometry and Applications, Ellis horwood, New York, 1990, ISBN-13: 0133179753, ISBN-10: 9780133179750 [3] De U. C., Singh R. N., Pandey S. K., On conharmonic curvature tensor of generalized Sasakian- space-forms, International Scholarly Research Network, 2012, 1-14 [4] Doric M., Petrovic-Turgasev M., Versraelen L., Conditions on the conharmonic curvature tensor of Kahler hypersurfaces in complex space forms, Publications De L’institut Mathematique, 1988, 44: 97-108 [5] Dwivedi M. K., Kim J. S. On conharmonic curvature tensor in K-contact and Sasakian manifolds, Bull Malays Math Sci Soc, 2011, 34: 171-180 [6] Ghosh S., De U. C., Taleshian A., Conharmonic curvature tensor on N(K)-contact metric manifolds, International Scholarly Research Network, 2011, 1-11 [7] Khan Q., On conharmonically and special weakly Ricci symmetric Sasakian manifolds, Navi Sad J Math, 2004, 34: 71-77 [8] Kirichenko V. F., Rustanov A. R., Shihab A. A., Geometry of conharmonic curvature tensor of almost Hermitian manifolds, Journal of Mathematical Sciences, 2011, 90: 79-93 [9] Kirichenko V. F., Shihab A. A., On geometry of conharmonic curvature tensor for nearly Kahler manifolds, Journal of Mathematical Sciences, 2011, 177:675-683 [10] Mishra R. S., Conharmonic curvature tensor in Riemannian, almost Hermite and Kahler manifolds, http://www.dli.gov.in/rawdataupload/upload/insa/INSA_2/20005a8a_330.pdf , 1969, 1: 330-335 [11] Shihab A. A., On geometry of conharmonic curvature tensor of nearly Kahler manifold, Journal of Researches (Sciences), 2011, 37: 39-48 [12] Szilasi J., Vincze C., A new look at Finsler connections and special Finsler manifolds, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, 2000, 16, 33-63 [13] Yalınız A. F., Caliskan N., Sasakian Finsler manifolds, Turk J Math, 2013; 37: 319-339.

18 Usak University, Faculty of Education, Department of Elementary Mathematics Education, 64200, Usak/Turkey, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Weyl-Euler-Lagrange Equations on Twistor Space for Tangent Structure

Zeki KASAP19

Abstract

Twistor spaces are certain complex 3-manifolds which are associated with special conformal Riemannian geometries on 4-manifolds. Also, classical mechanic is one of the major subfields for mechanics of dynamical system. A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space for classical mechanic. Euler-Lagrange equations are an efficient use of classical mechanics to solve problems using mathematical modeling. On the other hand, Weyl submitted a metric with a conformal transformation for unified theory of classical mechanic. The paper aims to introduce Euler-Lagrange partial differential equations (mathematical modeling, the equations of motion according to the time) for movement of objects on twistor space and the solution of differential equation systems will be made Maple software that to it will reach at the end of study. Additionally, the implicit solution of the equation will be obtained as a result of a special selection of graphics to be drawn.

Key Words: Twistor, Kähler, Mechanical System, Almost Complex, Lagrangian.

References

[1] Z. Kasap and M. Tekkoyun, Mechanical systems on almost para/pseudo-Kähler-Weyl manifolds, IJGMMP, vol. 10, no. 5, 2013, 1-8.

[2] R.G. Martín, Electromagnetic Field Theory for Physicists and Engineers: Fundamentals and Applications, Asignatura: Electrodinámica, Físicas, Granada, (2007).

[3] D.E. Soper, Classical Field Theory, Dover Books on Physics, 2008.

[4] R. Penrose, Twistor algebra. J. Math. Phys. 8, (1967), 345--366.

[5] R. Penrose, Twistor theory, its aims and achievements, Proceedings of Oxford Symposium on Quantum gravity, Clarendon Press, Oxford, 1975, 268-407.

[6] G.B. Folland, Weyl manifolds, J. Differential Geometry, 4, 1970, 145-153.

19 Pamukkale University, Faculty of Education, Elementary Mathematics Education Department, Denizli/ Turkey, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Spherical Circles Taxicab

Süleyman YÜKSEL20

Abstract

In this study, taxi unit circles which are on a unit sphere are defined and drawn using spherical taxicab metric. In addition the taxicab length of the circumference of the circle and arcs of a circle are calculated.

Key Words: Taxicab geometry, Euclidean geometry, Taxicab sphere, Taxicab Circle.

AMS 2010: 51K05, 51K99, 97G60, 53C22.

References

[1] H. Mınkowskı, Gesammelte Abhandlungen, Chelsa Publishing Co., New York, 1967. [2] K. Menger, You Will Like Geometry, Guildbook Of The Illinois Institute Of Technology Geometry Exhibit, Museum Of Science And Industry, Chicago, Il, 1952. [3] E.F. Krause, Taxicab Geometry; An Adventure İn Non-Euclidean Geometry, Dover Publications, Inc., New York, 1986. [4] Reynolds, B.E.: Taxicab Geometry, Pi Mu Epsilon Journal, 7 (1980), 77-88. [5] Bayar, A., And R. Kaya. "On A Taxicab Distance On A Sphere." Missouri J. Math. Sci 17 (2005): 41-51. [6] Gelişgen, Ö., And R. Kaya. "The Taxicab Space Group." Acta Mathematica Hungarica 122.1-2 (2008): 187-200. [7] Thompson, Kevin P. "The Nature Of Length, Area, And Volume İn Taxicab Geometry." Arxiv Preprint Arxiv:1101.2922 (2011). [8] Akca, Ziya, And Rüstem Kaya. "On The Norm İn Higher Dimensional Taxicab Spaces." Hadronic Journal Supplement 19.5 (2004): 491-501. [9] A. Korkmazoğlu, \Küresel Taksi Geometri Üzerine," Osmangazi Üniversitesi Fen Bilimleri Enstitüsü, 2000, Ph.D. Thesis. [10] Akca, Z. - Kaya, R.: On The Distance Formulae İn Three Dimensional Space, Hadronic Journal 27, 521- 532 (2004). [11] S. S. So, Recent Developments İn Taxicab Geometry, Cubo Matematica Educacional, Vol. 4 (2002), No. 2, 76-96. [12] Colakoğlu, H. Barıs, And Rüstem Kaya. "Regular Polygons İn The Taxicab Plane." Scientic And Professional Information Journal Of Croatian Society For Constructive Geometry And Computer Graphics (Kog) 12 (2008): 27-33.

20Gazi University, Ankara/ Turkey, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Characterizations for new partner curves in the Euclidean 3-space

Onur KAYA21 Mehmet ÖNDER22

Abstract In this paper, we give some new characterizations for new partner curves by the aid of integral curves of a reference curve. Also, we obtain some relationships between partner curves and some special curves such as slant helix.

Keywords: Alternative frame; partner curves; slant helix; integral curve.

References [1] Babaarslan, M., Yaylı, Y., On helices and Bertrand curves in Euclidean 3-space, Mathematical and Computational Applications, 18(1) (2013) 1-11. [2] Bertrand, J., Mémoire sur la théorie des courbes à double courbure, Comptes Rendus 36 (1850); Journal de Mathématiques Pures et Appliquées 15 (1850) 332–350. [3] Cheng, Y.M., Lin, C.C., On the generalized Bertrand curves in Euclidean N -spaces, Note di Matematica, 29 (2) (2009) 33–39. [4] Choi, J.H., Kang, T.H., Kim, Y.H., Mannheim curves in 3-dimensional space forms, Bull. Korean Math. Soc., 50(4) (2013) 1099–1108. [5] Choi, J.H., Kim, Y.H., Associated curves of a Frenet curve and their applications, Applied Mathematics and Computation, 218 (2012) 9116–9124. [6] Görgülü, A., Özdamar, E., A generalization of the Bertrand curves as general inclined curves in E n , Communications of the Faculty of Sciences of the University of Ankara, Series A1: Mathematics and Statistics, 35 (1–2) (1986) 53–60. [7] Izumiya, S., Takeuchi, N. Generic properties of helices and Bertrand curves, Journal of Geometry, 74 (2002) 97-109. [8] Izumiya, S., Takeuchi, N., New special curves and developable surfaces, Turk. J. Math. 28 (2004) 153-163. [9] Liu, H., Wang, F. Mannheim partner curves in 3-space. Journal of Geometry, 88(1-2) 2008 120-126. [10] Lucas, P., Ortega-Yagües, J.A., Bertrand curves in the three-dimensional sphere, Journal of geometry and physics, 62 (2012) 1903-1914. [11] Matsuda, H., Yorozu, S., Notes on Bertrand curves, Yokohama Mathematical Journal, 50 (2003) 41–58. [12] Monterde, J., Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion, Computer Aided Geometric Design 26 (2009) 271–278. [13] Pears, L.R., Bertrand Curves in Riemannian Space, Journal of the London Mathematical Society, s1-10 (3) (1935) 180–183. [14] Saint Venant, J.C., Mémoire sur les lignes courbes non planes, Journal d’Ecole Polytechnique 30 (1845) 1– 76. [15] Salkowski, E., Zur Transformation von Raumkurven, Mathematische Annalen 66 (4) (1909) 517–557. [16] Struik, D.J., Lectures on Classical Differential Geometry, 2nd ed. Addison Wesley, Dover, (1988). [17] Uzunoğlu, B., Gök, İ., Yaylı, Y., A new approach on curves of constant precession, arXiv:1311.4730 [math.DG]. [18] Wang, F., Liu, H. Mannheim partner curves in 3-Euclidean space, Mathematics in Practice and Theory, 37(1) (2007) 141-143. [19] Whittemore, J. K. Bertrand curves and helices, Duke Math. J., 6(1) (1940) 235-245. [20] Zhao, W., Pei, D., Cao, X., Mannheim Curves in Nonflat 3-Dimensional Space Forms, Advances in Mathematical Physics, Volume 2015 (2015), Article ID 319046, 9 pages.

21 1Manisa Celal Bayar University, Faculty of Arts and Sciences, Department of Mathematics, Muradiye Campus, 45140 Muradiye, Manisa, Turkey. E-mail: [email protected] 22 2Manisa Celal Bayar University, Faculty of Arts and Sciences, Department of Mathematics, Muradiye Campus, 45140 Muradiye, Manisa, Turkey. E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Some Notes on Almost Lorentzian r-Paracontact Structures on Tangent Bundle

Haşim ÇAYIR23

Abstract Lifting theory on tangent bundle T(M) has a considerable position ın modern differentiable geometry. Since, using lift method it is possible to generalize to differentiable structures on any manifold to its extensions. The paper is structured as follows. In section 2,3,4, some basic properties of the vertical, complete and horizontal lifts are given, respectively ([1],[4],[5],[6],[7],[8],[9]). In the main results section, firstly, we give some information about almost Lorentzian r-paracontact structures on tangent bundle T(M) secondly, we get some results about covarient derivatives with respect to XV,XC and XH of almost Lorentzian r-paracontact structures on tangent bundle T(M). In addition, this covarient derivatives which obtained shall be studied for some special values in almost Lorentzian r-paracontact structures.

Anahtar Kelimeler: Covarient Derivatives, Almost Lorentzian r-Paracontact Structure, Vertical Lift, Complete lifts, Horizontal Lift

References

[1] Blair, D.E. Contact Manifolds in Riemannian Geometry, Lecture Notes in Math, 509, Springer Verlag, New York (1976). [2] Das Lovejoy, S. Fiberings on almost r-contact manifolds, Publicationes Mathematicae, Debrecen, Hungary 43,(1993), 161-167. [3] Oproiu, V. Some remarkable structures and connexions, defined on the tangent bundle, Rendiconti di Matematica (3) (1973), 6 VI. [4] Omran,T., Sharffuddin,A., Husain,S.I. Lift of Structures on Manifolds, Publications de 1’Instıtut Mathematıqe, Nouvelle serie, 360 (50) ,(1984), 93 – 97. [5] Salimov, A.A. Tensor Operators and Their applications, Nova Science Publ, New York, (2013). [6] Sasaki, S. On The Differantial Geometry of Tangent Boundles of Riemannian Manifolds, Tohoku Math. J. 10, (1958), 338-358. [7] Salimov, A.A., Çayır, H. Some Notes On Almost Paracontact Structures, Comptes Rendus de 1’Acedemie Bulgare Des Sciences, tome 66 (3), (2013), 331-338. [8] Tekkoyun M. Lifts of Almost r-Contact and r-Paracontact Structures, arXiv:0902.4123v1[math.DS] 24 Feb. 2009 [9] Yano, K., Ishihara, S. Tangent and Cotangent Bundles, Marcel Dekker Inc, New York, (1973).

23 Giresun University, Faculty of Art and Science, Department of Mathematics, 28100, Giresun, Turkey, E-mail:[email protected] & [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Some Relationships between Darboux and Typ-2 Bishop Frames

Defined on Surface in Euclidean 3-space

Amine YILMAZ24 Emin ÖZYILMAZ25

Abstract

In this study, considering the Darboux and Type-2 Bishop frames in Euclidean 3-space we give some relationships between of them. Here, the geodesic curvature,normal curvature and geodesic torsion according to apparatus of the Type-2 Bishop frame of a unit speed curve on a surface are obtained. Also, we write transition matrix between the Darboux and Type-2 Bishop frames of the spherical images of the edges N1 , N2 and b . Finally, we have some interesting relations and illustrate of the examples by the aid of Maple programe.

Key Words: Type-2 Bishop frame,Darboux frame,geodesic curvature, spherical image curve

References

[1] E. Özyılmaz, S.Yılmaz, M. Turgut,``Relationships among Darboux and Bishop Frames'' In Gediz University,1st International Symposium on Computing in Science & Engineering, (2010), 378-383. [2] S.Yılmaz, M. Turgut,``A new version of Bishop frame and an application to spherical images'' J.Math.Anal.Appl.371, (2010), 764-776 . [3] L.R.Bishop,''There is more than one way to Frame a Curve''Amer.Math. Monthly, 82(3) ,(1975) 246-251. [4] B.Bükcü,M.K.Karacan,.''Special Bishop motion and Bishop Darboux rotation axis of the space curve'',J.Dyn.Syst:Geom.Theor.,6(1), (2008),27-34. [5] B.Bükcü,M.K.Karacan, ''The Slant Helices According to Bishop Frame''Int.J. Comput. Math.Sci.,3(2), (2009), 67-70. [6] M.Do Carmo, ''Differential Geometry of Curves and Surfaces'',New Jersey: Prentice-Hall Inc., (1976). [7] Andrew J.Hanson and Hui Ma,.''Parallel Transport Approach to Curve Framing'' Appl.Sci.,10, (1995), 115-120.

24 Ege Üniversitesi,Fen Fak,Matematik Böl.35100 Bornova /İZMİR, E-mail:[email protected]. 25 Ege Üniversitesi, Fen Fak,Matematik Böl.35100 Bornova /İZMİR, E-mail :[email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A new Type of Almost Contact Manifolds

Gülhan AYAR26, Alfonso CARRIAZO27 , Nesip AKTAN 28

Abstract

The purpose of this paper is to study the Singuler Semi-Riemannian Almost Contact manifolds. The geometry of manifolds with degenerate indefnite metrics has been studied by Demir Kupeli [1]. In that book it is shown that a manifold M with a degenerate indefinit metric g admits a geometric structure if and only if g is Lie parallel along the vector fields on M. In this case we call (M, g) a Singular Semi-Riemannian manifold. Then it is possible to attach a nondegenerate tangent bundle to (M, g) which admits a connection whose curvature tensor satisfies the usual identities of the curvature tensor of Levi Civita connection. We call this connection the Kozsul Connection of (M, g).

In this talk we will present Singuler Semi-Riemannian manifolds (introduced by Demir Küpeli in [1] ) with an adapted almost contact structure. We will study the main facts about such a structure, with some examples.

Key Words: Contact Manifolds, Almost Semi Riemannian Manifolds, Singular Manifolds, Singular Semi Riemannian Almost Contact Manifolds .

References

[1] Küpeli D. N., Degenerate manifolds geometry, Dedicata, 23(3) (1987), 259-290. [2] Erkekoğlu F., Degenerate hermitian manifolds, Mathematical Physics, Analysis and Geometry,8 (2005) 361- 387. [3] Sasaki S., On differentiable manifolds with certain structures which are closely related to almost contact structure, Tohoku Math. J., 2 (1960) 459-476. [4] Blair D. E., Riemannian Geometry of Contact and Symplectic Manifolds, Birkhauser, (2010). [5] Calvaruso G., Perrone D., Contact pseudo-metric manifolds, Differential Geom. Appl., 28 (2010) 615-634. [6] Küpeli D.N., Singular semi-Riemannian Geometry, Kluwer academic Publisher, (1996). [7] O’Neill B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, (1971).

26 Düzce University, Faculty of Art and science, Department of Mathematics, Konuralp Campus, 81620, Konuralp/Düzce, E-mail: [email protected] 27 Sevilla University, Faculty of Mathematics, Department of Geometry and Topology , 41012, Tarfia, Sevilla/Spain, E-mail: [email protected] 28 Konya Necmettin Erbakan University, Faculty of Science, Department of Mathematics-Computer Sciences,Meram Campus, 42060, Meram/Konya, E-mail:nesipaktan @gmail.com

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Geodesics on the Tangent Sphere Bundle of Pseudo Riemannian 3-Sphere İsmet AYHAN29

Abstract

3 4 In this paper, geodesics on a 3-sphere S1 in the 4 dimensional pseudo Euclidean space E1 have been

3 3 considered. Then the Sasaki semi-Riemann metric on the tangent sphere bundle with radius  T S1 of S1 has

3 been obtained and non-null geodesics on T S1 are classified into horizontal, vertical and oblique type. Moreover, the geodesics of oblique type have been classified with respect to the principle curvatures of

3 3 projected curve on S1 of the geodesics on T S1

Key Words: Tangent Sphere Bundle with Radius , Sasaki semi Riemann Metric, Geodesics.

References

[1] Ayhan, I., Geodesics On The Tangent Sphere Bundle of 3-Sphere, International Electronic Journal of Geometry, 6(2), 100-109 , 2013 [2] Ayhan, I, On The Sphere Bundle with The Sasaki semi Riemann Metric of a Space Form, Global Journal of Advanced Research on Classical and Modern Geometries, 3(1), 25-35 , 2014. [3] Ayhan, I., On The Tangent Sphere Bundle of The pseudo Hyperbolic two Space, Global Journal of Advanced Research on Classical and Modern Geometries, 3(2), 76-90, 2014. [4] Free, P., Introduction to General Relativity, http://personalpages.to.infn.it/~fre/PPT/ virgolect.ppt.3, 2003. [5] Kilingenberg, W., and Sasaki,S., On the tangent sphere bundle of a 2-sphere. Tohuku Math. Journ. 27(1975), 49--56. [6] Nagy, P.T., Geodesics on the tangent sphere bundle of a Riemann manifold, Geometriae Dedicata 7(1978), 233-243. [7] Sasaki,S., Geodesic on the tangent sphere bundles over space forms. Journ. Für die reine und angewandte math. 288(1976), 106-120. [8] Sasaki, S., On the Differential Geometry of Tangent Bundle of Riemann Manifolds II, Tohuku Math. Journ. 14(1962), 146-155.

29 Pamukkale University, Faculty of Education, Department of Mathematics Education, Kınıklı Campus, 20100, Kınıklı/Denizli, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Semi-Slant Riemannian Submersions From Locally Product Riemannian Manifolds

Hakan Mete TAŞTAN30, Fatma ÖZDEMİR31 , Cem SAYAR32

Abstract

In this paper, we study semi-slant submersions from locally product Riemannian manifolds onto Riemannian manifolds. We give necessary and sufficient conditions for the integrability and totally geodesicness of all distributions which are involved in the definition of the semi-slant submersion. Moreover, we give a characterization theorem for the proper semi-slant submersions with totally umbilical fibers. The paper ends with result for semi-slant submersions with parallel canonical structures.

Key Words: Riemannian submersion, semi-slant submersion, horizontal distribution, locally product Riemannian manifold

References

[1] O’Neill B., The fundamental equations of a submersion, Mich. Math. J. 13, 458-469, 1966. [2] Park K.S., Prasad R., Semi-slant submersions, Bull. Korean Math. Soc. 50(3), 951-962, 2013. [3] Şahin B., Slant submersions from almost Hermitian manifolds, Bull. Math. Soc. Sci. Math. Roumanie 54(102), No. 1, 93-105, 2011. [4] Taştan H.M., Şahin B., Yanan Ş., Hemi-slant submersions, Mediterr. J. Math. DOI:10.1007/s00009-015- 0602-7. [5] Yano K., Kon M., Structures on manifolds, World Scientiﬁc, Singapore, 1984.

30 Istanbul University, Faculty of Art and science, Department of Mathematics, Vezneciler, 34134, Istanbul, E-mail: [email protected] 31 Istanbul Technical University, Faculty of Science and Letters, Department of Mathematics, Maslak, 34469, Istanbul, E-mail: [email protected] 32 Istanbul Technical University, Faculty of Science and Letters, Department of Mathematics, Maslak, 34469, Istanbul, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On A New Type Of Framed Manifolds

Nesip AKTAN33, Mustafa YILDIRIM34 , Yavuz Selim BALKAN 35

Abstract The purpose of this paper is to introduce a new class of framed manifolds. Such manifolds are called almost framed f -cosymplectic manifolds . For some special cases of f and s , one obtains (almost) - cosymplectic, (almost) C -manifolds, and (almost) Kenmotsu -manifolds.

Key Words: -structure, almost -cosymplectic manifold, almost Kaehler manifold.

References

[1] Blair, D.E., Geometry of manifolds with structural group U (n)xO(s) , J. Diff. Geom., 4(1970), 155-167.

[2] Falcitelli, M. and Pastore, M., Almost Kenmotsu -manifolds, Balkan Journal of Geometry and Its Applications, No. 1, 12(2007), 32-43. [3] Falcitelli, M. and Pastore, M., -structure of Kenmotsu type, Mediterr. J. Math. 3 (2006), 549-564.

[4] Yıldırım, M., Aktan, N. and Murathan, C., Almost -Cosymplectic Manifolds, Mediterr. J. Math., 11(2014), 775-787. [5] Goldberg, S.I. and Yano, K., On normal globally framed -manifolds, Tohoku Math. J., 22(1970), 362- 370. [6] Goldberg, S.I. and Yano, K., Globally framed f-manifolds, Illinois J. Math., 15(1971), 456-474.

f 3  f  0 , Tensor N.S, 14(1963), 99-109. [7] Öztürk, H., Murathan, C., Aktan, N., and Vanlı, A. T., Almost alpha-cosymplectic f-manifolds, Annals of the AlexandruIoan Cuza University-Mathematics, Tom LX, S.I, f.1 (2014), 211-226. [8] Yano, K. and Kon, M., Structures on Manifolds, Series in Pure Math, Vol 3, World Sci, 1984.

[9] Yano, K., On a structure defined by a tensor field f of type (1,1) satisfing f 3  f  0 , Tensor (N.S.) 14 (1963), 99-109.

33 Necmettin Erbakan University, Faculty of Science, Department of Mathematics-Computer Sciences, Meram Campus, Meram/Konya, E-mail: [email protected] 34 Duzce University, Faculty of Art and science, Department of Mathematics, Konuralp Campus, Merkez/Düzce E-mail: [email protected] 35 Duzce University, Faculty of Art and science, Department of Mathematics, Konuralp Campus, Merkez/Düzce E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Euler-Lagrange and Hamilton-Jacobi Equations on a Riemann Almost Contact Model of a Cartan Space of order k

Ahmet MOLLAOĞULLARI36, Mehmet TEKKOYUN37

Abstract

Lagrangians and Hamiltonians have many applications in various fields, as: Mathematics, Physics, Optimal Control Theory, Dynamic Systems, Economy, Biology, etc[1]. Since one can construct geometries of higher-order Lagrange space and higher-order Hamilton space over the manifolds 푇푘푀 and 푇∗푘푀 of a manifold M respectively; manifold theory has an important role to describe "Euler-Lagrange and Hamilton (-Jacobi) equations" and also "Lagrangian and Hamiltonian mechanics" of a given manifold [2],[3]. Therefore, in this paper, we obtain Euler-Lagrange and Hamilton-Jacobi equations on a Riemann Almost Contact Model of a Cartan Space of order k. In the conclusion we discuss some results about related mechanical system.

Key Words: Cartan Manifold, Mechanical Systems, Lagrange and Hamilton Equations

References

[1] R. Miron, The Geometry of Higher-Order Hamilton Spaces: Applications toHamiltonian Mechanics , Kluwer Academic Publishers, Dordrecht, 2002. [2] Tekkoyun M., Civelek Ş., Görgülü A., "Higher Order Lifts Of Complex Structures", Rendiconti Rend.Istit. Mat. Univ. Trieste, vol.XXXVI, pp.85-95, 2004 [3] M. de Leon, P.R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies, New York,1989

36 Çanakkale Onsekiz Mart University, Faculty of Art and science, Department of Mathematics, Terzioğlu Campus, 17100, Çanakkale, E-mail: [email protected] 37 Çanakkale Onsekiz Mart University, Faculty of Economics and Administrative Sciences, Department of Management, Terzioğlu Campus, 17100, Çanakkale, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Isotropic Leaves Of Lightlike Hypersurfaces

Mehmet GÜLBAHAR38

Abstract

In this paper, isotropic leaves are investigated on a lightlike hypersurface of a Lorentzian manifold. Some results are obtained on screen conformal lightlike hypersurfaces. Furthermore, some relations involving curvature invariants of isotropic leaves are given.

Key Words: Lightlike hypersurface, Lorentzian manifold, Curvature.

References

[1] Bejan C. L. and Duggal K. L., Global lightlike manifolds and harmonicity, Kodai Math. J., 28(2005), 131- 145. [2] Duggal K. L. and Sahin B., Differential geometry of lightlike submanifolds, Birkhäuser, Basel, 2010. [3] O’Neill B., Isotropic and Kaehler immersions, Canad. J. Math., 17(1965), 907-915. [4] Vrancken L., Some remarks on isotropic submanifolds, Publ. Inst. Math. (Beograd), 51(1992), 94-100.

38 Siirt University, Faculty of Art and Science, Department of Mathematics, Kezer Campus, Siirt, E- mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Some Characterizations For Complex Lightlike Hypersurfaces

Erol KILIÇ39, Mehmet GÜLBAHAR40 , Sadık KELEŞ41

Abstract

In the present paper, we establish some inequalities involving curvature invariants of coisotropic lightlike submanifolds and improve these inequalities for complex lightlike hypersurfaces. Furthermore, we present some relations related to the holomorphic sectional curvature, anti-holomorphic sectional curvature and bi-sectional curvature for complex lightlike hypersurfaces.

Key Words: Lightlike submanifolds, Complex lightlike hypersurfaces, Curvature.

References

[1] Chen B.-Y., Pseudo-Riemannian geometry,  -invariants and applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. [2] Duggal K. L., On scalar curvature in lightlike geometry, J. of Geo. and Phys., 57(2)(2007), 473-478. [3] Duggal K. L. and Sahin B., Differential geometry of lightlike submanifolds, Birkhäuser, Basel, 2010. [4] Hong S., Matsumoto K. and Tripathi M. M., Certain basic inequalities for submanifolds of locally conformal Kaehlerian space forms, SUT J. Math., 4(2005), 75-95.

Acknowledgement: The first author of this work is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK). (113F388 coded project)

39 İnönü University, Faculty of Art and Science, Department of Mathematics, Malatya, E-mail: [email protected] 40 Siirt University, Faculty of Art and Science, Department of Mathematics, Kezer Campus, Siirt, E- mail: [email protected] 41 İnönü University, Faculty of Art and Science, Department of Mathematics, Malatya, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A compactness theorem by use of m-Bakry-Emery Ricci tensor

Yasemin SOYLU42, Murat LİMONCU43

Abstract

By using the m-Bakry-Emery Ricci tensor on a complete n-dimensional Riemannian manifold, we prove a compactness theorem including a diameter estimate.

Key Words: Distance function; Diameter estimate; Riccati comparison theorem.

References

[1] K. Kuwada, A probabilistic approach to the maximal diameter theorem, Math. Nachr. 286 (2013), 374-378. [2] M. Limoncu, Modifications of the Ricci tensor and applications, Arch. Math. 95 (2010), 191-199. [3] M. Limoncu, The Bakry-Emery Ricci tensor and its applications to some compactness theorems, Math. Z. 271 (2012), 715-722. [4] S.B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401-404. [5] Z. Qian, Estimates for Weighted Volumes and Applications, Quart. J. Math. Oxford 48 (1997), 235-242. [6] Q. Ruan, Two rigidity theorems on manifolds with Bakry-Emery Ricci curvature, Proc. Japan Acad. Ser. A 85 (2009), 71-74. [7] L.F. Wang, A Myers theorem via m-Bakry-´Emery curvature, Kodai Math. J. 37 (2014), 187-195. [8] S. Zhu, The comparison geometry of Ricci curvature, Comparison Geometry MSRI Publications. 30 (1997), 221-262.

42 Anadolu University, Faculty of Science, Department of Mathematics, Yunus Emre Campus, 26470, Eskişehir, E-mail: [email protected] 43 Anadolu University, Faculty of Science, Department of Mathematics, Yunus Emre Campus, 26470, Eskişehir, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A Special Connection On 3-Dimensional Quasi-Sasakian Manifolds

Azime ÇETİNKAYA44, AHMET YILDIZ45

Abstract Firstly we define a special quarter symmetric non-metric connection on almost contact metric manifolds. Using this connection, we inverstigate some curvature conditions on 3-dimensional quasi-Sasakian manifold with given this connection, e.g. (R̅(푋, 휉)푅̅)(푌, 푉)푊 =0, (R̅(푋, 휉)퐻̅)(푌, 푉)푊 = 0 , (P̅(푋, 휉)퐻̅)(푌, 푉)푊 = 0 , (P̅(푋, 휉)푅̅)(푌, 푉)푊 = 0 , (P̅(푋, 휉)푆̅)(푌, 푍) = 0 , (H̅(푋, 휉)푆̅)(푌, 푍) = 0 and (R̅(푋, 휉)푆̅)(푌, 푍) = 0 . Also we study cylic-parallel and 휂-parallel 3-dimensional quasi-Sasakian manifolds given with this connection. Finally we give an example for 3-dimensional quasi-Sasakian manifolds.

Key Words Quarter symmetric non-metric connection, 3-dimensional quasi-Sasakian manifold, cylic- parallel, η-parallel.

References

[1] Oubina J.A., New Classes of almost Contact metric structures, Publ.Math.Debrecen, 32(1985), 187-193. [2] Kuo K., On almost contact 3-structure, Tohoku Math. J., 22(1970), 325-332. [3] Yano K. ve Imai T., Quarter-symmetric metric connections and their curvature tensors, Tensor N. S., 38(1982), 13–18. [4] Yano K. ve Kon M.,Structures on manifolds, Series in Pure Mathematics, 3.World Scientic Publishing Corp., Singapore, 1984. [5] Kon, M., Invariant submanifolds in Sasakian manifolds, Mathematische Annalen 219, 277-290., 1975. [6] Kanemaki, S., Quasi-Sasakian manifolds, Tohoku Math. Journal, 29(1997), 227-233. [7] Kobayashi, S., ve Nomizu K., Foundations of differential geometry, John Wiley and Sons, Inc., New York,1996. [8] Tanno, S., The automorphism groups of almost contact Riemannian manifolds. Tohoku Math. J., 21(1969), 21–38. [9] Tanno, S., Quasi-Sasakian structure of rank 2푝 + 1, J. Differential Geom., 5(1971), 317-324. [10] Olszak, Z., Normal almost contact metric manifolds of dimension three, Ann. Polon. Math., 47(1986), 41- 50. [11] Olszak, Z., On three dimensional conformally flat quasi-Sasakian manifold, Period Math. Hungar., 33 (2), 105–113, 1996.

44 Piri Reis University, Faculty of Art and science, 34940, Tuzla/Istanbul, E-mail: [email protected] 45 Inonu University, Education Faculty, Department of Mathematics, 44000, Malatya, E- mail:[email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Getting an Hyperbolical Rotation Matrix by Using Householder’s Method in 3-Dimensional Space

Hakan ŞİMŞEK46, Mustafa ÖZDEMİR47

Abstract

Hyperbolical rotation is a linear map that represent hyperbolically the motion of a smooth object on the 2 2 2 2 general hyperboloids −푎1푥 + 푎2푦 + 푎3푧 = ±푟 , 푟 ∈ ℝ . In this paper, we use the Householder transformation in order to generate an hyperbolical rotation matrix which corresponds to hyperbolical rotation in 3-dimensional scalar product space.

Key Words: Hyperbolical Rotation Matrix, g-Householder Transformation, Scalar Product Space.

References [1] Aragón-González G., Aragón J.L., Rodríguez-Andrade M. A., The decomposition of an orthogonal transformation as a product of reflections, J. Math. Phys. 47 (2006), Art. No. 013509. [2] Aragón-González G., Aragón J.L., Rodríguez-Andrade M. A., Verde Star L., Reflections, Rotations, and Pythagorean Numbers, Adv. Appl. Clifford Algebras 19 (2009), 1-14. [3] Mackey D. S., Mackey N., Tisseur F., G-reflectors : Analogues of Householder transformations in scalar product spaces, Linear Algebra and its Applications Vol. 385 (2004), 187-213. [4] Özdemir M., An Alternative Approach to Elliptical Motion, Adv. Appl. Clifford Algebras, doi:10.1007/s00006-015-0592-3 (2015). [5] Özdemir M., Erdoğdu M., Şimşek H., On the Eigenvalues and Eigenvectors of a Lorentzian Rotation Matrix by Using Split Quaternions. Adv. Appl. Clifford Algebras 24 (2014), 179-192. [6] Simsek H., Özdemir M., Generating hyperbolical rotation matrix for a given hyperboloid, Linear Algebra and Its Applications, 496 (2016), 221-245. [7] Rodríguez-Andrade M.A., Aragón-González G., Aragón J.L., Verde-Star L., An algorithm for the Cartan- Dieudonné theorem on generalized scalar product spaces, Linear Algebra and Its Applications, Vol. 434, Issue 5 (2011), 1238-1254.

46 Akdeniz Üniversitesi, Fen Fakültesi, Matematik Bölümü, Merkez/Antalya, E-posta: [email protected] 47 Akdeniz Üniversitesi, Fen Fakültesi, Matematik Bölümü, Merkez/Antalya, E-posta: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Timelike Translation Surfaces According To Bishop Frame In Minkowski 3-Space

Zehra EKİNCİ48, Melike YAĞCI49

Abstract In this paper, we give timelike translation surfaces according to Bishop frames in Minkowski 3-space which is founded by using non-planar space curves and we find some properties of these surfaces. Firstly, we find first fundamental form, second fundamental form, Gaussian curvature and mean curvature of timelike translation surfaces. Then, we investigate Darboux frame of the generator curves of the timelike translation surfaces in Minkowski 3-space by considering Bishop frame of generator curves. Finally, we give the conditions for the generating curves to be geodesic, asymptotic line and principal line on the surface.

Key Words: Bishop frame, Darboux frame, fundamental forms, Minkowski 3-space, translation surface. References [1] Verstraelen L.; Walrave J.; Yaprak S., ‘ The Minimal Translation Surfaces in Euclidean Space’, Soochow J. Math. 1994,20(1):77-82. [2] Liu H., ‘Translation Surfaces with Constant Mean Curvature in 3-Dimensional Spaces’, J. Geometry, 1999, 64:141-149. [3] Yoon DW , "On the Gauss Map of Translation Surfaces in Minkowski 3-Space", Taiwan J. Math., 2002 ,6(3):389-398. [4] Munteanu M.; Nistor AI., "On the Geometry of the Second Fundamental Form of Translation Surfaces in 3 E " , Houston J. Math. , 2011, 37(4):1087-1102. [5] Çetin M.; Tunçer Y.; Ekmekçi N., "Translation Surfaces in Euclidean 3-Space", Int. J. Phys. Math. Sci., 2011, 2:49-56. [6] Çetin M.; Kocayiğit H.; Önder M., "Translation Surfaces acording to Frenet Frame in Minkowski 3-Space", Int. J. Phys. Math. Sci. Vol. , 2012, 7(47): 6135-6143. [7] Bishop L.R., "There is More Than One Way to Frame a Curve", Amer. Math. Monthly, 82(3), 1975, 246- 251. [8] Güler F.; Atalay G.S.; Kasap E.," Translation Surface According to Bishop Frame in Euclidean 3-Space", J. Math. Comput. Sci. 4, , 2014, No. 1, 50-57. [9] O’Neill B., "Semi-Riemannian Geometry with Applications to Relativity", Academic Press, London, 1983. [10] Beem J.K.; Ehrlich P.E., "Global Lorentzian Geometry", Marcel Dekker, New York, 1981. [11] Özdemir M.; Ergin A.A, "Parallel Frames of NonLightlike Curves", Missouri J. of Math. Sci., 2008,20(2), 127137. [12] Baba-Hamed C.; Bekkar M.; Zoubir H., "Translation Surfaces in the Three-Dimensional Lorentz- Minkowski Space Satisfying i i i r r ∆ = λ ", Int. J. Math. Anal, 2010, 4(17):797-808. [13] Tul, S.; Sarıoğlugil A., " On Bishop Frame of a Curve Lying on a Surface", Amer. J. of Mats. And Sci., 1/2013, Vol 2(1). [14] O’Neill B., "Elemantary Differential Geometry"Academic Press Inc. New York, 1966. [15] Gray A., "Modern Differential Geometry of Curves and Surfaces with Mathematica 2nd ed.", CRC Press, Washington, 1988.

48 Celal Bayar University, Faculty of Arts and Sciences, Department of Mathematics, Manisa,Turkey, E- mail: [email protected] 49 Instutition of Science and Technology, Celal Bayar University, Manisa, Turkey, E-mail: Melke-frkan- [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Hasimoto Surfaces in Minkowski 3-Space with Parallel Frame

Melek ERDOĞDUİ50, Mustafa ÖZDEMİRI51

Abstract

In this study, we investigate the Hasimoto surfaces in Minkowski 3-space. First, a great survey on the Hasimoto surfaces is given by using Frenet Frame. Then, Smoke Ring Equation is given by parallel frames. Finally, some differential geometric properties of Hasimoto surfaces are examined with parallel frames.

Key Words: Hasimoto surface, NLS Surface, Minkowski Space.

References

[1] Erdoğdu M., Özdemir M., Geometry of Hasimoto Surfaces in Minkowski 3-Space. Mathematical Physics, Analysis and Geometry, 17 (2014), 169-181. [2] Gürbüz N., Intrinsic Geometry of NLS Equation and Heat System in 3- Dimensional Minkowski Space, Adv. Studies Theor.,4 (2010), 557-564. [3] Gürbüz N., The Motion of Timelike Surfaces in Timelike Geodesic Coordinates, Int. Journal of Math. Analysis, 4 (2010), 349-356. [4] Hasimoto H., A Soliton on a vortex filament, J. Fluid. Mech., 51(1972), 477-485. [5] Özdemir M., Ergin A.A., Parallel Frames of Non-Lightlike Curves, Missouri Journal of Mathematical Sciences, 20 (2008), 127-137. [6] Rogers C., Schıef W.K., Intrinsic Geometry of the NLS Equation and its Backlund Transformation, Studies in Applied Mathematics, 101 (1998), 267-288. [7] Schıef W.K., Rogers C., Binormal Motion of Curves of Constant Curvature and Torsion. Generation of Soliton Surfaces, Proc. R. Soc. Lond. A., 455 (1999), 3163-3188.

50 Necmettin Erbakan University, Faculty of Sciences, Department of Mathematics- Computer Sciences, 42090, Meram/Konya, E-mail:[email protected] 51 Akdeniz University, Faculty of Science, Department of Mathematics, 07070, Kampüs/Antalya, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On the Line Congruences

Ferhat TAŞ52

Abstract

In this paper, we examined that differential geometric properties of the line congruences via its dual representation. Furthermore, the equations of principal, developable, central and focal surfaces of the line congruence are represented by coordinate functions.

Key Words: Line congruence, Developable surfaces.

References

[1] Blasckhe W., Diferensiyel Geometri Dersleri, Fırat Univ. Pub. of Fac. of Sci., İstanbul, 1980.

[2] Biran L., Diferensiyel Geometri Dersleri, Journal of Geo., Vol. 62(1998), 40-47.

[3] Pottmann H., Wallner J., Computational Line Geometry,

52 Istanbul University, Faculty of Science, Department of Mathematics, Istanbu/Turkey, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Minimal Surfaces and Harmonic Mappings

Hakan Mete TAŞTAN53, Sibel GERDAN54

Abstract

The projection on the base plane of a regular minimal surface S in 3 with isothermal parameters defines a complex-valued harmonic function f()()() z h z g z . The aim of this paper is to determine the

Gauss map and shape operator of the minimal surface in terms of analytic and co-analytic parts of the harmonic function .

Key Words: Minimal Surfaces, Harmonic mappings

References

[1] Taştan H.M., Polatoğlu Y., On quasiconformal harmonic mappings lifting to minimal surfaces, Turkish Journal of Mathematics, Vol. 37(2013), 267-277. [2] Duren P., Harmonic mappings in the plane, Cambridge University Press, 2004.

53 İstanbul University, Faculty of Science, Department of Mathematics, Vezneciler Campus, 34134, Fatih/İstanbul, E-mail: [email protected] 54 İstanbul University, Faculty of Science, Department of Mathematics, Vezneciler Campus, 34134, Fatih/İstanbul, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Cubical Cohomology Groups of Digital Images

Özgür EGE55

Abstract

Homology and cohomology theory for digital images have been very popular in recent times. The digital cubical homology groups were given in [6]. In this study, we present cubical cohomology groups of digital images. We compute cubical cohomology groups of some digital images. We get some results using Universal Coefficient Theorem for digital cubical cohomology groups.

Key Words: Digital image, digital cubical set, digital cubical cohomology group.

References

[1] Arslan, H., Karaca, I. and Oztel, A., Homology groups of n-dimensional digital images, XXI. Turkish National Mathematics Symposium, B(2008), 1-13. [2] Boxer, L., Karaca, I. and Oztel, A., Topological invariants in digital images, Journal of Mathematical Sciences: Advances and Applications, Vol. 11(2)(2011), 109-140. [3] Ege, O. and Karaca, I., Cohomology theory for digital images, Romanian Journal of Information Science and Technology, Vol. 16(1)(2013), 10-28. [4] Kaczynski, T., Mischaikow, K. and Mrozek, M., Computational Homology, Applied Mathematical Sciences, Springer-Verlag, NY, 2004. [5] Kaczynski, T. and Mrozek, M., The cubical cohomology ring: an algorithm approach, Foundations of Computational Mathematics, Vol. 13(5)(2013), 789-818. [6] Karaca, I. and Ege, O., Cubical homology in digital images, International Journal of Information and Computer Science, Vol. 1(7)(2012), 178-187. [7] Pilarczyk, P. and Real, P., Computation of cubical homology, cohomology, and (co)homological operations via chain contraction, Advances in Computational Mathematics, Vol. 41(1)(2015), 253-275.

55 Celal Bayar University, Faculty of Science and Letters, Department of Mathematics, Muradiye Campus, 45140, Yunusemre/Manisa, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Ruled Surface Reconstruction in Euclidean Space

Mustafa DEDE56, Cumali EKİCİ57

Abstract

In this paper, firstly we summarize some results concerning the differential geometry of the ruled surfaces. Then, we introduce the signature curve of ruled surfaces in Euclidean three-space. Furthermore, it is used to a simple algorithm for reconstruction of a ruled surface. Finally, some examples have been demonstrated the efficiency and accuracy of the algorithm.

Key Words: Ruled surface, Signature curve, Curvature, Reconstruction.

References

[1] Calabi E., Olver P.J., Shakiban C., Tannenbaum A. and Haker S., Differential and Numerically Invariant Signature Curves Applied to Object Recognition, Int. J. Computer Vision, Vol. 26(1998), 107-135. [2] Calabi E., Olver P. J. and Tannenbaum A., Affine Geometry, Curve Flows, and Invariant Numerical Approximations, Adv. Math., Vol. 124(1996), 154-196. [3] Boutin M., Numerically Invarint Signature Curves, Int. J. Comput. Vision, Vol. 40(2000), 235-248. [4] Hickman M. S., Euclidean Signature Curves, J. Math. Imaging. Vis., Vol. 43(2012), 206-213. [5] Wu S. and Li Y.F., Motion Trajectory Reproduction From Generalized Signature Description, Pattern Recognition, Vol. 43(2010), 204-221. [6] Wu S. and Li Y.F., On Signature Invariants for Effective Motion Trajectory Recognition, The International Journal of Robotics Research, Vol. 27(2008), 895-917. [7] Kühnel W., Ruled W-surfaces, Arch. Math., Vol. 62(1994), 475-480.

56 Kilis 7 Aralık University, Faculty of Art and science, Department of Mathematics, 79000, Kilis, E-mail: [email protected] 57 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-Computer, 26480 Eskişehir, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On the spacelike parallel ruled surfaces with Darboux frame

Muradiye ÇİMDİKER58 and Cumali EKİCİ59

Abstract

In this study, the spacelike parallel ruled surfaces with Darboux frame are introduced in Minkowski 3- space. Then some characteristic properties of the spacelike parallel ruled surfaces with Darboux frame such as developability, the striction point and distribution parameter are obtained in Minkowski 3-space. Key Words: Ruled surface, Parallel surface, Darboux frame.

References

[1] Gray A., Salamon S. and Abbena E., Modern Differential Geometry of Curves and Surfaces with Mathematica, Chapman and Hall/CRC, 2006. [2] Darboux G., Leçons Sur la Theorie Generale des Surfaces I-II-III-IV., Gauthier-Villars, Paris, 1896.

[3] Şentürk G. Y. and Yüce S., Characteristic Properties of the Ruled Surface with Darboux Frame in E 3 , Kuwait Journal of Science, Vol. 42(2)(2015), 14-33. [4] Hacısalihoğlu H. H., Diferensiyel Geometri, İnönü Üniv. Fen Edebiyat Fak. Yayınlar, 2, 1983. [5] Uğurlu H. H. and Kocayiğit H., The Frenet and Darboux Instantaneous Rotation Vectors of Curves on Timelike Surface, Mathematical and Computational Applications, Vol. 1(2)(1996), 133-141. [6] Özdemir M. and Engin A. A., Spacelike Darboux Curves in Minkowski Space, Differential Geometry- Dynamical Systems, Vol. 9(2007), 131-137. [7] Ravani T. and Ku S., Bertrand Offsets of Ruled Surface and Developable Surface, Computer-Aided Design, Vol. 23(2)(1991), 145-152. [8] Hlavaty V., Differentielle linien geometrie. Uitg P. Noorfhoff, Groningen, 1945. [9] Ünlütürk Y. and Ekici C., Parallel Surfaces Satisfying the Properties of Ruled Surfaces in Minkowski 3- Space, Global Journal of Science Frontier Research: F, Vol. 14(1)(2014). [10] Ünlütürk Y., Çimdiker M. and Ekici C., Characteristic Properties of the Parallel Ruled Surfaces with Darboux Frame in Euclidean 3-Space, to review, (2015). [11] Savcı Z., Görgülü A. and Ekici C., On Meusnier Theorem for Parallel Surfaces, Thai Journal of Mathematics (in press), (2016).

58 Kirklareli University, Faculty of Art and Science, Department of Mathematics, Kayalı Campus, 39000, Kayalı/Kirklareli, E-mail: [email protected] 59 Eskisehir Osmangazi University, Faculty of Art and Science, Department of Mathematics-Computer, Meselik Campus, 26480, Meselik/Eskisehir, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Triakis Octahedron Metric and Its Isometry Group

Gürol BOZKURT60, Temel ERMİŞ 61

Abstract

Mathematicians and geometers have recently studied on transformation geometry. It is understandable efforts to bring together geometry and algebra. Thus, one can easily analyze the mathematical system thanks to transformations on this systems. The transformation preserve designated features of the geometric structure. The set of transformations compose the groups consisting of the symmetries of geometric objects. Symmetry is a important concept in the study of mathematics. The excellent symmetry of the Platonic solids have made them perfect models for the studying on symmetries. The Platonic solids known as the regular polyhedrons, all of whose faces are congruent regular polygons, and where the same number of faces meet at every vertex. Similarly, Catalan and Archimedean solids have interesting symmetries. Also, Catalan and Archimedean solids called non-regular polyhedrons. In this work, we give the new metric which unit sphere is the triakis octahedron. Thus the triakis octahedron which is one of Catalan solids associated to metric geometry. Later, we have analytically showed that the group of isometries of the R³ with respect to the new metric.

Key Words: Metric Geometry, Distane Geometry, Polyhedrons, Isometry Group

References

[1] T. Ermiş, Düzgün Çokyüzlülerin Metrik Geometriler ile İlişkileri Üzerine, Esogü, PHD thesis, (2014). [2] T. Ermiş, R. Kaya, On The Isometries the of 3- Dimensional Maximum Space, KJM, Vol. 3, No. 1, 103-114, (2015). [3] O. Gelisgen, R. Kaya, Generalization of α-distance to n−dimensional space, KoG. Croat. Soc. Geom. Graph. 10, 33-35, (2006). [4] O. Gelisgen, R. Kaya, The Taxicab Space Group, Acta Mathematica Hungarica, Vol.122, No. 1-2, 187-200, (2009). [5] Z. Akca, R. Kaya, On the Distance Formulae In three Dimensional Taxicab Space, Hadronic Journal, 27, 521-532, (2006).

60 Eskişehir Osmangazi University, Faculty of Art and Science, Department of Mathematics and Computer Sciences, Meşelik Campus, 26480, Eskişehir, E-mail: gurolhoca @ gmail.com 61 Eskişehir Osmangazi University, Faculty of Art and Science, Department of Mathematics and Computer Sciences, Meşelik Campus, 26480, Eskişehir, E-mail: termis @ogu.edu.tr

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Umbilic Surfaces in Lorentz 3-Space

Esma DEMİR ÇETİN62, Yusuf YAYLI63

Abstract

As we all know for a surface M and a point P 2 M in Euclid 3-space, if H2 (P) - K(P) > 0 then the Weingarten map of M in P is diagonalizable. Here H is the mean curvature and K is the Gauss curvature of the surface. Also if H2 (P) - K(P)=0 then we say that P is an umbilic point of M. If all P 2 M is umbilic then we can say that M is an umbilic surface. In Lorentz 3-space the situation is different. The equation of H2 (P) - K(P) = 0 for a point P 2 M doesn’t mean that P is an umbilic point and Weingarten map of M in P can be diagonalizable. In this work we find the surfaces with the equation H2 - K = 0, whose generated by graph of a polynomial under homothetic motion groups in Lorentz 3-space.

Key Words: Gauss curvature, mean curvature, umbilic points, Lorentz space, homothetic motions

References

[1] Clelland, J. N. , Totally Quasi-Umbilical Timelike Surfaces in R12, arXiv:1006.4380vl (2010) [2] Hou, Z.H., Ji, F., Helicoidal Surfaces with H2=K in Minkowski 3-Space, J. Math Anal., Appl. 318 (2007), 101-113 [3]Lopez,R., Differential geometry of curves and surfaces in Lorentz Minkowski space, http://arxiv.org/abs0810. [4] Lopez, R. and Demir, E., Helicoidal Surfaces in Minkowski Space with Constant Mean Curvature and Constant Gauss Curvature, Cent. Eu. J. Math. 12(9), (2013), 1349-1361 [5] Tosun, M., Kucuk, A. and Gungor M. A., The homothetic motions in the Lorentz 3-space. Acta Mathematica Science 26B(4), (2006), 711-719.

62 Nevşehir Hacı Bektaş Veli University, Faculty of Science and Arts, Department of Mathematics, 2000 Evler Mah. Zübeyde Hanım Cad. 50300 Nevşehir, E-mail: [email protected] 63 Ankara University, Faculty of Science, Department of Mathematics, Dögol Cad 06100 Beşevler/ Ankara, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On the Mannheim curves in the three-dimensional sphere

Tanju KAHRAMAN64, Mehmet ÖNDER1

Abstract Mannheim curves are defined for immersed curves in 3-dimensional sphere S 3 . The definition is given by considering the geodesics of . First, two special geodesics, called principal normal geodesic and binormal geodesic, of are defined by using Frenet vectors of a curve  immersed in . Later, the curve  is called a

Mannheim curve if there exits another curve  in such that the principal normal geodesics of  coincide with the binormal geodesics of . Moreover, the relation between a Mannheim curve immersed in and a generalized Mannheim curve in E 4 is obtained.

Key Words: Spherical curves; generalized Mannheim curves; geodesics.

References [1] Barros, M., General helices and a theorem of Lancret, Proceedings of the American Mathematical Society 125 (1997) 1503–1509. [2] Blum, R., A Remarkable class of Mannheim-curves, Canad. Math. Bull., 9(1966), 223-228. [3] Choi, J., Kang, T. and Kim, Y., Mannheim Curves in 3-Dimensional Space Forms, Bull. Korean Math. Soc. 50(4) (2013) 1099–1108.

[4] Kim, C.Y., Park, J.H., Yorozu, S., Curves on the unit 3-sphere S 3 (1) in the Euclidean 4-space 4 , Bull. Korean Math. Soc., 50(5) (2013) 1599-1622. [5] Lucas, P., Ortega-Yagües, J., Bertrand Curves in the three-dimensional sphere, Journal of Geometry and Physics, 62 (2012) 1903–1914. [6] Mannheim, A., Paris C.R. 86 (1878) 1254–1256. [7] Matsuda, H., Yorozu, S., On Generalized Mannheim Curves in Euclidean 4-space, Nihonkai Math. J., 20 (2009) 33–56. [8] Saint-Venant, J.C., Mémoire sur les lignes courbes non planes, Journal d’Ecole Polytechnique 30 (1845) 1– 76. [9] Wang, F., Liu, H., Mannheim partner curves in 3-Euclidean space, Mathematics in Practice and Theory, 37(1) (2007) 141-143. [10] Wong, Y.C., Lai, H.F., A critical examination of the theory of curves in three dimensional differential geometry, Tohoku Math. J. 19 (1967) 1–31.

64Manisa Celal Bayar University, Faculty of Arts and Sciences, Department of Mathematics, Muradiye Campus, 45140, Muradiye, Manisa, TURKEY. E-mails: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Complete Lifts of Tensor Fields of Type (1,1) on Cross-Sections in a Special Class of Semi-Cotangent Bundles

Furkan YILDIRIM65, Kürşat AKBULUT66

Abstract

The main purpose of this paper is to investigate complete lift of tensor fields of type (1,1) from manifold M to its semi-cotangent bundle t*M. In this context cross-sections in semi-cotangent (pull-back) bundle t*M of cotangent bundle T*M by using projection (submersion) of the tangent bundle TM can be also defined.

Key Words: Vector field, complete lift, pull-back bundle, cross-section, semi-cotangent bundle.

References

[1]. D. Husemoller, Fibre Bundles, Springer, New York, 1994. [2]. C.J. Isham, "Modern differential geometry for physicists", World Scientific, 1999. [3]. H.B. Lawson and M.L. Michelsohn, Spin Geometry, Princeton University Press., Princeton, 1989. [4]. L.S. Pontryagin, Characteristic classes of differentiable manifolds, Transl. Amer. Math. Soc., (1962);7: 279-331. [5]. N. Steenrod, The Topology of Fibre Bundles, Princeton University Press., Princeton, 1951. [6]. V. V. Vishnevskii, Integrable affinor structures and their plural interpretations, Geometry, 7.J. Math. Sci., (New York) 108 (2002); 2: 151-187. [7]. K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Marcel Dekker, Inc., New York, 1973. [8]. F. Yıldırım, On a special class of semi-cotangent bundle, Proceedings of the Institute of Mathematics and Mechanics, (ANAS) 41 (2015), no. 1, 25-38. [9]. F. Yıldırım and A. Salimov, Semi-cotangent bundle and problems of lifts, Turk J. Math, (2014); 38: 325-339.

65 Atatürk University, Department of Mathematics, Faculty of Sci, Narman Vocational Training School, 25530, Narman/ Erzurum, E-mail: [email protected] 66 Atatürk University, Department of Mathematics, Faculty of Sci, 25240, Erzurum/Turkey, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Notes On The Curves According To Type-I Bishop Frame in Euclidean Plane

Süha YILMAZ67, Yasin ÜNLÜTÜRK68

Abstract In this study, position vector of a Euclidean plane curve is investigated .First, a system of differential equation whose solution gives the components of the position vector on the Bishop axis is constructed. By means of solution mentioned system, position vector of all such curves according to Type-I Bishop Frame is obtained. Thereafter, it is proven that, position vector and curvature of a Euclidean plane curve a satisfy a vector differential equation of third order. Moreover, we obtained characterizations curves of constant breadth according to Type-I Bishop Frame in Euclidean plane in terms of Bishop vector fields. Finally, we characterized Smarandache curves Type-I Bishop Frame in Euclidean plane.

Key Words: Euclidean plane, Type-I Bishop Frame, curves of constant breadth, Smarandache curves.

References

[1]Bükçü B.,Karacan M.K., Special Bishop Motion and Bishop Darboux Rotation Axis of the Space Curve, J.Dyn.Syst.Geom.Theor., 6(1),2008,27-34. [2]Bükçü B., Karacan M.K., The Slant Helices According to Bishop Frame, Int.J. Math. Comput. Sci. 3(2),2009,67-70. [3]Bishop L.R.,There is more than one way to frame a curve, Amer. Math. Monthly, Vol. 82(3), (1975), 246- 251. [4] Kose Ö.,On space curves of constant breadth, Doğa Math., 1986, 10,11-14. [5] Köse Ö.,Some properties of ovals and curves of constant width in a plane. Doğa Math., 1984, 8,119-126. [6] Yilmaz S.,Turgut M., Some Characterizations of Isotropic Curves in the Euclidean Space, Int. J. Comput. Math. Sci. 2(2), 2008, 107-109. [7] Yılmaz S.,Position Vectors of Some Special Space-like Curves according to Bishop frame in Minkowski Space E₁³, Sci. Magna, 5 (1), 48--50, 2009. [8] Yılmaz S., Turgut M., A new version of Bishop frame and an application to spherical images. J Math Anal Appl 2010; 371: 764-776.

67 Dokuz Eylül University, Faculty of Education , Department of Mathematic Education, Buca Campus, 35150, Buca/İzmir, E-mail: [email protected] 68 Kırklareli University Faculty of Science,Department of Mathematic, Kayalı Campus, 39020, Kavaklı / Kırklareli E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Semi-invariant semi-Riemannian submersions from para-Kahler manifolds

Yılmaz GÜNDÜZALP69, Mehmet Akif AKYOL70

Abstract

In this paper, we introduce semi-invariant semi-Riemannian submersions from almost para-Kahler manifolds onto semi-Riemannian manifolds. We give some examples, investigate the geometry of foliations that arise from the definition of a semi-Riemannian submersion and check the harmonicity of such submersions. We also find necessary and sufficient conditions for a semi- invariant semi-Riemannian submersion to be totally geodesic. Moreover, we obtain curvature relations between the base manifold and the total manifold.

Key Words: Para-Kahler manifold, semi-Riemannian submersion, anti-invariant semi-Riemannian submersion, semi-invariant semi-Riemannian submersion.

References [1] Falcitelli, M., Ianus, S. and Pastore, A.M. Riemannian Submersions and Related Topics, World Scientific, 2004. [2] Gündüzalp, Y. and Şahin, B. Paracontact semi-Riemannian submersions, Turkish J.Math. 37(1), 114-128, 2013. [3] Gündüzalp, Y. Anti-invariant semi-Riemannian submersions from almost para-Hermitian manifolds, Journal of Function Spaces and Applications, ID 720623, 2013. [4] Ivanov, S. and Zamkovoy, S. Para-Hermitian and para-quaternionic manifolds, Diff. Geom. and Its Appl. 23, 205-234, 2005. [5] O`Neill, B. Semi-Riemannian Geometry with Application to Relativity, Academic Press, New York, 1983. [6] Şahin, B. Semi-invariant Riemannian submersions from almost Hermitian manifolds, Canad. Math. Bull. 56, 173-183, 2013.

69Dicle University, Faculty of Art and science, Department of Mathematics, 21280, Diyarbakır/TURKEY E-mail: [email protected] 70 Bingöl University, Faculty of Art and science, Department of Mathematics, 12000, Bingöl/TURKEY E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Lagrangian Dynamics on Matched Pairs

Oğul ESEN71, Serkan SÜTLÜ72

Abstract

Given a matched pair of Lie groups, we show that the tangent bundle of a matched pair group is isomorphic to the matched pair of the tangent groups. We thus obtain the Euler-Lagrange equations on the trivialized matched pair of tangent groups, as well as the Euler-Poincaré equations on the matched pair of Lie algebras. We show explicitly how these equations cover those of the semi-direct product theory. In particular, we study the trivialized, and the reduced Lagrangian dynamics on the Lorentz group SO(3, 1).

Key Words: matched pair of Lie groups and Lie algebras, Euler-Lagrange equations, Euler-Poincaré equations

References

[1] O. Esen and S. Sütlü, Lagrangian Dynamics on Matched Pairs, arxiv: 1512.06770 (2015)

71 Gebze Technical University, Faculty of Art and Science, Department of Mathematics, Çayırova Campus, 41400, Gebze/KOCAELİ, E-mail: [email protected] 72 Işık University, Faculty of Art and Science, Department of Mathematics, 34980, Şile/İSTANBUL, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Reduction of Tulczyjew’s Triplet

Oğul ESEN73, Hasan GÜMRAL74

Abstract

Choosing the configuration space as a Lie group G, the trivialized and reduced Tulczyjew’s triplets are contructed. The trivialized Euler-Lagrange and Hamilton’s equations are derived and presented as Lagrangian submanifolds of the trivialized Tulczyjew’s symplectic space. Euler-Poincaré and Lie-Poisson equations are presented as Lagrangian submanifolds of the reduced Tulczyjew’s symplectic space. Tulczyjew’s generalized Legendre transformations for trivialized and reduced dynamics are constructed.

Key Words: Lagrangian Dynamics, Hamiltonian Dynamics, Reduction, Legendre transformation, Lie groups, Lie algebras.

References

[1] O. Esen and H. Gümral, (2014), Tulczyjew's Triplet for Lie Groups I: Trivializations and Reductions, Journal of Lie Theory, Volume: 24, pp. 1115-1160. [2] O.Esen and H. Gümral, (2015), Tulczyjew's Triplet for Lie Groups II: Dynamics, arXiv:1503.06566. [3] O.Esen and H. Gümral, (2015), Reductions of Dynamics on Second Iterated Bundles of Lie Groups arXiv:1503.06568.

73 Gebze Teknik University, Faculty of Science, Department of Mathematics, Gebze-Kocaeli 41400, Turkey, E-mail: [email protected] 74 (On leave of absence from Department of Mathematics, Yeditepe University ) Australian College of Kuwait, West Mishref, 13015 Safat, Kuwait, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Spherical Motions And Dual Frenet Formulas Aydın ALTUN75 Abstract In this study, Some theorems is given and make interpretations with related to them. The real unit spherical representation of the roselike curve being generated by ep(t;4,1,3,0) which is presented and some properties of the developable ruled surface is obtained. Furthermore; the curvature and torsion functions of the curve is given. The results written in this manuscript imply that, at regular points the Gaussian curvature of a developable ruled surface is identically zero. The dual geodesic trihedron, the dual Frenet-Serret frame, the dual form of usual Frenet-Serret equations, the dual curvature and torsion functions have been computed and interpreted. Key Words: Developable ruled surface, Real spherical motion, Dual spherical motion, Dual curvature, Dual torsion, Dual angle. AMS 2010: 14J26, 43A90, 49M29, 16S90, 32Q10. References [1] Coventy, J., Page, W., The Fundamental Periods Of Sums Of Periodic Functions, The College Mathematics Journal, 20 (1989), 32-41. [2] S.Goldenberg, S., H. Greenwald, Calculus Applications İn Engineering And Science, D.C. Heath, Lexington, Ma, 1990. [3] J.D. Lawrence, A Catalog Of Special Plane Curves, Dower, New York, 1972. [4] E.H. Lockwood, A Book Of Curves, Cambridge University Press, Cambridge, 1961. [5] Morıtz, R.E., On The Construction Of Curves Given İn Polar Coordinates, American Mathematical Monthly, 24 (1917): 213-220. [6] Nash, D.H., Rotary Engine Geometry, Mathematics Magazine, 50 (1977): 87-89. [7] Rıgge, W.F., A Compound Harmonic Motion Machine I, Iı, Scientific American Supplement 2197, 2198 (1918): 88-91, 108-110. [8] Rıgge, W.F., Concerning A New Method Of Tracing Cardioids, American Mathematical Monthly, 26 (1919): 21-32. [9] Rıgge, W.F., Cuspidal Rosettes, American Mathematical Monthly, 26 (1919): 332-340. [10] Rıgge, W.F., Envelope Rosettes, American Mathematical Monthly, 27 (1920): 151-157. [11] Rıgge, W.F., Cuspidal Envelope Rosettes, American Mathematical Monthly, 29 (1922): 6-8. [12] .E. Taylor, Advanced Calculus, Ginn, New York, 1955. [13] Hall, L.M., Throchoids, Roses And Thorns - Beyond The Spirograph, The College Mathematics Journal, 23 (1992): 20-35. [14] Altın, A., Dual Spherical Motions And The Ruled Rose And Ellipse Surfaces, I. Turkish National Geometry Symposium Journal, 6, 2003. [15] Altın, A., Some General Propositions For The Edge Of Regressions Of Developable Ruled İn En, Hacettepe Bulletin Of Natural Sciences And Engineering, Faculty Of Science, 16 (1988): 13-23. [16] Altın, A., Özdemir, H.B., Spherical Images And Higher Curvatures, Uludağ University Journal, 3 (1988): 103-110. [17] Hacısalihoğlu, H.H., On Closed Spherical Motions, Q. App. Math., 29 (1971): 269-276. [18] Hacısalihoğlu, H.H., On The Rolling Of One Curve Or Surface Upon Another, Mechanism And Machine Theory, 7 (1972): 291-305. [19] Altın, A., Plane Mechanism And Dual Spherical Special Motions, Xvı. Turkish National Mathematics Symposium Journal, 25 (2003): 31-32. [20] Müller, H.R., Sphärische Kinematik, Veb Deutscher Verlag, Wissenschaften, Berlin, 5-20, 1962. [21] Do Carmo, M.P., Differential Geometry Of Curves And Surfaces, Prentice-Hall, 188-213, 1976. [22] O'neıll, B., Elmentary Differential Geometry, Academic Press, 232-242, 1966. [23] Glück, H., Higher Curvatures Of Curves İn Euclidean Space, Amer. Math. Month., 73 (1966), 699-704. [24] Morgan, F., Riemannian Geometry, Ak Peters Ltd, 99-113, 1998.

75 Dokuz Eylül University, P.K. 746, 06100, Yenişehir – Ankara / Turkey, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

The Timelike Bezier Spline in Minkowski 3 Space

Hatice KUŞAK SAMANCI76, Özgür BOYACIOĞLU KALKAN77 , Serkan ÇELİK 78

Abstract The purpose of this study is to develop a Bezier spline in Minkowski 3 space called by the Timelike Bezier Spline. In this paper firstly, we investigate the Frenet frame, curvatures and derivative formulations at the starting and end points of the Timelike Bezier Spline. Moreover, we obtain the derivative formulas of the Bishop frame and the curvatures according to the Bishop frame at starting and end points of the Timelike Bezier spline in Minkowski 3 space. Consequently we give some examples for this concept.

Key Words: Timelike Bezier spline, Frenet and Bishop frame, Minkowski 3 space.

References

[1] López, R., 2008. Differential geometry of curves and surfaces in Lorentz-Minkowski space, arXiv preprint arXiv:0810.3351. [2] Uğurlu H.H.,Çalışkan A., Darboux Ani Dönme Vektörleri ile Spacelike Spacelike ve timelike Yüzeyler Geometrisi Kitabı, 2012. [3] Farin G., Curves and Surfaces for Computer-Aided Geometric Design, Academic Press,1996 . [4] Incesu, M. And Gürsoy, O., “Bezier Eğrilerinde Esas Formlar ve Eğrilikler”, XVII Ulusal Matematik Sempozyumu, Bildiriler, Abant İzzet Baysal Üniversitesi,2004:146-157. [5] G.H. Georgiev, Spacelike Bezier curves in the three-dimensional Minkowski space, Proceedings of AIP Conference 1067 (1) (2008). [6] P. Chalmoviansky and B. Pokorna “Quadratic spacelike Bezier Curves in the three dimensional Minkowski Space”. Proceeding of Symposium on Computer Geometry, 20:104-110, 2011. [7] Pokorná, Barbora, and Pavel Chalmovianský. "Planar Cubic Spacelike Bezier Curves in Three Dimensional Minkowski Space.", Proceeding of Syposium on Computer Geometry, SCG 2012, Vol.21, pp.93-98. [8] Bishop L.R., There is more than one way to frame a curve, Amer. Math. Monthly, 82(1975), pp 246-251.

76 Bitlis Eren University, Faculty of Art and science, Department of Mathematics, 13000, Bitlis/Turkey, E-mail: [email protected] 77Afyon Kocatepe University,Faculty of Art and Science, Department of Mathematics,03200,Afyon Turkey, E-mail: [email protected] 78 Bitlis Eren University, Faculty of Art and science, Department of Mathematics, 13000, Bitlis/Turkey, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

The Geometric Approach of Yarn Surface and Weft Knitted Fabric

Hatice KUŞAK SAMANCI79, Filiz YAĞCI80 , Ali ÇALIŞKAN 81

Abstract

In this paper we investigate some geometric properties of yarn surface and weft knitted fabric by using some of curves and surfaces that used in Computer Aided Geometric Design (CAGD). CAGD, which includes the mathematical representations of shapes using computer graphics was discovered in 1974 by R.E. Barnhill and R.F. Riesenfeld. In particularly Bezier curves and surfaces provide a geometric understanding of many CAGD facts. Therefore, we used the Bezier curves and surfaces for modelling of the yarn surface and the knitted fabric.

Key Words: CAGD, Bezier curves, Yarn, Knitted fabric

References

[1] G., Farin, Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, 3rd Edition, Academic Press Inc., San Diego, 1993. [2] B. Güngör, Tekstil Mekaniğinin Temelleri, Dokuz Eylül Ün. Müh. Basım Ün., 2008. [3] O. Goktepe, Use of Non-Uniform Rational B-Splines for Three-Dimensional Computer Simulation of Warp Knitted Structures, Turk J Engin Environ Sci, 369-378, 2001. [4] D.F. Rogers, and J.A. Adams. Mathematical Elements for Computer Graphics, McGraw- Hill. New York NY. USA., 1976. [5] Goktepe O 2001 Use of non-uniform rational B-spline for threedimensional computer simulation of warp knitted fabric Turkey J. Eng. Environ. Sci. 25 (2001) 369-378. [6] Kurbak, Arif. "Geometrical models for balanced rib knitted fabrics part I: conventionally Knitted 1×1 rib fabrics." Textile Research Journal 79.5 (2009): 418-435.

79 Bitlis Eren University, Faculty of Art and science, Department of Mathematics, 13000, Center/Bitlis, E-mail: [email protected] 80 Uludağ University, Faculty of Art and science, Department of Mathematics, 16059, Görükle/Bursa, E-mail: [email protected] 81 Ege University, Faculty of Science, Department of Mathematics, 35100, Bornova/İzmir, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Some Solutions of the Non-minimally coupled electromagnetic fields to gravity

Özcan SERT82

Abstract

Einstein-Maxwell theory is known as a minimally coupled theory between electromagnetic fields and gravitation. We consider gravitational models which involves non-minimally coupled electromagnetic fields to gravitational fields in Y (R)F2 form. The gravitational models for various non-minimal function Y(R) were studied in order to explain the late-time acceleration and inflation of the universe[1]. Additionally some non- minimal Y (R)F2 gravitational models can explain the rotational curves of test particles around galaxies [2,3,4]. We look at the non-minimally coupled models and field equations using the algebra of exterior differential forms. We give some static, spherically symmetric, solutions with electric and magnetic charge.

Key words: Gravitation, Non-minimal coupling, Einstein-Maxwell.

References

[1] Bamba, K., Nojiri, S., Odintsov, S.D., The future of the universe in modified gravitational theories: approaching a finite-time future singularity, JCAP 10, 045, 2008. [2] Dereli, T., Sert, Ö., Non-minimal ln(R)F2 couplings of electromagnetic fields to gravity: static, spherically symmetric solutions, Eur. Phys. J. C 71, 1589, 2011. [3] Sert, Ö., Electromagnetic duality and new solutions of the non-minimally coupled Y(R)-Maxwell gravity, Mod. Phys. Lett. A, 28, 12, 1350049, 2013. [4] Sert, Ö., Gravity and Electromagnetism with Y(R)F2-type Coupling and Magnetic Monopole Solutions, Eur. Phys. J. Plus, 127: 152, 2012.

82 Pamukkale University, Faculty of Art and science, Department of Mathematics, Kınıklı Campus, 20070, Kınıklı/Denizli, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Differantial Equations of Motion Objects with An Almost Paracontact Metric Structure

Oğuzhan ÇELİK83, Zeki KASAP84

Abstract

The geometry of almost paracontact manifolds is a natural extension in the odd dimensional case of almost Hermitian geometry. In additional, the paracontact geometry as symplectic geometry has large and comprehensive applications in physics, geometrical optics, classical mechanics, thermodynamics, geometric quantization, differential geometry and applied mathematics. the Euler-Lagrange differential equations one of the common ways of solving problems in classical and analytical mechanics. In the study, we consider Euler- Lagrange differential equations with almost paracontact metric structure for motion objects. Also, implicit solutions of the differential equations found in this study will be solved by Maple computation program and a graphic example will be drawn.

Key Words: Paracontact Manifold, Mechanical System, Dynamic Equation, Lagrangian Formalism.

References

[1] Tripathi M.M., Kilic E., Perktas S.Y. and Keles S., Indefinite almost paracontact metric manifolds, International Journal of Mathematics and Mathematical Sciences, 2010, 1-19. [2] Srivastava S.Kr., Narain D. and Srivastava K., Properties of ε-S paracontact manifold, VSRD-TNTJ, Vol. 2 (11), 2011, 559-569. [3] Girtu M., An almost 2-paracontact structure on the cotangent bundle of a Cartan space, Hacettepe Journal of Mathematics and Statistics, Volume 33, 2004, 15-22. [4] Ahmad M. and Jun J-B., Submanifolds of an almost r-paracontact Riemannian manifold endowed with a semi-symmetric non-metric connection, Journal of The Chungcheong Mathematical Society, Volume 22, No 4, 2009, 653-664, [5] Kupeli Erken I., Some classes of 3-dimensional normal almost paracontact metric manifolds, Honam Mathematical J., 37, No: 4, 2015, 457-468. [6] Kasap Z. and Tekkoyun M., Mechanical systems on almost para/pseudo-Kähler--Weyl manifolds, IJGMMP, Vol. 10, No.5, 2013, 1-8. [7] Tekkoyun M., Çelik O., "Mechanical Systems On An Almost Kähler Model Of Finsler Manifold", International Journal of Geometric Methods in Modern Physics (IJGMMP), vol.10, 2013,18-27

83 Çanakkale Eighteenmart University, Institute of Science, Department of Mathematics, Çanakkale / Turkey, E-mail: [email protected] 84 Pamukkale University, Faculty of Education, Elementary Mathematics Education Department, Denizli/ Turkey, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Characterizations of Some Special Time-like Curves In Lorentzian Plane

Abdullah MAĞDEN85, Süha YILMAZ86 , Yasin ÜNLÜTÜRK 87 Abstract

In this paper,we give the properties of the time-like curves of constant breadth Lorentzian plane. Moreover ,we define Smarandache curves for time-like curves in Lorentzian plane and characterized this curves. Additionally, we circular indicatrices of time-like-like curves in Lorentzian plane.

Key Words:.Circular indicatrices,time-like curve, curves of constant breadth, Smarandache curves.

References

[1] A.T. Ali, Special Smarandache Curves in The Euclidean Space, math. Cobin. Bookser,Vol.2(2), 2010,30- 36. [2] A.T. Ali, R. Lopez, Slant helices in Minkowski space E₁³, J. Korean Math. Soc. 48, 2011,159-167. [3] Akbulut F.,Vector Calculus, Ege University Press, İzmir, 1981. [4] Cetin M.,Tuncer Y.,and Karacan M.K.,Smarandache Curves According to Bishop Frame in Euclidean 3- Space, Gen. Math. Notes, 2014; 20: 50-56. [5] Izumiya S.,Takeuchi N.,New special curves and developable surfaces, Turkish J. Math. 28(2), 2004,531-537. [6] Kose Ö.,On space curves of constant breadth, Doğa Math., 1986, 10,11-14 [7] Turgut M.,Yılmaz S., Smarandache Curves in Minkowski SpaceTime, International J. Math. Combin. 2008; 3,: 51-55. [8]Turgut, M., Smarandache Breadth Pseudo Null Curves in Minkowski Space-Time, International J. Math. Combin. 2009; 1: 46-49. [9] Yılmaz S., Turgut M., A new version of Bishop frame and an application to spherical images. J Math Anal Appl 2010; 371: 764-776. [10] S. Yilmaz, Spherical Indicators of Curves and Characterizations of Some Special Curves in four dimensional Lorentzian Space L4, Dissertation, Dokuz Eylul University,2001.

85 Atatürk University, Faculty of Science, Department of Mathematics, Atatürk University Campus, 25400/Erzurum E-mail: [email protected] 86 Dokuz Eylül University, Faculty of Education, Department of Mathematic Education, Buca Campus, 35150, Buca/İzmir, E-mail: [email protected] 87 Kırıkkale University, Faculty of Science, Department of Mathematics, Kayalı Campus, 39020, Kavaklı / Kırklareli E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Contributions to Differential Geometry of Space-like Curves In Lorentzian Plane Yasin ÜNLÜTÜRK88 Süha YILMAZ89

Abstract

In this study,we investigated the properties of the space-like curves of constant breadth Lorentzian plane. Later, this paper devoted to the study of Smarandache curves for tangent and normal vectors of space-like curves in Lorentzian plane and characterized this curves. Moreover, we give circular indicatrices of space-like curves in same plane.

Key Words: Circular indicatrices, space-like curve, curves of constant breadth, Smarandache curves.

References [1] A.T. Ali, Special Smarandache Curves in The Euclidean Space, math. Cobin. Bookser,Vol.2(2), 2010,30-36. [2] A.T. Ali, R. Lopez, Slant helices in Minkowski space E₁³, J. Korean Math. Soc. 48, 2011,159-167. [3] Akbulut F.,Vector Calculus, Ege University Press, İzmir, 1981. [4] Cetin M.,Tuncer Y.,and Karacan M.K.,Smarandache Curves According to Bishop Frame in Euclidean 3- Space, Gen. Math. Notes, 2014; 20: 50-56. [5] Izumiya S.,Takeuchi N.,New special curves and developable surfaces, Turkish J. Math. 28(2), 2004,531-537. [6] Kose Ö.,On space curves of constant breadth, Doğa Math., 1986, 10,11-14. [7] Köse Ö.,Some properties of ovals and curves of constant width in a plane. Doğa Math., 1984, 8,119-126. [8] Kula L.,Yaylı Y., On Slant Helix and Its Spherical Indicatrix, Appl. Math. Comput. 169 (1), 2005,600-607. [9] Şemin F.,Differential Geometry I, Istanbul University, Science Faculty Press, 1983. [10] Turgut M.,Yılmaz S., Smarandache Curves in Minkowski SpaceTime, International J. Math. Combin. 2008; 3,: 51-55. [11] Turgut, M., Smarandache Breadth Pseudo Null Curves in Minkowski Space-Time, International J. Math. Combin. 2009; 1: 46-49. [12] Yılmaz S., Turgut M., A new version of Bishop frame and an application to spherical images. J Math Anal Appl 2010; 371: 764-776.

88 Kırklareli University, Faculty of Science, Department of Mathematics, Kayalı Campus, 39020, Kavaklı Kırklareli E-mail: [email protected] 89 Dokuz Eylül University, Faculty of Education, Department of Mathematic Education, Buca Campus, 35150, Buca/İzmir, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

n1 On The Massey Theorem in E

Cumali EKİCİ 90 and Ali GÖRGÜLÜ91

Abstract In this paper, firstly, we define the generalized (k+1)-dimensional semi-ruled surface whose directional

n1+ surface is a semi-subspace in the semi-Euclidean space E.n Then we investigate the sufficient and necessary conditions for these surfaces are to be totally developable. In addition, we give the generalization of Massey theorem, which is well-known for the ruled surfaces defined in 3-dimensional Euclidean space, for the (k+1)-- dimensional ruled surfaces in the semi-Euclidean space

Key Words: Ruled surface, Massey Theorem, Semi-Euclidean space.

References [1] Çöken A. C. and Görgülü A., On The Joachimsthal's Theorems in semi-Euclidean Spaces, Nonlinear Analysis: Theory, Methods & Applications,, 70(11)(2009), 3932-3942. [2] Çöken A. C., On Euler's Theorem in semi-Euclidean Spaces, International journal of Geometric Methods in Modern Physics, 8(5)(2011), 1117-1129. [3] O'Neill B., Semi-Riemannian Geometry, Acedemic Press. New York, London, 1983. [4] Ekici C., Generalized semi-ruled surfaces in semi-Euclidean Spaces, (in Turkish). PhD Thesis, Eskişehir Osmangazi Univ. Grad. Sch. Nat. Sci., Eskisehir, 1998.

[5] Thas C., Properties of Ruled Surfaces in The Euclidean Space En , Bull. Inst. Math. Acedemica Sinica,

6(1)(1978), 133-142. [6] Thas C., Minimal Monosystems, Yokohama Math. Journal, 26(2)(1978), 157-167. [7] Frank H. and Giering O., Verallgemeinerte Regelflachen, Math. Zeit. 150(1976), 261-271. [8] Juza M., Ligne de Striction Sur Une Generalisetion a Plusreurs Dimensiona d’ une Surface Reglee, Czechosl. Math. J., 12(87)(1962), 243-250.

n [9] Tosun M. and Kuruoğlu N., On (k+1)-dimensional time-like ruled surfaces in the Minkowski space R1 , J. Inst. Math. Comput. Sci. Math., Ser. 11(1)(1998), 1-9.

[10] Tosun M. and Aydemir İ., On (k+1)-dimensional space-like ruled surfaces in the Minkowski space ,

Commun Fac. Sci. Univ. Ank., Ser.A1 Math. 46, 1-2(1998), 27-36.

[11] Keleş S. and Kuruoğlu N., Properties of Generalized Ruled Surfaces in the Euclidean n-Space and

Massey’s Theorem, Karadeniz University Mathematical Journal, VI(1983), 41-54.

90 Eskisehir Osmangazi University, Faculty of Art and Science, Department of Mathematics-Computer, Meselik Campus, 26480, Meselik/Eskisehir, E-mail: [email protected] 91 Eskisehir Osmangazi University, Faculty of Art and Science, Department of Mathematics-Computer, Meselik Campus, 26480, Meselik/Eskisehir, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Statistical Manifolds: New Approaches and Results

Muhittin Evren AYDIN92, Mahmut ERGUT93

Abstract

A statistical manifold is a triple (M,g,D), where (M,g) is a Riemannian manifold, D is a torsion-free affine connection such that Dg is symmetric. In this talk, we present some new results for submanifolds of statistical manifolds.

Key Words: Probability distribution function, statistical manifold, torsion-free affine connection.

References

[1] Aydin M.E., Mihai I., Wintgen inequality for statistical surfaces, arxiv 1511.04987 [math.DG], 2015. [2] Furuhata H., Hypersurfaces in statistical manifolds, Diff. Geom. Appl., Vol. 27 (2009), 420-429. [3] Mihai A., Geometric inequalities for purely real submanifolds in complex space forms, Results Math., Vol. 55 (2009), 457-468. [4] Opozda B., A sectional curvature for statistical structures, arXiv:1504.01279v1 [math.DG], 2015. [5] Vilcu A.E., Vilcu G.E., Statistical manifolds with almost quaternionic structures and quaternionic Kähler-like statistical submersions, Entropy, Vol. 17 (2015), 6213-6228.

92 Firat University, Faculty of Science, Department of Mathematics, 23119, Elazig, Turkey, E-mail: [email protected] 93 Namik Kemal University, Faculty of Art and Science, Department of Mathematics, 59000, Tekirdag, Turkey, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Similarity and Semi-similarity Relations on Generalized Quaternions

Abdullah İNALCIK94

Abstract

In this paper, the concept of similarity and semi-similarity for elements of generalized quaternions is given by solving ax xb and xay b ; ybx a , respectively.

Key Words: quaternions, generalized quaternions, similarity, semi-similarity, generalized inverse.

References

[1] Hamilton, W.R., Lectures on Quaternions, Hodges and Smith, Dublin (1853). [2] Yaglom, I.M., Comlex Numbers in Geometry, Academic Press, New York (1968). [3] Agrawal, O.P., Hamilton operators and dual-number-quaternions in spatial kine-matics, Mech. Mach. Theory, 22 (6), 569-575 (1987). [4] Flaut, C., Some equation in algebras obtained by Cayley-Dickson process, An. St. Univ. Ovidius Constanta, 9 (2), 45-68 (2001). [5] Tian, Y., Universal factorization equalities for quaternionic matrices and their Applications, Math. J. Okayama Univ., 42, 45-62 (1999). [6] Hartwig, R.E., Putcha, M.S., Semisimilarity for matrices over a division ring, Linear Algebra Appl., 39, 125- 132 (1981). [7] Tian, Y., Solving Two Pairs of Quaternionic Equations in Quaternions, Adv. Appl. Clifford Algebras, 20, 185-193 (2010). [8] Jafari, M., Yaylı, Y., Generalized quaternions and their algebraic properties, Commun. Fac. Sci. Univ. Ank. Seriers A1, 64 (1), 15-27 (2015). [10] Yildiz, O.G., Kosal, H.H., Tosun,M., On the Semisimilarity and Consemisim-ilarity of Split Quaternions, Adv. Appl. Clifford Algebras, doi: 10.1007/s00006-015-0633-y.

94 Department of Elementary Education, Faculty of Education, Artvin Çoruh University, Artvin/TURKEY E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Examples of Curves which Spherical Indicatrices are Spherical Conics

Mesut ALTINOK95, Levent KULA96

Abstract

In this study, we investigate relations of general helix (slant helix) and T-conical helix (N-conical helix). Moreover, we obtain examples for the curves. Also related examples and their illustrations are drawn with Mathematica.

Key Words. T-conical helix, N-conical helix, B-conical helix, spherical curves. AMS 2010. 53A04, 14H52.

References

[1] Altunkaya, B., Spherical Conics and Application, Doktora tezi, Ankara Üniversitesi, Fen Bilimleri Enstitüsü, 2012. [2] Altunkaya, B., Yayli, Y., Hacısalihoğlu, H. H. and Arslan, F., Equations of the spherical conics, Electronic Journal of Mathematics and Technology, 5(2011), 3, 330-341. [3] Dirnbock, H., Absolute polarity on the sphere; conics; loxodrome; tractrix, Mathematical Communication, 4(1999), 225-240. [4] Maeda, Y., Spherical conics and the fourth parameter, KMITL Sci. J., 5(2005), 1, 165-171. [5] Namikawa, Y., Spherical surfaces and hyperbolas, Sugaku, 11(1960), 22-24. [6] Kopacz, P., On geometric properties of spherical conics and generalization of Pi in navigation and mapping, Geodesy and cartography, 38(2012), 4, 141-151. [7] Sykes, G., S. and Peirce, B., Spherical Conics, Proceedings of the American Academy of Arts and Sciences, 13(1878), 375-395.

This work is supported by Ahi Evran University Scientific Research Project Coordination Unit. Project number: PYOFEN.4003.13.002 and “Tübitak 2211-A Genel Yurtiçi Doktora Burs Programı”

95 Ahi Evran University, Faculty of Art and science, Department of Mathematics, Kırsehir, E-mail: [email protected] 96 Ahi Evran University, Faculty of Art and science, Department of Mathematics, Kırsehir, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On The Special Smarandache Curves

Pelin POŞPOŞ TEKİN97, Erdal ÖZÜSAĞLAM98

Abstract

In this work, we introduce some special Smarandache curves in Euclidean space according to new version of Bishop frame. Also we give some differential geometric properties of this curves.

Key Words: Smarandache Curves, Type-2 Bishop Frame.

References

[1] A. Gray, Modern Diferential Geometry of Curves and Surfaces with Mathematica (2nd Edition), CRC Press, (1998). [4] A.T. Ali, Special Smarandache curves in the Euclidean space, International Journal of Mathematical Combinatorics, 2(2010), 30-36. [5] B. Bükçü and M.K. Karacan, On the slant Helices according to Bishop frame, International Journal of Computational and Mathematical Sciences, 3(2) (Spring) (2009), 1039-1042. [6] B. Bükçü and M.K. Karacan, Special Bishop motion and Bishop Darboux rotation axis of the space curve, Journal of Dynamical Systems and Geometric Theories, 6(2008), 27-34. [7] E. Turhan and T. Körpınar, Biharmonic slant Helices according to Bishop frame in E³, International Journal of Mathematical Combinatorics, 3(2010), 64-68. [8] L.R. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, 82(3) (1975), 246-251. [9] M. Çetin, Y. Tunçer, M.K. Karacan, Smarandache Curves According to Bishop Frame in Euclidean 3-Space, Gen. Math. Notes, Vol. 20, No.2, (Feb. 2014) 50-66. [10] M. Turgut and S. Yilmaz, Smarandache curves in Minkowski space-time, International Journal of Mathematical Combinatorics, 3(2008), 51-55. [11] S. Yilmaz and M. Turgut, On the diferential geometry of the curves in Minkowski space-time I, Int. J. Contemp. Math. Sciences, 3(27) (2008), 1343-1349.

97 Aksaray University, Faculty of Art and science, Department of Mathematics, 68100, Aksaray, E-mail: [email protected] 98 2Aksaray University, Faculty of Art and science, Department of Mathematics, 68100, Aksaray, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Generalized Beltrami Surfaces in Euclidean Spaces

Didem KOSOVA99, Kadri ARSLAN100 , Betül BULCA 101

Abstract In the present study we consider the generalized rotational surfaces in Euclidean spaces. This study consists of third parts. In the first part we give some basic concepts of surfaces in Euclidean n-space피푛. In the second part we introduce generalized tractrix curves in Euclidean (n+1)-space피푛+1 . We also give some examples. In the final section we consider generalized rotational surfaces in 피3 and 피4 respectively. We also calculate the Gauss and mean curvatures of these kind of surfaces. Finally we give some curvature properties of (generalized) Beltrami surfaces in 피3 and 피4.

Key Words: Rotational surfaces, generalized tractrix, generalized Beltrami surfaces

References [1] Bulca, B., Arslan, K., Bayram, B.K. and Öztürk, G. Spherical product surfaces in E⁴. An. St. Univ. Ovidius Constanta, 20(2012), 41-54. [2] Bulca, B., Arslan, K., Bayram, B.K., Öztürk, G. and Ugail, H. Spherical product surfaces in E3. IEEE Computer Society, Int. Conference on CYBERWORLDS, 2009. [3] Arslan, K., Bulca, B. and Milousheva, V. Meridian Surfaces in E⁴ with Pointwise 1-type Gauss map. Bull. Korean Math. Soc., 51(2014), 911-922. [4] Öztürk, G., Bayram, B.K., Bulca, B. and Arslan, K. Meridian Surfaces of Weingarten Type in Four Dimensional Euclidean Spaces E⁴, Accepted in Konuralp J. Math. [5] Chen, B.Y. Geometry of Submanifolds, Dekker, New York, 1973. [6] Chen, B.Y. Pseudo-umbilical surfaces with constant Gauss curvature, Proceedings of the Edinburgh Mathematical Society (Series 2), 18(2) (1972), 143-148. [7] Chen, B.Y. Geometry of Submanifolds and its Applications, Science University of Tokyo, 1981. [8] Ganchev, G. and Milousheva, V. On the Theory of Surfaces in the Four-dimensional Euclidean Space. Kodai Math. J. 31 (2008), 183-198. [9] Ganchev, G. and Milousheva, V. Invariants and Bonnet-type theorem for surfaces in R⁴, Cent. Eur. J. Math., 8 (2010), no. 6, 993-1008. [10] Dursun, U. and Turgay, N.C. General rotational surfaces in Euclidean space E⁴ with pointwise 1-type Gauss map. Math. Commun., 17(2012), 71-81.

99 Uludag University, Faculty of Art and Science, Department of Mathematics, Görükle Campus, 16059, Bursa, E-mail: [email protected] 100 Uludag University, Faculty of Art and Science, Department of Mathematics, Görükle Campus, 16059, Bursa, E-mail: [email protected] 101 Uludag University, Faculty of Art and Science, Department of Mathematics, Görükle Campus, 16059, Bursa, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On the second order involute curves in 푬ퟑ

Şeyda KILIÇOĞLU 102 and Süleyman ŞENYURT103

Abstract

In this study we worked on the involute of involute curve of curve 훼. We called them the second order involute of curve 훼 in E3. All Frenet apparatus of the second order involute of curve 훼 are examined in terms of

Frenet apparatus of the curve 훼. Further we show that; Frenet vector fields of the second order involute curve 훼2 can be written based on the principal normal vector field of curve 훼. Besides, we illustrate examples of our results.

Key Words: involute curve, Frenet apparataus.

References

[1] Bilici M. and Çalışkan, M., Some characterizations for the pair of involute evolute curves is Euclidian 퐸3, Bulletin of Pure and Applied Sciences, Vol. 21E(2) (2002), 289-294. [2] Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 205, 1997. [3] Hacısalihoğlu H.H., Differential Geometry (in Turkish), Academic Press Inc. Ankara, 1994. [4] Fenchel, W., On The Differential Geometry of Closed Space Curves, Bull. Amer. Math. Soc. Vol. 57 (1951), 44-54. [5] Lipschutz M.M., Differential Geometry, Schaum's Outlines, 1969.

102 Başkent University, Faculty of Education, Department of Mathematics, Ankara, Turkey E-mail: [email protected] 103Ordu University, Faculty of Art and science, Department of Mathematics, 52200, Ordu, Turkey.E- mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Rational Surfaces Generated From The Split Quaternion Product ퟒ of Two Rational Space Curves in 푬ퟐ

Veysel Kıvanç KARAKAŞ104 , Levent KULA105 , Mesut ALTINOK106

Abstract

In this work, a split quaternion rational surface is a surface generated from two rational space curves by 4 split quaternion multiplication in 퐸2 . The goal this presentation is to demonstrate how to apply syzygies to analyze split quaternion rational surfaces. We show that we can easily construct three special syzygies for a split quaternion rational surfaces from a µ-basis for one of the generating rational space curves. Also releated examples are given.

Key Words: quaternion rational surfaces, syzygy, µ-basis

References

[1] Wang, X., Goldman, R. Quaternion rational surfaces:Rational surfaces generated from the quaternion product of two rational space curves, Journal of Graphical Models, 2015, no.81, 18-32. [2] Kula, L., Bolunmuş Kuaterniyonlar ve Geometrik Uygulamaları, Doktora Tezi, Ankara Universitesi Fen Bilimleri Enstitusu, Ankara, 2003. [3] Chen, F., Wang, X., The μ-basis of a planar rational curve-properties and computation, Journal of Graphical Models, 2003, no.2, 368-381. [4] Cox, D., Sederberg, T., Chen, F., The moving line ideal basis of a planar rational curve, Journal of Computer Aided Geometric Design, 1998, no.15, 803-827.2

104 Ahi Evran University, Faculty of Art and Science, Department of Mathematics, Kırsehir, E-mail: [email protected] 105Ahi Evran University, Faculty of Art and Science, Department of Mathematics, Kırsehir, E-mail: [email protected] 106Ahi Evran University, Faculty of Art and Science, Department of Mathematics, Kırsehir, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Contact Pseudo-Slant Submanifolds of a Kenmotsu Manifold

Süleyman DİRİK107, Mehmet ATÇEKEN108, Ümit YILDIRIM109

Abstract

In this study, the geometry of the contact pseudo-slant submanifolds of a Kenmotsu manifold were studied. The necessary and sufficient conditions were given for a contact pseudo-slant submanifold to be contact pseudo-slant product.

Key Words: Kenmotsu manifold, contact pseudo-slant altmanifold, contact pseudo-slant product.

References

[1] Atçeken M. and Dirik S., On the geometry of pseudo-slant submanifolds of a Kenmotsu manifold, Gulf joural of mathematics., Vol. 2(2014), 51-66. [2] Chen B. Y., Slant immersions, Bull. Austral. Math. Soc. Vol. 41(1990), 135-147.

[3] De U. C. and Sarkar A., On pseudo-slant submanifolds of trans sasakian manifolds, Proceedings of th Estonian. A. S. 60,1-11.2011.doi:10.3176\ proc.2011.1.01.

[4] Cabrerizo J. L., Carriazo A., Fernandez, L. M. and Fernandez M. Slant submanifolds in Sasakian manifolds, Glasgow Math. journal., Vol. 42(2000), 125-138.

[5] Khan V. A. and Khan M. A., Pseudo-slant submanifolds of a Sasakian manifold, Indian J. prue appl.Mathematics., Vol. 38(2007), 31-42.

107 Amasya University, Faculty of Arts and Sciences, Department of Statistic, 05100, Amasya-Turkey E-mail: [email protected] 108 Gaziosmanpasa University, Faculty of Arts and Sciences, Department of Mathematics, 60100, Tokat- Turkey E-mail: [email protected] 109 Gaziosmanpasa University, Faculty of Arts and Sciences, Department of Mathematics, 60100, Tokat- Turkey E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

f-Biharmonicity Conditions for Curves

Fatma KARACA110, Cihan ÖZGÜR111

Abstract

In this study, we obtain necessary and sufficient conditions for curves in Sol spaces, Cartan-Vranceanu 3-dimensional spaces and homogeneous contact 3-manifolds to be f-biharmonic.

Key Words: f-biharmonic, Sol space, Cartan-Vranceanu 3-dimensional space, homogeneous contact 3- manifold.

References

[1] Ou, Y-L., On f -biharmonic maps and f -biharmonic submanifolds, Pacific J. Math. 271 (2014), 461-477.

[2] Ou, Y-L. and Wang, Z-P., Biharmonic maps into Sol and Nil spaces, arXiv preprint math/0612329 (2006)

[3] Caddeo, R., Montaldo, S., Oniciuc, C., Piu, P., The classification of biharmonic curves of

Cartan-Vranceanu 3 -dimensional spaces, Modern trends in geometry and topology, 121-131,

Cluj Univ. Press, Cluj-Napoca, (2006).

[4] Inoguchi, Jun-ichi, Biminimal submanifolds in contact 3-manifolds, Balkan J. Geom. Appl. 12(2007), no. 1, 56-67.

110 Balıkesir University, Faculty of Art and Science, Department of Mathematics, Cagıs Campus, 10145, Balıkesir, E-mail: [email protected] 111 Balıkesir University, Faculty of Art and Science, Department of Mathematics, Cagıs Campus, 10145, Balıkesir, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Rotational Surfaces in 3-Dimensional Isotropic Space

Alper Osman ÖRENMİŞ112

Abstract

In this talk, we present the rotational surfaces obtained by rotating a curve around a isotropic line in the 3-dimensional isotropic space. We derive several classification results on such surfaces satisfying some curvature conditions.

Key Words: Isotropic space, rotational surface, isotropic mean curvature, relative curvature.

References

[1] Aydin M. E., A generalization of translation surfaces with constant curvature in the isotropic space, J. Geom., (2015), DOI 10.1007/s00022-015-0292-0.

[2] Chen B.-Y., Decu S. and Verstraelen L., Notes on isotropic geometry of production models, Kragujevac J. Math. 37(2) (2013), 217-220.

[3] Sachs, H., Isotrope Geometrie des Raumes, Vieweg Verlag, Braunschweig, 1990.

[4] Sipus Z.M., Translation surfaces of constant curvatures in a simply isotropic space, Period. Math. Hung. 68 (2014), 160-175

112 Firat University, Faculty of Science, Department of Mathematics, 23119, Elazig, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On the Generalization of Geometric Design and Analysis of a MMD Machine

Engin CAN113, Hellmuth STACHEL114

Abstract

This study focuses on the geometric analysis of MMD (Multi Motion Drive) Machine, in general, of a planar parallel 3-RRR robot with three synchronously driven cranks. Graphical methods for velocity and acceleration analysis turn out that these constructions are not as straightforward as one might expect. Therefore, it can be reduced to a problem of projective geometry. There are simple geometric characterizations for both by coplanar carrier lines of the arms or additionally by particular coplanar parallels.

Key Words: Planar mechanism, completely turnable, representation of constrained motion, simulation of movement References

[1] Can, E. (2012). Analyse und Synthese eines schnelllaufenden ebenen Mechanismus mit modifizierbaren Zwangläufen. PhD Thesis, Vienna University of Technology.

[2] Can, E. & Stachel H. (2014). A planar parallel 3-RRR robot with synchronously driven cranks. Mechanism and Machine Theory, Vol. 79 (pp. 29-45).

[3] Can, E. (2015). The geometric design of currently polplan and velocity vectors of a planar parallel robot. Sakarya University Journal of Science, Vol. 19 (pp. 151-156).

[4] SAM 6.1. ARTAS Engineering Software, Holland. www.artas.nl

[5] Wunderlich, W. (1970). Ebene Kinematik. BI–Hochschultaschenbücher, Bd. 447. Bibliographisches Institut, Mannheim.

113 Sakarya University, Kaynarca School of Applied Sciences, 54650, Kaynarca/Sakarya, E-mail: [email protected] 114 Vienna University of Technology, Institute of Discrete Mathematics and Geometry, A-1040, Vienna/Austria, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

About The Generated Spacelike Bezier Spline with a Spacelike Principal Normal in Minkowski 3-Space

Hatice KUŞAK SAMANCI115, Serkan ÇELİK116

Abstract

In this paper, we study the spacelike Bezier spline with a spacelike principal normal which include Frenet frame, curvatures and derivative equations. Then we focus on the Bishop frame of the spacelike Bezier curve with a spacelike principle normal in Minkowski 3 space.

Key Words: Spacelike Bezier spline with a spacelike principal normal, Minkowski 3 space, Bishop frame.

References

[1] López, R., 2008. Differential geometry of curves and surfaces in Lorentz-Minkowski space, arXiv preprint arXiv:0810.3351.

[2] Uğurlu H.H.,Çalışkan A., Darboux Ani Dönme Vektörleri ile Spacelike Spacelike ve timelike Yüzeyler Geometrisi Kitabı, 2012.

[3] Farin G., Curves and Surfaces for Computer-Aided Geometric Design, Academic Press,1996 .

[4] Incesu, M. And Gürsoy, O., “Bezier Eğrilerinde Esas Formlar ve Eğrilikler”, XVII Ulusal Matematik Sempozyumu, Bildiriler, Abant İzzet Baysal Üniversitesi,2004:146-157.

[5] G.H. Georgiev, Spacelike Bezier curves in the three-dimensional Minkowski space, Proceedings of AIP Conference 1067 (1) (2008).

[6] P. Chalmoviansky and B. Pokorna “Quadratic spacelike Bezier Curves in the three dimensional Minkowski Space”. Proceeding of Symposium on Computer Geometry, 20:104-110, 2011.

[7] Pokorná, Barbora, and Pavel Chalmovianský. "Planar Cubic Spacelike Bezier Curves in Three Dimensional Minkowski Space.", Proceeding of Syposium on Computer Geometry, SCG 2012, Vol.21, pp.93-98.

[8] Ören I. “The Equivalence Problem for Vectors in the two dimensional Minkowski spacetime and its application to Bezier Curves”, J.Math. Comput.Sci.,6, 2016, No.1,1-21, ISSN: 1927-5307.

[9] Bishop L.R., There is more than one way to frame a curve, Amer. Math. Monthly, 82(1975), pp 246-251.

[10] Bukcu B., Karacan M., Bishop frame of the Spacelike curve with a spacelike principal normal in minkowski 3-space ,Commun.Fac. Sci. Univ. Ank. Series, A1 (2008), Vol 57, Number 1, pp 13-22.

115 Bitlis Eren University, Faculty of Art and Science, Department of Mathematics, 13000, Center/Bitlis, E-mail: [email protected] 116Bitlis Eren University, Faculty of Art and Science, Department of Mathematics, 13000, Center/Bitlis, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

3 Constant Ratio Quaternionic Curves in Euclidean 3-Space E

Günay ÖZTÜRK117, İlim KİŞİ118 , Sezgin BÜYÜKKÜTÜK119

Abstract

In this paper, we give some characterizations of spatial quaternionic curves in Euclidean 3-space E 3 . We consider a quaternionic curve in E 3 whose position vector satisfies the parametric equation

xs  m0sts m1sn1s m2sn2s for some differentiable functions mi s, 0  i  2 . We characterize such curves in terms of their curvature functions mi s and give the necessary and sufficient conditions to become constant ratio, T-constant and N- constant.

Key Words: Position vectors, Frenet equations, quaternionic curves.

References

[1] Bharathi K. and Nagaraj M., Quaternion Valued Function of a Real Variable Serret-Frenet Formulae, Indian Journal of Pure and Applied Mathematics, Vol. 18(1987), 507-511. [2] Chen B. Y., Constant Ratio Hypersurfaces, Soochow Journal Math., Vol. 28(2001), 353-362. [3] Chen B. Y., Geometry of Warped Products as Riemannian Submanifolds and Related Problems, Soochow Journal Math., Vol. 28(2002), 125-156. [4] Chen B. Y., More on convolution of Riemannian manifolds, Beitrage Algebra Geom., Vol. 44(2003), 9-24. [5] Chen B. Y., When Does the Position Vector of a Space Curve Always Lies in its Rectifying Plane?, Amer. Math. Monthly, Vol. 110(2003), 147-152. [6] Güngör M. A. and Tosun M., Some Characterizations of Quaternionic Rectifying Curves, Differential Geometry-Dynamical Systems, Vol. 13(2011), 89-100. [7] Gürpınar S., Arslan K. and Öztürk G., A Characterization of Constant-ratio Curves in Euclidean 3-space

E 3 , Acta Universitatis Apulensis, Vol. 44(2015), 39-51. [8] Şenyurt S. and Grilli L., Spherical Indicatrix Curves of Spatial Quaternionic Curves, Applied Mathematical Sciences, Vol. 9(2015), 4469-4477. [9] Ward, J. P., Quaternions and Cayley Numbers, Kluwer Academic Publishers, Boston, Lonndon 1997. [10] Yoon D. W., On the Quaternionic General Helices in Euclidean 4-Space, Honam Mathematical Journal, Vol. 34(2012), 381-390.

117 Kocaeli University, Faculty of Art and science, Department of Mathematics, Umuttepe Campus, 41380, İzmit/Kocaeli, E-mail: [email protected] 118 Kocaeli University, Faculty of Art and science, Department of Mathematics, Umuttepe Campus, 41380, İzmit/Kocaeli, E-mail: [email protected] 119 Kocaeli University, Faculty of Art and science, Department of Mathematics, Umuttepe Campus, 41380, İzmit/Kocaeli, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Tube Surfaces with Type-2 Bishop Frame

Ali ÇAKMAK120 Sezai KIZILTUĞ121

Abstract

In this paper, we study tube surfaces with type-2 Bishop frame instead of Frenet frame in Euclidean 3-space E3. Besides, we have discussed Weingarten and linear Weingarten conditions for tube surfaces with the

Gaussian curvature K, the mean curvature H and the second Gaussian curvature KII.

Key Words: Tube surfaces, Weingarten property, Type-2 Bishop frame,

Mean and Gaussian curvatures, Second Gaussian curvature.

References

[1] A. Gray, Modern Diﬀerential Geometry of Curves and Surfaces with Mathematica, USA, 1999.

[2] B. Bukcu, and M.K. Karacan, The Bishop Darboux Rotation Axis of The Spacelike Curve in Minkowski 3- Space, JFS, 30(2007), 1-5.

[3] B. O’ Neill, Semi-Riemannian Geometry with Applications to Relativity, New York, 1983.

[4] F. Dogan and Y. Yaylı, Tubes with Darboux Frame, Int. J. Contemp. Math. Sciences, 7(2012), 751-758.

[5] J.S. Ro and D.W. Yoon, Tubes of Weingarten Types in a Euclidean 3- spaces, Journal of the Chungcheong Mathematical Society, 22(2009), 359-366.

[6] S. Kiziltug and Y. Yayli, Timelike tubes with Darboux frame in Minkowski 3-space, Internatioal Journal of Physical Sciences, 8 (2013), 31-36.

120 Bitlis Eren University, Faculty of Art and science, Department of Mathematics, Bitlis, 13000 E-mail: [email protected] 121 Erzincan University, Faculty of Art and science, Department of Mathematics, Erzincan. E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

1 Some Characterizations of Curves in Pseudo-Galilean 3-Space G3

İlim KİŞİ122, , Sezgin BÜYÜKKÜTÜK123, Günay ÖZTÜRK124

Abstract

In this paper, we consider unit speed timelike curves whose position vectors can be written as linear

1 combination of theirs Serret-Frenet vectors in pseudo-Galilean 3-space G3 . We obtain some results of constant ratio curves and give an example of these type curves. Further, we show that there is no T-constant curve in and we obtain some results of N-constant curves in .

Key Words: Position vectors, Frenet equations, pseudo-Galilean 3-space.

References

[1] Akyiğit M., Azak A. Z. and Tosun M., Admissible Manheim Curves in Pseudo Galilean Space , African Diaspora Journal of Mathematics, Vol. 10(2010), 75-80. [2] Bektaş M., The Characterizations of General Helices in the 3-Dimensional Pseudo-Galilean Space, Soochow Journal of Mathematics, Vol. 31(2015), 441-447. [3] Chen, B. Y., Constant Ratio Spacelike Submanifolds in Pseudo-Euclidean Space, Houston Journal of Mathematics, Vol. 2(2003), 281-294. [4] Chen, B. Y., Geometry of Position Functions of Riemannian Submanifolds in Pseudo- Euclidean Space, J. Geom., Vol. 74(2002), 61-77. [5] Dijivak B., Curves in Pseudo Galilean Geometry, Annales Univ. Sci. Budapeşt., Vol. 41(1998), 117-128. [6] Erjavec Z., On Generalization of Helices in the Galilean and the Pseudo-Galilean Space, Journal of Mathematics Research, Vol. 6(2014), 39-50. [7] Gürpınar S., Arslan K. and Öztürk G., A Characterization of Constant-ratio Curves in Euclidean 3-space E 3 , Acta Universitatis Apulensis, Vol. 44(2015), 39-51. [7] Külahcı M., Characterizations of a Helix in the Pseudo Galilean Space , International Journal of Physical Sciences, Vol. 9(2010), 1438-1442. [8] Öztekin H. And Öğrenmiş A. O., Normal and Rectifying Curves in Pseudo-Galilean Space and Their Characterizations, Journal of Mathematical and Computational Science, Vol. 2(2012), 91-100. [9] Röschel O., "Die Geometrie Des Galileischen Raumes", Forschungszentrum Graz Research Centre, Austria, 1986. [10] Yaglom, I. M., A Simple Non-Euclidean Geometry and Its Physical Basis, Springer-Verlag Inc., New York, 1979.

122 Kocaeli University, Faculty of Art and science, Department of Mathematics, Umuttepe Campus, 41380, İzmit/Kocaeli, E-mail: [email protected] 123 Kocaeli University, Faculty of Art and science, Department of Mathematics, Umuttepe Campus, 41380, İzmit/Kocaeli, E-mail: [email protected] 124 Kocaeli University, Faculty of Art and science, Department of Mathematics, Umuttepe Campus, 41380, İzmit/Kocaeli, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Factorable Surfaces in Euclidean 4-Space E4

Sezgin BÜYÜKKÜTÜK125, Günay ÖZTÜRK126

Abstract

4 In the present study, we consider the factorable surfaces in Euclidean 4-space IE . We characterize such surfaces in terms of their Gaussian curvature, Gaussian torsion and mean curvature. Further, we give the 4 necessary and sufficient condition for a quadratic triangular Bezier surface in IE to become a factorable surface.

Key Words: Factorable surface, Euclidean 4-space, Bezier surface

References

[1] Bulca B., A characterization of surfaces in E4 , Phd Thesis, Uludağ University, 2012.

[2] Bulca B., Arslan K., Surfaces Given with the Monge Patch in E4 , Journal of Mathematical Physics, Analysis, Geometry, Vol. 9(4) (2013), 435-447. [3] Bekkar B, Senoussi B., Factorable Surfaces in the three-dimensional Euclidean and Lorentzian spaces

satisfying ri  iri , J. Geom., Vol. 103 (2012), 17-29. [4] Chen B. Y., Geometry of Submanifolds, Dekker, Newyork, 1973. [5] Chen B. Y., Pseudo-umbilical surfaces with constant Gauss curvature, Proceedings of the Edinburgh Mathematical Society (Series 2), Vol. 18(2) (1972), 143-148. [6] Gutierrez Nunez J. M., Romero Fuster M. C., Sanchez-Bringas F., Codazzi Fields on Surfaces Immersed in Euclidean 4-space, Osaka J. Math., Vol. 45 (2008), 877-894. [7] Lopez R., Moruz M., Translation and Homotethical Surfaces in Euclidean Spaces with Constant Curvature, J. Korean Math. Soc., Vol. 52(3) (2015), 523-535. [8] Meng H., Liu H., Factorable Surfaces in 3-Minkowski Space, Bull Korean Math. Soc., Vol. 5 (1985), 23-36. [9] Woestyne I. V., A new characterization of helicoids, Geometry and topology of submanifolds, World Sci. Publ., River Edge, N.J., (1993), 267-273. [10] Woestyne I. V., Minimal homothetical hypersurfaces of a semi-Euclidean space, Results Math, Vol. 27(3) (1995), 333-342.

125 Kocaeli University, Art and Science Faculty, Department of Mathematics, Kocaeli, TURKEY, E- mail: [email protected] 126 Kocaeli University, Art and Science Faculty, Department of Mathematics, Kocaeli, TURKEY, E- mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Some Notes on Tachibana and Vishnevskii Operators

Seher ASLANCI127, Haşim ÇAYIR128

Abstract

The main purpose of the present paper is to study Tachibana and Vishnevskii Operators Applied to XV and XH in almost paracontact structure on tangent bundle T(M). In addition, this results which obtained shall be studied for some special values in almost paracontact structure.

Key Words: Tachibana Operators,Vishnevskii Operators, Almost Paracontact Structure, Horizontal Lift, Vertical Lift

References

[1] Blair, D.E. Contact Manifolds in Riemannian Geometry, Lecture Notes in Math, 509, Springer Verlag, New York (1976).

[2] Das Lovejoy, S. Fiberings on almost r-contact manifolds, Publicationes Mathematicae, Debrecen, Hungary 43,(1993), 161-167.

[3] Oproiu, V. Some remarkable structures and connexions, defined on the tangent bundle, Rendiconti di Matematica (3) (1973), 6 VI.

[4] Omran,T., Sharffuddin,A., Husain,S.I. Lift of Structures on Manifolds, Publications de 1’Instıtut Mathematıqe, Nouvelle serie, 360 (50) ,(1984), 93 – 97.

[5] Salimov, A.A. Tensor Operators and Their applications, Nova Science Publ, New York, (2013).

[6] Sasaki, S. On The Differantial Geometry of Tangent Boundles of Riemannian Manifolds, Tohoku Math. J. 10, (1958), 338-358.

[7] Salimov, A.A., Çayır, H. Some Notes On Almost Paracontact Structures, Comptes Rendus de 1’Acedemie Bulgare Des Sciences, tome 66 (3), (2013), 331-338.

[8] Yano, K., Ishihara, S. Tangent and Cotangent Bundles, Marcel Dekker Inc, New York, (1973).

127 Kocaeli University, Art and Science Faculty, Department of Mathematics, Kocaeli, TURKEY, E-mail: [email protected] (Eksik) 128 Kocaeli University, Art and Science Faculty, Department of Mathematics, Kocaeli, TURKEY, E-mail: [email protected] Eksik

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Isometry Groups of CO and TO Spaces

Zeynep CAN129, Özcan GELİŞGEN130 , Rüstem KAYA 131

Abstract

The history of man’s interest in symmetry goes back many centuries. Symmetry is the primary matter of aesthetic thus it has been worked on, in various fields.

Polyhedra have attracted the attention because of their symmetries. Just as regular polygons were the most “uniform” polygons possible, man wanted to find polyhedra that are as “uniform” as possible. So Platonic, Archimedean and Catalan solids was found.

3-dimensional analytical space which is covered by a metric is called Minkowski geometry. In the Minkowski geometries the unit balls are symmetric, convex closed sets. Up to present some mathematicians have studied and improved metric geometry ([2],[3],[4],[5]). According to these studies it is shown that unit spheres of Minkowski geometries which are covered by some metric are some convex polyhedra. So metrics and convex polyhedra are releated.

One of the fundamental problem in geometry for S, which is a space with d metric, is to define the G group of isometries. In [6] truncated octahedron and cuboctahedron metrics are defined. Each one of these solids is an Archimedean solid. For Archimedean solids there are three kinds of symmetry groups; tetrahedral symmetry (Td), octahedral symmetry (Oh, O) and icosahedral symmetry (Ih, I). In this study we show that groups of isometries of the 3-dimensional space with respect to the TO and CO metrics are the semi-direct products of G(TO) and T(3), and G(CO) and T(3) respectively. Here G(TO) is the (Euclidean) symmetry group of the truncated octahedron and G(CO) is the (Euclidean) symmetry group of the cuboctahedron, and T(3) is the group of all translations of 3-dimensional analytical space.

Key Words: Archimedean solids, Isometry Group, Polyhedra, Truncated Octahedron, Cuboctahedron

References

[1] Cromwell, P., Polyhedra, Cambridge University Press, 1999 [2] Can Z., Gelişgen Ö., Kaya R., On the Metrics Induced by Icosidodecahedron and Rhombic Triacontahedron, Scientific and Professional Journal of the Croatian Society for Geometry and Graphics (KoG), Vol. 19, 17- 23,2015 [3] Ermiş T., 2014, Düzgün Çokyüzlülerin Metrik Geometriler İle İlişkileri Üzerine, ESOGÜ, PhD Thesis [4] Gelişgen, Ö., Kaya, R., Ozcan, M., Distance Formulae in The Chinese Checker Space, Int. J. PureAppl. Math., Vol. 26, no.1, 35-44, 2006. [5] Gelişgen, Ö., Kaya, R., The Taxicab Space Group, Acta Mathematica Hungarica, DOI:10.1007/s10474-008- 8006-9, Vol.122, No.1-2, 187-200, 2009. [6] Gelişgen Ö., Can Z., On The Family of Metrics For Some Platonic And Archimedean Polyhedra, ESOGU preprint, 1-9, 2016. [7] Gelişgen Ö., Kaya R., The Isometry Group of Chinese Checker Space, International Electronic Journal of Geometry, Vol.8, No.2, 82-96, 2015. [8] Thompson, A.C. Minkowski Geometry, Cambridge University Press, Cambridge, 1996.

129 Aksaray University, Faculty of Art and science, Department of Mathematics, 68100, Aksaray, E- mail: [email protected] 130 Eskişehir Osmangazi University, Faculty of Mathematics And Computer, Meşelik Campus, 26480, Eskişehir, E-mail:[email protected] 131 Eskişehir Osmangazi University, Faculty of Mathematics And Computer, Meşelik Campus, 26480, Eskişehir, E-mail:[email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Some Ruled Surfaces Related To W-Direction Curves

İlkay ARSLAN GÜVEN132, Semra KAYA NURKAN133 , Filiz ÖZSOY 134

Abstract

In this study, some special ruled surfaces are identified which are formed by using the base curve as the W-direction curves. We give the developable and minimal properties of these ruled surfaces. We investigate the relation between the main curve and the base curve of being geodesic curve, asymptotic line and principal line.

Key Words: normal surface, binormal surface, geodesic curve

References

[1] Ali AT, Aziz HS, Sorour AH. Ruled surfaces generated by some special curves in Euclidean 3-space. J of the Egyp Math Soc 2013; 21: 285-294.

[2] Choi JH, Kim YH. Associated curves of a Frenet curve and their applications. Applied Math and Comp 2012; 218: 9116-9124.

[3] Gray A. Modern differential geometry of curves and surfaces with mathematica. Second ed, Boca Raton, FL: Crc Press, 1993.

[4] Izumiya S, Takeuchi N. Special curves and ruled surfaces. Beitrage zur Alg und Geo Contributions to Alg and Geo 2003; 44(1): 203-212.

[5] Izumiya S, Takeuchi N. New special curves and developable surfaces. Turk J Math 2004; 28: 153-163.

[6] Macit N, Düldül M. Some new associated curves of a Frenet curve in E³ and E⁴. Turk J Math 2014; 38: 1023-1037.

132 Gaziantep University, Faculty of Art and Science, Department of Mathematics, Şehitkamil/Gaziantep, 27310 E-mail: [email protected] 133 Uşak University, Faculty of Art and Science, Department of Mathematics, Uşak E-mail: [email protected] 134 Gaziantep University, Faculty of Art and Science, Department of Mathematics, Şehitkamil/Gaziantep, 27310 E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Screen Semi-invariant Half-lightlike Submanifolds of a Semi-Riemannian Product Manifold

Oğuzhan Bahadır135

Abstract

In this paper, we study half-lightlike submanifolds of a semi-Riemannian product manifold. We introduce a classes half-lightlike submanifolds of called screen semi-invariant half-lightlike submanifolds. We defined some special distribution of screen semi-invariant half-lightlike submanifold. We give some equivalent conditions for integrability of distributions with respect to the Levi-Civita connection of semi-Riemannian manifolds and some results.

Key Words: Half-lightlike submanifold, Product manifolds, Screen semi-invariant.

References

[1] Duggal, Krishan L. and Bejancu, A., Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers, Dordrecht, 1996. [2] Duggal, K. L. and Bejancu, A. Lightlike submanifolds of codimension two, Math. J. Toyama Univ.,15(1992), 59-82. [3] Duggal, K.L. and Jin, D.H.: Null Curves and Hypersurfaces of Semi-Riemannian manifolds, World Scientific Publishing Co. Pte. Ltd., 2007. [4] Duggal, K. L. Riemannian geometry of half lightlike submanifolds, Math. J. Toyama Univ., 25,(2002), 169- 179. [5] Duggal, K. L. and Sahin, B. Screen conformal half-lightlike submanifolds, Int.. J. Math., Math. Sci.,68, (2004), 3737-3753. [6] Duggal, K. L. and Sahin, B. Screen Cauchy Riemann lightlike submanifolds, Acta Math. Hungar.,106(1-2) (2005), 137-165 [7] Duggal, K. L. and Sahin, B. Generalized Cauchy Riemann lightlike submanifolds, Acta Math. Hungar., 112(1-2), (2006), 113-136. [8] Duggal, K. L. and Sahin, B. Lightlike submanifolds of indefinite Sasakian manifolds, Int. J. Math. Math. Sci., 2007, Art ID 57585, 1-21.[162] [9] Duggal, K. L. and Sahin, B. Contact generalized CR-lightlike submanifolds of Sasakian submanifolds.Acta Math. Hungar., 122, No. 1-2, (2009), 45-58. [10] Atceken, M. and Kilic, E., Semi-Invariant Lightlike Submanifolds of a Semi- Riemannian Product Manifold, Kodai Math. J., Vol. 30, No. 3, (2007), pp. 361-378. [11] Duggal K. L., Sahin B., Diferential Geometry of Lightlike Submanifolds, Birkhauser Veriag AG Basel- Boston-Berlin (2010). [12] Kilic, E. and Sahin, B., Radical Anti-Invariant Lightlike Submanifolds of a Semi-Riemannian Product Manifold, Turkish J. Math., 32, (2008), 429-449. [13] Kilic, E. and Bahadir, O., Lightlike Hypersurfaces of a Semi-Riemannian Product Manifold and Quarter- Symmetric Nonmetric Connections, Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 178390, 17 pages.

135 K.S.U. , Faculty of Arts and Sciences ,Department of Mathematics, Kahramanmaras, Turkey, Avsar Campus, 46100, Onikisubat/Kahramanmaras, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Hamilton Equations of Frenet-Serret Frame on Minkowski Space

Zeki KASAP136 Emin OZYILMAZ137

Abstract

The Frenet-Serret trihedron (frame) consisting of the tangent T, normal N and binormal B collectively forms an orthonormal basis of 3-space. (T(t),N(t),B(t)) is referred to as trio Frenette trihedron. The Frenet-Serret trihedron plays a key role in the differential geometry of curves. Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems such that usually by employing differential equations. Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics. In this study, we have established Hamilton equations of Frenet-Serret frame on Minkowski space and we considered a relativistic for an electromagnetic field that it is moving under the influence of its own Frenet- Serret curvatures. Also, we will be obtain the Hamilton equations of motion for several curvatures dependent actions of interest in physics.

Key Words: Frenet-Serret Curvature, Mechanical System, Minkowski Space, Hamiltonian Equation

References

[1] Bini D., de Felice F. and Jantzen R.T., Absolute and Relative Frenet-Serret Frames and Fermi-Walker Transport, Class. Quantum Grav., 16 (1999), 2105-2124.

[2] Arreaga G., Capovilla R. and Guven J., Frenet-Serret Dynamics, Class. Quantum Grav., 18 (2001), 5065- 5083.

[3] Selig J. M., Characterisation of Frenet-Serret and Bishop Motions, Robotica, Vol.31, (2013), 981-992.

[4] Yilmaz S., Ozyilmaz E., Yayli Y. and Turgut M., Tangent and Trinormal Spherical Images of A Time-Like Curve on the Pseudohyperbolic Space H₀³, Proceedings of the Estonian Academy of Sciences, 59, 3, (2010), 216-224.

[5] Kasap Z. and Tekkoyun M., Mechanical Systems on Almost Para/Pseudo-KhlerWeyl Manifolds, IJGMMP, Vol.10, No.5, (2013); 1-8.

136 Pamukkale University, Faculty of Education, Elementary Mathematics Education Department, Denizli/Turkey, E-mail: [email protected] 137 Department of Mathematics, Faculty of Science, Ege University,Bornova Izmir/Turkey, E-mail:[email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On a Novel Formula for Reidemeister Torsion of Orientable Σg,n,b Riemann Surfaces

Esma DİRİCAN138, Yaşar SÖZEN139

Abstract

Let Σg,n,b denote compact orientable Riemann surfaces with genus g ≥ 2, bordered by n ≥ 1 curves homeomorphic to circle, and b ≥ 1 points removed. In this study, we consider the pants-decomposition of this type of surfaces, where two such pair of pants are glued along only one common boundary circle. Using the notion of symplectic chain complex, we establish a novel formula for computing the Reidemeister torsion of surfaces Σg,n,b .

Key Words: Reidemeister torsion, Symplectic chain complex, Pants-decomposition of Riemann surfaces. References

[1] Dirican E., Reidemeister Torsion and Pants Decomposition of Oriented Surfaces, Master Tezi, Hacettepe Univ., YÖK Ulusal Tez Merkezi Tez No: 415246, (2015), 15-102. [2] Dirican E., Sözen Y., Reidemeister Torsion of Some Surfaces, AIP Conf. Proc. 1676 (2015) 020006- 1020006-4, doi: 10.1063/1.4930432. [3] Porti J., Torsion de Reidemeister pour les Varieties Hyperboliques, Mem. Amer. Math. Soc. 128 (612), Amer. Math. Soc., Providence, 1997. [4] Reidemeister K., Homotopieringe und Linsenraume, Abh. Math. Sem. Hansischen Univ., 11 (1925), 102- 109. [5] Sözen Y., On Reidemeister Torsion of a Symplectic Complex, Osaka J. Math., Vol. 45 (2008), 1-39. [6] Turaev V., Introduction to Combinatorial Torsions, Lectures in Math. ETH Zurich, Birkhauser, 2001. [7] Witten E., On Quantum Gauge Theories in Two Dimensions, Comm. Math. Phys., Vol. 141 (1991), 153-209.

*Research was supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK) under the project number 114F516.

138Hacettepe University, Faculty of Science, Department of Mathematics, Beytepe Campus, 06800, Çankaya/Ankara, E-mail: [email protected] 139Hacettepe University, Faculty of Science, Department of Mathematics, Beytepe Campus, 06800, Çankaya/Ankara, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Prime Decomposition of 3-Manifolds and Reidemeister Torsion

Yaşar SÖZEN140, Esma DİRİCAN 141

Abstract We consider the building blocks of compact orientable 3-manifolds, namely Prime Decomposition Theorem for compact orientable 3-manifolds. Combining this and symplectic chain complex method, we establish a formula computing the Reidemeister torsion (R-torsion) of compact orientable 3-manifolds in terms of R-torsion of prime 3-manifolds in the decomposition of such 3-manifolds.

Key Words: Prime decomposition of compact 3-manifolds, Symplectic chain complex, Reidemeister torsion.

References [1] Dirican E., Reidemeister Torsion and Pants Decomposition of Oriented Surfaces, Master Tezi, Hacettepe Univ., YÖK Ulusal Tez Merkezi Tez No: 415246, (2015), 15-102. [2] Hempel J., 3-Manifolds, Ann. of Math. Stud. 86, Princeton Univ. Press, Ewing, NJ, 1976. [3] Porti J., Torsion de Reidemeister pour les Varieties Hyperboliques, Mem. Amer. Math. Soc. 128 (612), Amer. Math. Soc., Providence, 1997. [4] Reidemeister K., Homotopieringe und Linsenraume, Abh. Math. Sem. Hansischen Univ., 11 (1925), 102- 109. [5] Sözen Y., On Reidemeister Torsion of a Symplectic Complex, Osaka J. Math., 45 (2008), 1-39. [6] Thurston W., Three-Dimensional Geometry and Topology, Princeton University Press, 1997. [7] Turaev V., Torsions of 3-Dimensional Manifolds, Progr. Math. 208, Birkhauser, Basel, 2002. [8] Witten E., On Quantum Gauge Theories in Two Dimension, Comm. Math. Phys., 141 (1991), 153-209.

*Research was supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK) under the project number 114F516.

140 Hacettepe University, Faculty of Science, Department of Mathematics, Beytepe Campus, 06800, Çankaya/Ankara, E-mail: [email protected] 141 Hacettepe University, Faculty of Science, Department of Mathematics, Beytepe Campus, 06800, Çankaya/Ankara, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A Note On Reidemeister Torsion of G-Anosov Representations

Hatice ZEYBEK142, Yaşar SÖZEN143

Abstract

In this study, we consider the G-Anosov representations of a closed oriented Riemann surface Σ with genus at least 2, where G is the Lie group PSp(2n,IR), PSO(n,n) or PSO(n,n+1). We prove that topological invariant Reidemeister torsion (R-torsion) associated to Σ with coefficients in the adjoint representations of such representations is also well-defined. Moreover, using symplectic chain complex method, we establish a novel formula for R-torsion of such representations in terms of the Atiyah-Bott-Goldman symplectic form corresponding to Lie group G. Furthermore, we apply our results to Hitchin component, in particular, Teichmüller space, which both have geometric importance.

Key Words: Anosov representation, Reidemeister torsion, Symplectic chain complex, Hitchin component, Teichmüller space, Atiyah-Bott-Goldman symplectic form.

References [1] Labourie F., Anosov Flows, Surface Groups and Curves in Projective Space, Invent. Math., 165 (1) (2016), 51-114. [2] Hitchin N., Lie Groups and Teichmüller Spaces, Topology, 31 (3) (1992), 449-473. [3] Porti J., Torsion de Reidemeister pour les Varieties Hyperboliques, Mem. Amer. Math. Soc. 128 (612), Amer. Math. Soc., Providence, 1997. [4] Reidemeister K., Homotopieringe und Linsenraume, Abh. Math. Sem. Hansischen Univ., 11 (1925), 102- 109. [5] Sözen Y., Bonahan F., The Weil-Petersson and Thurston Symplectic Forms, Duke Math. Journal, 108 (2001), 581-597. [6] Sözen Y., On Reidemeister Torsion of a Symplectic Complex, Osaka J. Math., 45 (2008), 1-39. [7] Sözen Y., On a Volume Element of Hitchin Component, Fund. Math., 217 (2012), 249-264. [8] Turaev V., Torsions of 3-Dimensional Manifolds, Progr. Math. 208, Birkhauser, Basel, 2002. [9] Witten E., On Quantum Gauge Theories in Two Dimension, Comm. Math. Phys., 141 (1991), 153-209.

142 Hacettepe University, Faculty of Science, Department of Mathematics, Beytepe Campus, 06800, Çankaya/Ankara, E-mail: [email protected] 143 Hacettepe University, Faculty of Science, Department of Mathematics, Beytepe Campus, 06800, Çankaya/Ankara, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Some Characterizations of a Timelike Curve in R^3_1

M. Aykut AKGÜN144 and A. İhsan SVRİDAĞ145

Abstract

Investigating of special curves is one of the most attractive topic in differential geometry. Some of these special curves are spacelike curves, timelike curves and null curves. Spacelike curves and timelike curves were investigated and developed by several authors. Later, this topic drew attention of several authors and they 3 4 studied different kinds of curves in the Lorentzian manifolds 푅1 and 푅1 .

3 In this paper, we study the position vectors of a timelike curve in the Minkowski 3-space 푅1 . We give 3 some characterizations for timelike curves to lie on some subspaces of 푅1 .

Key Words: Timelike curve, Frenet frame, Minkowski space

References

[1] Coken A. C, Ciftci U., On the Cartan Curvatures of a Null Curve in Minkowski Spacetime, Geometriae Dedicate 114 (2005), 71-78.

[2] Ali A.T., Onder M., Some Characterizations of Rectifying Spacelike Curves in the Minkowski Space-Time, Global J of Sciences Frontier Research Math, Vol 12, Is 1, (2012) 2249-4626.

[3] H.H. Ugurlu, On The Geometry of Timelike Surfaces , Communi-cation, Ankara University, Faculty of Sciences, Dept.of Math., Series Al, Vol.46, (1997) pp. 211-223.

[4] Ilarslan K. and Boyacioglu O., Position vectors of a timelike and a null helix in Minkowski 3-space, Chaos, Solitions and Fractals (2008),1383-1389.

[5] Ilarslan K., Spacelike Normal Curves in Minkowski E^3_1, Turk. J. Math. 29(2005), 53-63.

[6] Akgun M. A., Sivridag A. I., Some Characterizations of a Spacelike Curve in R^4_1, Pure Mathematical Sciences, Hikari Ltd., Vol.4, No.1-4, (2015), 43-55.

144 Inonu University, Faculty of Art and science, Department of Mathematics, 44000, Malatya, E-mail: [email protected] 145 Inonu University, Faculty of Art and science, Department of Mathematics, 44000, Malatya, E-mail: [email protected]

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Semi-invariant submanifolds of almost α-cosymplectic f-manifolds

Selahattin BEYENDİ146, Nesip AKTAN147, Ali İhsan SİVRİDAĞ148

Abstract

In this paper, we have and study several properties of semi-invariant submanifolds of an almost α- cosymplectic f-manifold. We give an example and investigate the integrability conditions for the distributions involved in the definition of a semi-invariant submanifold of an almost α-cosymplectic f-manifold.

Key Words: Almost α-cosypmlectic f-manifolds, Semi-invariant submanifolds,integrability conditions.

References

[1] Öztürk H., Murathan C., Aktan N., and Vanlı A. T. 2014. Almost 훼 – cosymplektic f- manifolds. Analele ştııntıfıce ale unıversıtatıı ‘AI.I Cuza’ Dın ıaşı (S.N.) Matematica, Tomul LX, f.1.

[2] Bejan C. L., Almost α-semi-invariant submanifolds of a cosymplectic manifold, An. Ti. Univ. ‘Al. I. Cuza’ Ias Sect. I a Mat. 31, 149-156, 1985.

[3] Bejancu A., Geometry of CR-Submanifolds, D.Reidel Publ. Co., Holland, 169p. 1986.

[4] Blair D.E., Geometry of Manifolds with structural group U(n)x O(s), J.Differential Geometry, 4, 155-167, 1970.

[5] Erken K.I, Dacko P., and Murathan C. Almost α-Paracosymplectic Manifolds, arXiv: 1402.6930v1.

[6] Kobayashi M., CR-submanifolds of a Sasakian manifold, Tensor N. S., 35, 297-307, 1981.

[7] Yano K., Kon M., Structures on Manifolds, World Scientific, Singapore. 1984.

146 Inönü University, Deparment of Mathematics, 44000, Malatya, Turkey E-Posta : [email protected] 147 Konya Necmettin Erbakan University, Faculty of Scinence, Department of Mathematics and Computer Sciences, Konya, Turkey, E-Posta: [email protected] 148 Inönü University, Deparment of Mathematics, 44000, Malatya, Turkey, E-Posta : [email protected]

100

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Nearly Trans-Sasakian Manifolds With Quarter- Symmetric Non-Metric Connection

Oğuzhan BAHADIR149, Ertuğrul AKKAYA150

Abstract

In this study, firstly we define almost contact manifolds and give an example of such manifolds. Later, trans-Sasakian and nearly trans-Sasakian manifolds are studied. Finally we obtain some result for nearly trans- Sasakian manifolds with quarter- symmetric non-metric connection.

Key Words: Almost contact manifolds, Almost contact metric manifolds, Trans-Sasakian manifolds , Nearly trans-Sasakian manifolds

References [1] Ahmad M., Jun J. B. ve Siddiqi M. D., On Some Properties of Semi-İnvariant Submanifolds of a Nearly Trans-Sasakian Manifolds Admitting a Quarter-Simetric Non-Metric Connection, Journal Of The Chungcheong Math. Soc., Vol. 25, no. 1, 2012.

[2] De U. C. ve De K., On a Class of Three-Dimensional Trans-Sasakian Manifolds, Commun. Korean Math. Soc., Vol. 27, no. 4, pp. 795-808, 2012.

[3] Kim J. S., Prasad R. ve Tripathi M. M., On Generalized Ricci-Recurrent Trans-Sasakian Manifolds, J. Korean Math. Soc., Vol. 39, no. 6, pp. 953-961, 2002.

[4] Öztürk U., Sasakian Manifoldlarda Eğriler Teorisi,Yüksek Lisans Tezi, Ankara: Ankara Üniversitesi Fen Bilimleri Enstitüsü Matematik Anabilim Dalı, 2006.

[5] Yano K. ve Kon M., Structures on Manifolds, Singapore: World Scientific Publishing co pte ltd, 1984.

149 Kahramanmaras Sutcu İmam University, Faculty of Science and Letters, Department of Mathematics, Avsar Campus, 46100, Onikisubat/Kahramanmaras, E-mail: [email protected] 150 Kahramanmaras Sutcu İmam University, Faculty of Science and Letters, Department of Mathematics, Avsar Campus, 46100, Onikisubat/Kahramanmaras, E-mail: [email protected]

101

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Generalized Spherical Surfaces in Euclidean Spaces

Bengü BAYRAM151, Kadri ARSLAN152 , Betül BULCA 153

Abstract

In the present study we consider the generalized rotational surfaces in Euclidean spaces. This study consists of third parts. In the first part we give some basic concepts of surfaces in Euclidean n-space피푛. In the second part we introduce generalized spherical curves in Euclidean (n+1)-space피푛+1. In the final section we consider generalized spherical surfaces in 피3 and 피4 respectively. We have shown that generalized spherical surfaces in 피4 are considered in two kinds. The first kind generalized spherical surfaces are also known as rotational surfaces and the generalized spherical of second kind are known as meridian surfaces. We also calculate the Gauss and mean curvatures of these kind of surfaces. Finally, we give some examples.

Key Words: Rotational surfaces, meridian surface, generalized spherical surfaces

References

[1] Bulca, B., Arslan, K., Bayram, B.K. and Öztürk, G. Spherical product surfaces in E⁴. An. St. Univ. Ovidius Constanta, 20(2012), 41-54. [2] Bulca, B., Arslan, K., Bayram, B.K., Öztürk, G. and Ugail, H. Spherical product surfaces in E3. IEEE Computer Society, Int. Conference on CYBERWORLDS, 2009. [3] Arslan, K., Bulca, B. and Milousheva, V. Meridian Surfaces in E⁴ with Pointwise 1-type Gauss map. Bull. Korean Math. Soc., 51(2014), 911-922. [4] Öztürk, G., Bayram, B.K., Bulca, B. and Arslan, K. Meridian Surfaces of Weingarten Type in Four Dimensional Euclidean Spaces E⁴, Accepted in Konuralp J. Math. [5] Chen, B.Y. Geometry of Submanifolds, Dekker, New York, 1973. [6] Chen, B.Y. Pseudo-umbilical surfaces with constant Gauss curvature, Proceedings of the Edinburgh Mathematical Society (Series 2), 18(2) (1972), 143-148. [7] Chen, B.Y. Geometry of Submanifolds and its Applications, Science University of Tokyo, 1981. [8] Ganchev, G. and Milousheva, V. On the Theory of Surfaces in the Four-dimensional Euclidean Space. Kodai Math. J. 31 (2008), 183-198. [9] Ganchev, G. and Milousheva, V. Invariants and Bonnet-type theorem for surfaces in R⁴, Cent. Eur. J. Math., 8 (2010), no. 6, 993-1008. [10] Dursun, U. and Turgay, N.C. General rotational surfaces in Euclidean space E⁴ with pointwise 1-type Gauss map. Math. Commun., 17(2012), 71-81.

151 Balıkesir University, Faculty of Art and Science, Department of Mathematics, Çağış Campus, Balıkesir, E-mail: [email protected] 152 Uludag University, Faculty of Art and Science, Department of Mathematics, Görükle Campus, 16059, Bursa, E-mail: [email protected] 153 Uludag University, Faculty of Art and Science, Department of Mathematics, Görükle Campus, 16059, Bursa, E-mail: [email protected]

102

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

H-curvature Tensors of IK-Normal Complex Contact Metric Manifold

Aysel TURGUT VANLI 154, İnan ÜNAL155

Abstract

H-projective , H-conformal, H-concircular and H-conharmonic tensors of a Kähler manifold were studied by Sinha [7]. In this paper, we study on these tensors for IK-Normal complex contact metric manifolds.

Key Words: Complex contact metric manifold, H-projective, H-conformal, H-concircular and H- conharmonic References

[1] Blair,D.E and Molina,V.M. Bochner and conformal flatness on normal complex contact metric manifolds, Ann. Glob. Anal. Geom (2011)39:249—258.

[2] Blair,D.E and Mihai,A. Symmetry in complex Contact Geometry, Rocky Mountain J. Math. 42, Number 2, (2012). [3] Blair, D. E. and Turgut Vanli, A., Corrected Energy of Distributions for 3-Sasakian and Normal Complex Contact Manifolds, , Osaka J. Math 43 , 193--200 (2006). [4] Hawley , N. S. Constant holomorphic curvature, Canadian J. Math., 5:53--56, 1953. [5] Ishihara, S and Konishi , M. (1980) Complex almost contact manifolds, Kōdai Math. J. 3; 385-396. [6] Korkmaz, B. Normality of complex contact manifolds. Rocky Mountain J. Math. 30, 1343--1380 (2000). [7] Kobayashi, S. On compact Kähler manifolds with positive definite Ricci tensor. Ann. of Math. (2), 74:570- -574, 1961. [8] Kobayashi, S. Remarks on complex contact manifolds. Proc. Amer. Math. Soc. 10, 164--167 (1959). [9] Sinha, B.B. On H-curvature tensors in Kähler manifold, Kyungpook Math. J. Vol.13, No.2 , (1973) [10] Turgut Vanli, A. and Blair, D. E., The Boothby-Wang Fibration of the Iwasawa Manifold as a Critical Point of the Energy, Monatsh. Math. v.147, 75--84 (2006). [11] Turgut Vanli, A and Unal, I. Curvature properties of normal complex contact metric manifolds, preprint, (arXiv: 1510.05916 ) (2015)

154 Gazi University, Faculty of Art and science, Department of Mathematics, ANKARA 06500, TURKEY, E-mail: [email protected] 155 Tunceli University, Faculty of Engineering, Department of Computer Engineering , 62000, TUNCELİ, TURKEY , E-mail: inanunal @tunceli.edu.tr

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Quaternionic Bertrand Direction Curves

Burak ŞAHİNER 156, Mehmet ÖNDER157

Abstract

In this study, we give definitions of quaternionic integral curves of quaternionic vector fields. Then, we examine spatial quaternionic and quaternionic Bertrand-direction curves in E3 and E 4 , respectively. We obtain relationships about Frenet vectors and curvatures of quaternionic Bertrand-direction curves. By using these relationships, we give some corollaries concerning general helix, slant helix, B1 -slant and B2 -slant helix.

Key Words: Bertrand-direction curves; direction curve; helix; quaternionic curve; slant helix.

References

[1] Barros, M., General helices and a theorem of Lancret, Proc. Amer. Math. Soc., Vol. 125(5), (1997), 1503- 1509. [2] Bharathi, K., Nagaraj, M., Quaternion valued function of a real Serret-Frenet formulae, Indian J. Pure Appl. Math. Vol. 16 (1985), 741-756. [3] Burke, J.F., Bertrand curves associated with a pair of curves, Mathematics Magazine, Vol. 34(1), (1960), 60-62. [4] Choi, J.H., Kim, Y.H., Associated curves of a Frenet curve and their applications, Applied Mathematics and Computation, Vol. 218 (2012), 9116-9124.

[5] Gök, İ., Okuyucu, O.Z., Kahraman, F., Hacısalihoğlu, H.H., On the Quaternionic B2 -slant helix in the Euclidean space E 4 , Adv. Appl. Clifford Algebras, Vol. 21 (2011), 707–719. [6] Hacısalihoğlu, H.H., Hareket Geometrisi ve Kuaterniyonlar Teorisi, Gazi Üniversitesi, Fen-Edebiyat Fakültesi Yayınları, No: 2, 1983. [7] Izumiya, S., Takeuchi, N., New special curves and developable surfaces, Turk. J. Math., Vol. 28 (2004), 153-163. [8] Izumiya, S., Takeuchi, N., Generic properties of helices and Bertrand curves, Journal of Geometry, 74 (2002) 97-109. [9] Keçilioğlu, O., İlarslan, K., Quaternionic Bertrand curves in Euclidean 4-space, Bulletin of Mathematical Analysis and Applications, Vol. 5(3) (2013), 27-38. [10] Önder, M., Kazaz, M., Kocayiğit, H., Kılıç, -slant helix in Euclidean 4-space , Int. J. Cont. Math. Sci. Vol. 3(29) (2008), 1433-1440. [11] Struik, D.J., Lectures on Classical Differential Geometry, 2nd ed. Addison Wesley, Dover, 1988.

156 Celal Bayar University, Faculty of Arts and Sciences, Department of Mathematics, Muradiye Campus, 45140, Manisa, E-mails: [email protected] 157 Celal Bayar University, Faculty of Arts and Sciences, Department of Mathematics, Muradiye Campus, 45140, Manisa, E-mails: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Some Results About Harmonic Curves On Lorentzian Manifolds

Sibel SEVİNÇ158, Gülşah AYDIN ŞEKERCİ159 , A. Ceylan ÇÖKEN 160

Abstract

In this paper, we characterize the harmonic curves on Lorentzian manifolds. Particularly, we obtain the conditions for being “transversal harmonic curve”. We give some properties about such curves and research the relations between biharmonic and harmonic curves. After that we find some results for ∇-transversal harmonic curves that are given by the Laplacian and provide the condition Δ∇H = 0. Finally we explore some surfaces on Lorentzian manifolds which we can say they are ∇-transversal harmonic and give some examples for these surfaces.

Key Words: Harmonic curves, transversal harmonic curves, harmonic surfaces, Lorentzian manifold.

References

[1] Duggal K. L., Bejancu A., Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers, 346, 1996.

[2] Duggal, K. L., Jin D. H., Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific Publishing, 2007.

[3] Kılıç B., -Harmonic Curves and Surfaces in Euclidean Space, Commun. Fac. Sci. Univ. Ank. Series A1. 54(2) (2005), 13-20.

[4] Kocayiğit H., Önder M., and Arslan K., Some Characterizations of Timelike and Spacelike Curves with Harmonic 1-Type Darboux Instantaneous RotationVector in the Minkowski 3-Space E³, Commun. Fac. Sci. Univ. Ank. Series A1. 62(1) (2013), 21-32.

[5] Matea S., K-Harmonic Curves into a Riemannian Manifold with Constant Sectional Curvature, arXiv: 1005.1393v2 [math.DG]8Jun2010.

158 Cumhuriyet University, Faculty of Science, Department of Mathematics, 58000, Sivas, E-mail: [email protected] 159 Süleyman Demirel University, Faculty of Arts and Science, Department of Mathematics, 32000, Isparta, E-mail: [email protected] 160 Akdeniz University, Faculty of Science, Department of Mathematics, 07000, Antalya, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Relations Among Lines of Complex Hyperbolic Space

Ramazan ŞİMŞEK161

Abstract

2 2,1 The Complex hyperbolic space H is the projectivisation the set of negative vectors in , that is 2 H  P8V  . [1,2]. Let p1,,, p 2 p 3 p 4 be four pairwise points in the boundary of complex hyperbolic 2- 2 space H and any three points do not lie in the same C-circle. Thus, Lij denote the complex line spanned by pi and p j for ij [3]. In this study, I show that using second Hermitian form will describe the relationship between two complex lines Lij and Luk of complex hyperbolic space [3,4].

Key Words: Complex hyperbolic space, Hermitian cross-product, Complex lines.

References

[1] W.M.Goldam, Complex Hyperbolic Geometri, Oxford University Press, 1999.

[2] Parker, J.R., Notes on Complex Hyperbolic Geometry, Preliminary version, 2003.

[3] Parker, J.R. and Platis, I.D., Global, Geometric coordinates on Fabel’s Cross-Ratio variety, Canad. Math. Bull.,52 (2009), 285-294.

2,1 [4] Xiao, Y. and Jiang, Y. Complec lines in complec hyperbolic space H , Indian J. Pure Appl. Math., 42

(5): 279-289, 2011

161 Bayburt University, Bayburt Vocational College, Bayburt, Turkey, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Elastic Strips with Null Directrix

Gözde ÖZKAN TÜKEL162, Ahmet YÜCESAN163

Abstract

In this work, we firstly give a functional belongs to the variational problem which gives elastic strips with null directrix in Minkowski 3-space. Then, we characterize critical points of the functional by three Euler- Lagrange equations. We see that critical points without torsion of the functional correspond to null elastic curves in Minkowski 3-space. We secondly give conservation laws of elastic strips with null directrix by using two different variations including Lorentz translations and rotations. Finally, we define two new types of the elastic strips by means of these laws. So, we establish a connection between elastic curves on null cone and elastic strips with null directrix.

Key Words: Elastic strip, elastic curve, conservation laws, null cone.

References

[1] Chubelaschwili D., Pinkall U., Elastic Strips, Manuscripta Mathematica, Vol.133(2010), 307-326. [2] Duggal K.L., Bejancu A., Lightlike Submanifolds of Semi Riemannian Manifolds and Applications, Kluwer Academic Publishers, Netherlands, 1996. [3] Honda K., Inoguchi J., Deformation of Cartan Framed Null Curves Preserving the Torsion, Differential Geometry-Dynamical Systems, Vol.5(2003), 31-37. [4] Liu H., Curves in the Lightlike Cone, Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry, Vol.44(2004), no. 1, 291-303. [5] Lopez R., Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space, International Electronic Journal of Geometry, Vol.7(2014), no.1, 44-107. [6] O' Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1993. [7] Sager I., Abazari N., Ekmekci N., Yaylı Y., The Classical Elastic Curves in Lorentz-Minkowski Space, International Journal of Contemporary Mathematical Sciences, Vol.6 (2011), no.7, 309-320. [8] Tükel Özkan G., Elastic Strips in Minkowski 3-space, Süleyman Demirel University Graduate School of Natural and Applied Science, PhD Thesis, Isparta, 2014. [9] Tükel Özkan G., Yücesan A., Elastic Curves in a Two-dimensional Lightlike Cone, International Electronic Journal of Geometry, Vol.8(2015), no.2, 1-8.

162 E-mail: [email protected] 163 Süleyman Demirel University, Faculty of Art and Science, Department of Mathematics, 32600, Çünür/Isparta, E-mail: [email protected] * This work was supported by the Unit of Scientific Research Projects Coordination of Suleyman Demirel University under project 3356-D1-12.

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Bézier Geodesic-like Curves on 2-dimensional Pseudo-hyperbolic Space

Ayşe AKINCI164, Ahmet YÜCESAN165

Abstract

In this work, we derive the system of equations characterizing Bézier geodesic-like curves on 2- dimensional pseudo-hyperbolic space. Then we find a spacelike Bézier geodesic-like curve on 2-dimensional pseudo-hyperbolic space by means of this system of equations. Finally, we see that this curve is the geodesic of 2-dimensional pseudo-hyperbolic space.

Key Words: Bézier curve, geodesic, variational calculus, pseudo-hyperbolic space.

References [1] Chen S-G., Geodesic-like Curves on Parametric Surfaces, Computer Aided Geometric Design, Vol.27(2010), no.10, 106-117. [2] Chen S-G., Chen W-H., Computations of Bezier Geodesic-like Curves on Spheres, World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, Vol.4(2010), no.5, 544-547. [3] Farin G., Curves and Surfaces for CAGD, Academic Press, 2002. [4] Lopez R., Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space, International Electronic Journal of Geometry, Vol.7(2014), no.1, 44-107. [5] Marsden, J. E., Hoffman, M. J., Elementary Classical Analysis, W. H. Freeman, 1993. [6] O'Neill B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1993. [7] Weinstock R., Calculus of Variations with Applications to Physics and Engineering, Dover Publications, Inc., 1974. [8] Yücesan, A., Akıncı, A., Bézier Geodesic-like Curves on 2-dimensional de Sitter Space, The 4th Abu Dhabi University Annual International Conference: Mathematical Science & It's Applications, December 23-25 2015, Abu Dhabi, UAE.

164 Süleyman Demirel University, Graduate School of Art and Science, 32600, Çünür/Isparta, E-mail: [email protected] 165 Süleyman Demirel University, Faculty of Art and Science, Department of Mathematics, 32600, Çünür/Isparta, E-mail: [email protected] *This work is supported by the Unit of Scientific Research Projects Coordination of Süleyman Demirel University under project 4606-YL1-16.

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On the Kinematics of the Hyperbolic Spinors and Split Quaternions

Mustafa TARAKÇIOĞLU166, Tülay ERİŞİR167, Mehmet Ali GÜNGÖR 168, Murat TOSUN169

Abstract

In this study, the split quaternions and the hyperbolic spinors are derived from the vector formulation of

3 the Euler’s theorem on the general displacement of a rigid body with a fixed point in the Minkowski space R 1 . Then, the relationship between the hyperbolic spinors and the split quaternions is given by this vector formulation. Finally, the hyperbolic spinor formulation of rotations in the Minkowski space is obtained.

Key Words: Kinematics, Hyperbolic Spinors, Split Quaternions

References

[1] Cartan E., The Theory of Spinors, M.I.T. Press, Cambridge, MA, 1966.

[2] Vivarelli M. D. , Development of Spinors Descriptions of Rotational Mechanics from Euler’s Rigid Body Displacement Theorem, Celestial Mechanics, Vol.32(1984), 193-207.

[3] Brauer R. and Weyl H., Spinors in n -dimensions, Am. J. Math, Vol.57(1935), 425-449.

[4]Erisir T., Gungor M.A. and Tosun M., Geometry of the Hyperbolic Spinors Corresponding to Alternative Frame, Adv. in Appl. Cliff. Algebr., Vol.25(2015), no.4, 799-810.

[5] Ketenci Z., Erişir T., GÜNGÖR M. A., A Construction of Hyperbolic Spinors According to Frenet Frame in Minkowski Space, Journal of Dynamical Systems and Geometric Theories, Vol.13(2015), no.2, 179-193.

[6] Balcı Y., Erişir T., Güngör M. A., Hyperbolic Spinor Darboux Equations of Spacelike Curves in Minkowski 3-Space, Journal of the Chungcheong Mathemarical Society, Vol.28(2015), no.4, 525-535.

166 Sakarya University, Faculty of Art and Science, Department of Mathematics, Esentepe Campus, 54187, Serdivan/Sakarya, E-mail: [email protected] 167 Sakarya University, Department of Mathematics, 54187, Serdivan/Sakarya, E-mail: [email protected] 168 Sakarya University, Department of Mathematics, 54187, Serdivan/Sakarya, E-mail: [email protected] 169 Sakarya University, Department of Mathematics, 54187, Serdivan/Sakarya, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A Survey on Rectifying Curves in Lorentz n-Space

Tunahan TURHAN170, Vildan ÖZDEMİR171 , Nihat AYYILDIZ 172

Abstract

In this work, we study null and spacelike rectifying curves in Lorentz n-space. Considering the structure of a rectifying curve, we give some generalizations of such curves in Lorentz n -space and we characterize some properties of these curves in terms of their curvature functions.

Key Words: Curvature, Lorentz n -space, null rectifying curve, spacelike rectifying curve.

References

[1] Ali A.T., Önder M., Some characterizations of space-like rectifying curves in the Minkowski space-time, Glob. J. Sci. Front Res. Math. Decision Sci., Vol. 12(2012), 57-64. [2] Cambie S., Goemans W., Van Den Bussche I., Rectifying curves in n -dimensional Euclidean space, Turk. J. Math., Vol. 40(2016), 210-223. [3] Chen B.Y., When does the position vector of a space curve always lie in its rectifying plane?, Am. Math. Mon., 110(2003), 147-152. [4] Chen B.Y., Dillen F., Rectifying curves as centrodes and extremal curves, Bull. Inst. Math. Acad. Sinica, 33(2005), 77-90. [5] İlarslan K., Nesovic E., Some characterizations of null, pseudo null and partially null rectifying curves in Minkowski space-time, Taiwanese J. Math., 12(2008), 1035-1044.

[6] İlarslan K., Nesovic E., Some characterizations of rectifying curves in the Euclidean space E 4 , Turk. J. Math., 32(2008), 21-30. [7] İlarslan K., Nesovic E., On rectifying curves as centodes and extremal curves in the Minkowski 3-space, Novi Sad. J. Math., 37(2007), 53-64. [8] İlarslan K., Nesovic E., Petrovic-Torgasev M., Some characterizations of rectifying curves in the Minkowski 3-space, Novi Sad. J. Math., 33(2003), 23-32. [9] İlarslan K., Some special curves on non-Euclidean manifolds. PhD, Ankara University, Ankara, Turkey, 2002. [10] O'neill B., Semi-Riemann Geometry with application to relativity. New York: Academic Press, 1983.

170 Necmettin Erbakan University, Seydişehir Vocational School, 42370, Konya/Turkey, E-mail: [email protected] 171 Selçuk University, Science Faculty, Department of Mathematics, 42250, Konya/Turkey, E-mail: [email protected] 172 Süleyman Demirel University, Art and Science Faculty, Department of Mathematics, 32260, Isparta/Turkey, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Applications of Complex Form of Instantaneous Invariants to Planar Path-Curvature Theory

Kemal EREN173, Soley ERSOY 174

Abstract

The objective of this study is to take advantage of exploiting the complex numbers for instantaneous geometric properties of planar motion of rigid bodies. It is a conventional method to reinvestigate a convenient formulation for problems in planar kinematics by taking planar position vectors as complex numbers. From this point view, we give Bottema's instantaneous invariants in complex forms and make use of this formulation to study the kinematic geometry of infinitesimally separated positions of a moving complex plane. By using the complex forms of the instantaneous invariants we provide a straightforward way to obtain order properties of motions in the complex plane and obtain the complex forms of the inflection circle and cubic stationary curvature. Moreover, we give the existence conditions of Ball and Ball-Burmester points.

Key Words: Instantaneous invariants, Ball and Ball-Burmester points

References

[1] Bottema, O. and Roth, B., Theoretical Kinematics, New York, Dover Publications, 1990. [2] Freudenstein, F. Higher path-curvature analysis in plane kinematics, J. Manuf. Sci. Eng., 1965; 87: 184- 190. [3] Freudenstein, F., and Sandor, G.N., On the Burmester points of a plane, J Appl Mech 1961; 28: 41-49. [4] Veldkamp, GR. Some remarks on higher curvature theory, J. Eng. Ind., 1967, 89(1), 84-86. [5] Kamphuis, HJ. Application of spherical instantaneous kinematics to the spherical slider-crank mechanism, J. Mech., 1969; (4) 43-56. [6] Roth, B. and Yang, A.T., Application of Instantaneous Invariants to the Analysis and Synthesis of Mechanisms, ASME Journal of Engineering for Industry, 1997, 99(1), 97-103. [7] Veldkamp, G.R., Curvature Theory in Plane Kinematics, Doctoral dissertation, T.H. Delft, Groningen, 1963.

173 Sakarya University, Faculty of Art and science, Department of Mathematics, Sakarya, E-mail: [email protected] 174 Fatsa Science High School, Fatsa, Ordu, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Some Solutions on the Flux Surfaces

Zehra ÖZDEMİR175, İsmail GÖK176 , F. Nejat EKMEKCİ 177, Yusuf YAYLI 178

Abstract

In this study, we give some geometric approach to Killing magnetic flux surfaces in Euclidean 3-space and some solutions for differential equations which expressed the mentioned surfaces. Furthermore we give some examples and draw their pictures by using the programme Mathematica.

Key Words: Special curves, Killing vector field, magnetic flows, Euclidean space, differential equations

References

[1] Bird RB, Stewart WE, Lightfoot EN (1960) Transport Phenomena. John Wiley&Sons. ISBN 0-471-07392- X. [2] Hazeltine RD, Meiss J D (2003) Plasma Confinement. Dover publications, inc. Mineola, New York. [3] Boozer AH (2004) Physics of magnetically confined plasmas. Rev Mod Phys 76:1071-1141. [4] Barros M, Romeo A (2007) Magnetic vortices. EPL 77: 1-5. [5] Barros M, Cabrerizo JL, Fernández M, Romero A (2007) Magnetic vortex filament flows, J Math Phys, 48: 082904. [6] Hasimoto HA (1972) Soliton on a vortex filament, J Fluid Mech 51: 477-485.

3 [7] Drut-Romanius SL, Munteanu MI (2011) Magnetic curves corresponding to Killing magnetic fields in E , J Math Phys 52: 113506.

Department of Mathematics, Faculty of Science, Ankara University, 06100, Ankara, Turkey 175 E-mail: [email protected] 176 E-mail: [email protected] 177 E-mail: [email protected] 178 E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A Note on Warped Product Manifolds With Certain Curvature Conditions

Sinem GÜLER179, Sezgin ALTAY DEMİRBAĞ180

Abstract

In Riemannian geometry, warped product manifolds have been used to construct new examples with interesting curvature properties, [1]. Also, in Lorentzian geometry, some well-known solutions of Einstein’s field equations can be expressed as Lorentzian warped products [2]. Because of these significiant applications, the study of warped product manifolds plays an important role in differential geometry as well as in general relativity. For this reason, in this talk we aim to give some classifications of warped product manifolds satisfying certain curvature conditions.

Key Words: warped product manifold, warping function, generalized quasi Einstein manifold, N(k)- quasi Einstein manifold.

References

[1] Dobarro F., Ünal B., Curvature of Multiply Warped Products, J. Geom. Phys., vol.55(2005), no.1, 75-106 . [2] Beem J. K., Ehrlich P., Global Lorentzian Geometry, Markel-Deccer, New York, 1981.

[3] Bishop R. L., O'Neill B., Manifolds of negative curvature, Transactions of the American Mathematical Society, vol. 145 (1969), pp. 1–49. [4] O’Neill B., Semi-Riemannian Geometry with Applications to Relativity New York, Academic Press, 1983. [5] Arslan K., Deszcz R., Ezentaş R., Hotlos R., Murathan C., “On Generalized Robertson-Walker Spacetimes Satisfying Some Curvature Condition”, Turk. J. Math. 38(2014), 353-373. [6] Chojnacka-Dulas J., Deszcz R., Glogowska M., Prvanovic M., “On warped product manifolds satisfying some curvature conditions” J. Geo. and Phys. vol.74(2013), 328-341 [7] Dobarro F., Ünal F., Curvature in special base conformal warped products, Acta Appl. Math., vol.104(2008), 1-46.

179 Istanbul Technical University, Faculty of Science and Letter, Department of Mathematics,34469, İstanbul, Turkey, E-mail: [email protected] 180 Istanbul Technical University, Faculty of Science and Letter, Department of Mathematics,34469, İstanbul, Turkey, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Submanifolds with Finite Type Spherical Gauss Map in Sphere

Burcu BEKTAŞ181, Uğur DURSUN182

Abstract

In this talk, we present some results about spherical submanifolds with finite type spherical Gauss map. First, we prove that a submanifold of a sphere has mass symmetric 1-type spherical Gauss map if and only if it is an open part of a small n-sphere of a totally geodesic (n+1)-sphere. Then, we study a non totally umbilical spherical hypersurface with constant mean curvature in a sphere which has mass symmetric 2-type spherical Gauss map. In particular, we give characterization theorem for surfaces with mass symmetric 2-type spherical Gauss map.

Key Words: Finite type map, Spherical Gauss map and Mean curvature

References

[1] Bektaş B. and Dursun U., On Spherical Submanifolds with Finite Type Spherical Gauss Map, Advance in Geometry, 2016, DOI:10.1515/advgeom-2016-0005.

181 Istanbul Technical University, Faculty of Science and Letters, Department of Mathematics, Maslak Campus, 34469, Maslak/Istanbul, E-mail: [email protected] 182 Işık University, Faculty of Art and Sciences, Department of Mathematics, Şile Campus, 34980, Şile/Istanbul, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Rectifying Salkowski Curves with Serial Approach in Minkowski 3-Space

Beyhan YILMAZ183, İsmail GÖK184 , Yusuf YAYLI 185

Abstract

The aim of the paper is to find rectifying Salkowski curves of polynomial parametric equations with serial approach in Minkowski 3-space. We characterize these curves in which the curvature function κ is a constant and the harmonic curvature function H=(τ/κ) is a linear function. Finally, we obtained the equation of the rectifying Salkowski curve via serial solutions of third-order polynomial coefficients differential equations.

Key Words: Rectifying curve, Salkowski curve, Harmonic Curvature

References

[1] Chen BY. When does the position vector of a space curve always lie in its rectifying plane ?. Amer. Math. Monthly 2003; 110: 147-152.

[2] İlarslan K, Nesovic E,Petrovic-Torgasev M. Some characterizations of rectifying curves in the Minkowski 3- space. Novi. Sad. J. Math 2003; 33: 2, 23-32.

[3] Monterde J. Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion. Computer Aided Geometric Design 2009; 26: 271-278.

[4] Özdamar E, Hacisalihoğlu H.H. A characterization of inclined curves in Euclidean n-space. Communication de la facult´e des sciences de L'Universit´e d'Ankara 1975; 24: 15-22.

[5] Salkowski E. Zur transformation von raumkurven. Mathematische Annalen 1909; 66(4): 517-557.

[6] Yun MO, Ye LS. A Curve Satisfying τ/κ=s with constant κ>0. American Journal of Undergraduate Research 2015; 2-12.

183 Ankara University, Faculty of Science, Department of Mathematics, Tandoğan/ANKARA, E-mail: [email protected] 184 Ankara University, Faculty of Science, Department of Mathematics, Tandoğan/ANKARA, E-mail: [email protected] 185 Ankara University, Faculty of Science, Department of Mathematics, Tandoğan/ANKARA, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Normal Section Curves on Semi-Riemannian Manifolds

Feyza Esra ERDOĞAN186, Selcen YÜKSEL PERKTAŞ187

Abstract

In this study, we investigate curvatures of normal section curves on semi-Riemannian manifolds. We find some necessary and sufficient conditions for a curve in terms of curvatures which is assumed to be a normal section curve and classify such curves. Moreover, we give some characterizations for null curves of 3 4 4 R1 , R1 as well as R2 to be normal section curves.

Key Words: Semi Riemann Manifold, Null Curve, Normal Section Curve, Curvature, Planar Normal Section.

References

[1] Blomstrom C., Planar geodesic immersions in pseudo-Euclidean Space, Math. Ann. 274(1986),585- 589.

[2] Chen B.Y., Geometry of Submanifolds. Pure and Apllied Mathematics, No.22, Marcell Dekker.,Inc., New York, (1973).

[3] Chen B.Y., Submanifolds with planar normal sections, Soochow J. Math. 7(1981),19-24.

[4] Chen B.Y., Differential geometry of submanifolds with planar normal sections, Ann. Mat. Pura Appl.130 (1982), 59-66.

[5] Chen B.Y., S. J. Li, Classification of surfaces with pointwise planar normal sections and its application to Fomenko's conjecture, J.Geom. 26 (1986), 21-34.

[6] Chen B.Y., Classification of surfaces with planar normal sections, J. of Geometry 20 (1983), 122-127.

[7] Chen B.Y., P. Verheyen. Submanifolds with geodesic normal sections, Math.Ann.269 (1984) 417-429.

[8] Hong Y., On submanifolds With planar normal Sections, Mich. Math. J. 32 (1985), 203-210.

[9] Kim Y.H., Surfaces in a pseudo-Euclidean space with planar normal sections, J. Geom. 35(1989).

186 Adıyaman University, Faculty of Education, Department of Elementary Education, Adıyaman, E-mail: [email protected] 187 Adıyaman University, Faculty of Arts and Sciences, Department of Mathematics, Adıyaman, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Generalized D-Conformal Deformations of Some Classes of Almost Contact Metric Manifolds

Nülifer ÖZDEMİR188

Abstract

In this work, we study the generalized D-conformal deformations of the almost contact metric manifolds, in particular we investigate deformations of 훽-Kenmotsu manifolds. The new Levi-Civita covariant derivative of the new metric corresponding to deformed 훽-Kenmotsu manifold is written in terms of old one. Under some restrictions, deformed 훽̃-Kenmotsu manifolds are obtained. By the same method, deformations of some other classes of almost contact metric manifolds are analyzed.

Key Words: Almost contact metric manifold, generalized D-conformal deformation.

References

[1] Alegre, P. and Carriazo, A., Generalized Sasakian Space Forms and Conformal Changes of the Metric, Results. Math., Vol. 59(2011), 485-493.

[2] Alexiev, V. A. and Ganchev, G. T., On the Classification of the Almost Contact Metric Manifolds, Math. and Educ. in Math., Proc. of the XV Spring Conf. of UBM, Sunny Beach, 155(1986).

[3] Blair, D. E., The theory of quasi-Sasakian structures, J. Differential Geometry, Vol. 1 (1967), 331-345.

[4] Blair, D.E., Riemannian Geometry of Contact and Symplectic Manifolds, Birkhauser, Switzerland, 2002.

[5] Blair, D.E. and Oubina, J. A., Conformal and related changes of metric on the product of two almost contact metric manifolds, Publ. Mat., Vol. 34(1), (1990) 199-207.

[6] Boyer, C. P and Galicki, K., Sasakian Geometry, Oxford Mathematical Monogrphs, Oxford University Press, 2008.

[7] Chinea, D. and Gonzales, C., A Classification of Almost Contact Metric Manifolds, Ann. Mat. Pura Appl., Vol. (4) 156 (1990), 15-36.

188 Anadolu University, Faculty of Sscience, Department of Mathematics, Yunus Emre Campus, 26470, Eskişehir, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Semi-invariant   -Riemannian submersions from almost contact manifolds

Mehmet Akif AKYOL189, Ramazan SARI190, Elif AKSOY191,

Abstract

As a generalization of anti-invariant -Riemannian submersions, we introduce the notion of semi- invariant -Riemannian submersions from almost contact manifolds onto Riemannian manifolds. We give an example and investigate the geometry of foliations that arise from the definition of a Riemannian submersion and find necessary and sufficient condition for total manifold to be a locally product manifold. Moreover, we investigate necessary and sufficient condition for a semi-invariant -Riemannian submersion to be totally geodesic and harmonic.

Key Words: Sasakian manifold, Riemannian submersion, anti-invariant -Riemannian submersion, semi-invariant -Riemannian submersion.

References

[1] Blair, D. E., Contact manifold in Riemannian geometry, Lecture Notes in Math., 509, Springer-Verlag, Berlin-New York, 1976. [2] Baird, P., Wood, J. C., Harmonic Morphisms Between Riemannian Manifolds, London Mathematical Society Monographs, 29, Oxford University Press, The Clarendon Press, Oxford, 2003. [3] Erken, İ. K., Murathan, C., Anti-invariant Riemannian submersions from Sasakian manifolds, arxiv:1302.4906v1. [4] Falcitelli, M., Ianus, S. and Pastore, A. M., Riemannian Submersions and Related Topics, World Scientific, 2004. [5] Gündüzalp, Y., Anti-invariant semi-Riemannian submersions from almost para-Hermitian manifolds, Journal of Function Spaces and Applications, Vol. 2013, (2013), 720623, 7 pages. [6] Lee, J. W., Anti-invariant -Riemannian submersions from almost contact manifolds, Hacettepe Journal of Mathematics and Statistic, 42(3), (2013), 231-241. [7] Murathan, C., Erken, İ. K., Anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds, Filomat, 29(7), (2015), 1429-1444. [8] O’Neill, B., The fundamental equations of a submersion, Mich. Math. J.,13, 458–469, 1966. [9] Şahin, B., Anti-invariant Riemannian submersions from almost hermitian manifolds, Cent. Eur. J. Math., 8(3), 437–447, 2010. [10] Şahin, B., Semi-invariant Riemannian submersions from almost Hermitian manifolds, Canad. Math. Bull. 56, 173-183, 2013. [11] Şahin, B., Riemannian submersions from almost Hermitian manifolds, Taiwanese J.Math. 17(2), 629-659, 2013. [12] Taştan, H. M., Anti-holomorphic semi-invariant submersions from Kählerian manifolds. arxiv: 1404.2385. [13] Watson, B., Almost Hermitian submersions, J. Differential Geometry, 11(1), 147–165, 1976.

189 Bingöl University, Faculty of Art and science, Department of Mathematics, 12000, Bingöl/TURKEY E-mail: [email protected] 190 Amasya University, Merzifon Vocational Schools, 05300, Amasya/TURKEY E-mail: [email protected] 191 Amasya University, Merzifon Vocational Schools, 05300, Amasya/TURKEY E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Lucas Collocation Method to Determination Spherical Curves in Euclidean 3-Space

Muhammed ÇETİN192, Hüseyin KOCAYİĞİT193, Mehmet SEZER194

Abstract

In this study, we give a necassary and sufficient condition for an arbitrary-speed regular space curve to lie on a sphere centered at origin. Also, we obtain the position vector of any regular arbitrary-speed space curve lying on a sphere centered at origin satisfies a third-order linear differential equation whose coefficients is related to speed function, curvature and torsion. Then, a collocation method based on Lucas polynomials is developed for the approximate solutions of this differential equation. Morover, by means of the Lucas collacation method, we approximately obtain the parametric equation of the spherical curve by using this differential equation. Furthermore, an example is given to demonstrate the efficiency of the method and the results are compared with figures and tables.

Key Words: Spherical curves, Frenet frame, Lucas polynomial and series, collocation points, differential equation

References

[1] Wong, Y.C., A Global Formulation of the Condition for a Curve to Lie in a Sphere, Monatsh Math, 1963, 67, 363-365. [2] Breuer, S., Gottlieb, D., Explicit Characterization of Spherical Curves, Proceedings of the American Mathematical Society, 1971, 27(1), 126-127. [3] Wong, Y.C., On an Explicit Characterizations of Spherical Curves, Proceedings of the American Mathematical Society, 1972, 34(1), 239-242. [4] Mehlum, E., Wimp, J., Spherical Curves and Quadratic Relationships for Special Functions, J. Austral. Math. Soc. Ser. B, 1985, 27, 111-124. [5] Koshy, T., Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication, John Wiley &Sons, Inc., 2001.

192 E-mail: [email protected] 193 Celal Bayar University, Faculty of Art and Science, Department of Mathematics, Muradiye Campus, Muradiye, Manisa, Turkey, E-mail: [email protected] 194 Celal Bayar University, Faculty of Art and Science, Department of Mathematics, Muradiye Campus, Muradiye, Manisa, Turkey, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Dual Euler-Rodrigues Formula

Derya KAHVECİ195, Yusuf YAYLI196 , İsmail GÖK 197

Abstract

Dual Euler-Rodrigues formula is known as matrix representation of dual rotation. The aim of the paper is to give the geometrical interpretations of dual Euler-Rodrigues formula. We show that rotations in dual plane corresponds to screw motion in R^3.

Key Words: dual Euler-Rodrigues formula, dual rotation, screw motion

References

[1] J. M. McCarthy, An Introduction to Theoretical Kinematics, MIT Press, Cambridge, 1990.

[2] J.S. Dai, Euler-Rodrigues formula variations, quaternion conjugation and intrinsic connections, Mech. Mach. Theory, 92(2015), 144-152.

195 Ankara University, Faculty of Science, Department of Mathematics, Tandoğan Campus, 06100, Çankaya/Ankara, E-mail: [email protected] 196 Ankara University, Faculty of Science, Department of Mathematics, Tandoğan Campus, 06100, Çankaya/Ankara, E-mail: [email protected] 197 Ankara University, Faculty of Science, Department of Mathematics, Tandoğan Campus, 06100, Çankaya/Ankara, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On The Isometry Group of Deltoidal Hexacontahedron Space

Zeynep ÇOLAK198, Özcan GELİŞGEN199

Abstract

Polyhedra have interesting symmetries. Therefore they have attracted the attention of scientists and artists from past to present. Thus polyhedra are discussed in a lot of scientific and artistic works. Minkowski geometry is a non-Euclidean geometry in a finite number of dimensions. Here the linear structure is the same as the Euclidean one but distance is not uniform in all directions. According to studies on polyhedra, there are some Minkowski geometries in which unit spheres of these spaces furnished by some metrics are associated with convex solids. For example, unit spheres of maximum space and taxicab space are cubes and octahedrons, respectively, which are Platonic Solids. And unit sphere of CC-space is a deltoidal icositetrahedron which is a Catalan solid. Also in recent studies we show that there are some Minkowski geometries which their unit sphere some Archemedian and Catalan solids([1],[2],[3],[5]). Three essential methods geometric investigations; synthetic, metric and group approach. The group approach involves isometry groups of a geometry and convex sets plays an substantial role in indication of the group of isometries of geometries. Those properties are invariant under the group of motions and geometry studies those properties. In the recent years, there are a lot of studies about isometry group of various spaces([4],[6],[7],[8],[9]). In this work, we give the isometry group of the 3-dimensional analytical space furnished by Deltoidal Hexacontahedron metric.

Key Words: Catalan Solids, Deltoidal hexacontahedron, Isometry Group, Metric,

References [1] Can, Z., Gelişgen, Ö., Kaya, R., On the Metrics Induced by Icosidodecahedron and Rhombic Triacontahedron, Scientific and Professional Journal of the Croatian Society for Geometry and Graphics (KoG), Vol.19, 17-23, 2015. [2] Can, Z., Çolak Z., Gelişgen, Ö., A Note On The Metrics Induced By Triakis Icosahedron And Disdyakis Triacontahedron, Eurasian Academy of Sciences Eurasian Life Sciences Journal / Avrasya Fen Bilimleri Dergisi Vol.1, 1 - 11, 2015. [3] Çolak Z., Gelişgen, Ö., New Metrics for Deltoidal Hexacontahedron and Pentakis Dodecahedron, SAU Fen Bilimleri Enstitüsü Dergisi, Vol.19, No.3, 353-360, 2015 [4] Gelişgen, Ö., Kaya, R., The Isometry Group of Chinese Checker Space, International Electronic Journal Geometry, Vol.8, No:2, 82-96, 2015. [5] Gelişgen, Ö., Çolak Z., A Family of Metrics for Some Polyhedra, Automation Computers Applied Mathematics Scientific Journal, Vol.24, No.1, 3-15, 2015. [6] Gelişgen, Ö., Kaya, R., The Taxicab Space Group, Acta Mathematica Hungarica, DOI:10.1007/s10474- 008-8006-9, Vol.122, No.1-2, 187-200, 2009. [7] Kaya, R., Gelişgen, Ö., Ekmekçi, S. ve Bayar, A., 2006, Group of Isometries of CC-Plane, Missouri J. of Math. Sci., 18, 3, 221-233. [8] Kaya, R., Gelişgen, Ö., Ekmekçi, S. ve Bayar, A., On The Group of Isometries of The Plane with Generalized Absolute Value Metric, Rocky Mountain Journal of Mathematics, Vol. 39, No.2, 591-603, 2009. [9] Schattschneider, D. J. , The Taxicab Group, Amer. Math. Monthly, 91, 423-428, 1984.

198 Çanakkale Onsekiz Mart University. Faculty of Economics and Admin Sciences, Department of Management and Information Systems, E-mail: [email protected] 199 Eskişehir Osmangazi University, Faculty of Arts and Sciences, Department of Mathematics- Computer , E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Seiberg-Witten Equations on 6-Dimensional manifold Without Duality

Serhan EKER200, Nedim Değirmenci201

Abstract

Although Seiberg-Witten equations that consist of Dirac and Curvature equation are meaningfull in 4- dimension with duality concept, the generelaziation of these equations in high dimensions have been studied similar to the 4-dimension, depending on self-duality concept. In this work on 6-dimensional manifold without needing self-duality concept Seiberg- Witten equations were obtained. Then non-trivial solution of these equations were given.

Keywords. Seiberg-Witten equations, Dirac operator, Without-duality

AMS 2010. 53A40, 20M15.

References

[1] Bilge, A.H., Dereli, T., Koçak, S., Monopole equations on 8-manifolds with Spin(7) holonomy, Commun Math Phys 1999; 203: 21–30.

[2] Salamon, D., Spin Geometry and Seiberg-Witten Invariants,1996 (preprint). [3] Değirmenci N., N. Özdemir, "Seiberg-Witten Like Equations on 7-Manifolds with G2-Structure",Journal of Nonlinear Mathematical Physics. 12,457-461 (2005).

[4] Değirmenci, N., Özdemir N., Seiberg-Witten Like Equations On 8-Manifold With Structure Group Spin(7), Journal of Dynamical Systems and Geometric Theories, Vol.7(2009) - No.1 – May.

[5] Friedrich, T., Dirac Operators in Riemannian Geometry, Providence, RI, USA: AMS, 2000.

[6]Karapazar, Ş., Seiberg-Witten equations on 8-dimensional SU(4)-structure, International Journal of Geometric Methods in Modern Physics, Vol. 10, No. 3 (2013).

[7] Tanaka, Y., Monopole type equations on compact symplectic 6-manifolds, arXiv:1407.1934.

[8] Witten, E., Monopoles and four manifolds, Math Res Lett 1994; 1: 769–796.

200 Anadolu University, Faculty of Science, Department of Mathematics, Yunus Emre Campus, 26470, Eskişehir, E-mail: [email protected] 201 Anadolu University, Faculty of Science, Department of Mathematics, Yunus Emre Campus, 26470, Eskişehir, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On the Concircular Curvature Tensor of a Normal Paracontact Metric Manifold

Ümit YILDIRIM202, Mehmet ATÇEKEN203 , Süleyman DİRİK 204

Abstract

We classify normal paracontact metric manifolds which satisfy the curvature conditions

ZXR( , ) 0, ZXS( , ) 0, ZXP( , ) 0, ZXZ( , ) 0 and ZXC( , ) 0, where Z is concircular curvature tensor, P is projective curvature tensor, S is Ricci tensor, R is Riemannian curvature tensor and C is quasi-conformal curvature tensor.

Key Words: Normal paracontact metric manifold, concircular curvature tensor, projective curvature tensor, quasi-conformal curvature tensor.

References

[1] Atçeken M. and Yıldırım Ü., On almost C()  manifold satisfying certain conditions on concircular curvature tensor, Pure and Applied Mathematics Journal, 9(2015), 4(1-2), 31-34. [2] Kaneyuki S. and Williams F. L., Almost paracontact and parahodge structures on manifolds, Nagoya Math. J. Vol. 99(1985), 173-187. [3] Olszak Z., Normal almost contact metric manifolds of dimension three, Ann. Polon. Math. XLVII(1986), 41–50. [4] Wełyczko J., On basic curvature identities for almost (para)contact metric manifolds, Available in Arxiv: 1209.4731v1 [math.DG]. [5] Zamkovoy S., Canonical connections on paracontact manifolds, Ann Glob Anal Geom., 36(2009), 37-60.

202 Gaziosmanpasa University, Faculty of Arts and Sciences, Department of Mathematics, 60100, Tokat-Turkey, E-mail: [email protected] 203 Gaziosmanpasa University, Faculty of Arts and Sciences, Department of Mathematics, 60100, Tokat-Turkey E-mail: [email protected] 204 Amasya University, Faculty of Arts and Sciences, Department of Statistic, 05100, Amasya-Turkey, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Sierpinski-type Fractals in Galilean Plane

Elif Aybike BÜYÜKYILMAZ205, Yusuf YAYLI206 İsmail GÖK207

Abstract

In this work, we study Sierpinski triangle in Galilean-2 plane with using deterministic algorithm 10 iteration method. We investigate the effects of rotation matrix  ,called shear transformation, to  1 cos sin Sierpinski triangle in Galilean plane and compare with  in Euclidean plane. Then we give sin cos the definitions of Galilean self-similarity and Galilean box-counting dimension which are essential properties of a fractal object.

Keywords: Galilean transformation, Fractal, Dimension, Iteration

References

[1] Akar, M., Yüce, S. and Kuruoglu, N., One-Parameter-Planar Motion in the Galilean Plane, International Electronic Journal of Geometry, Volume 6, no.1, pp. 79-88, 2003.

[2] Barnsley, M.F., Fractal Everywhere, 2nd ed., Academic Press, San Diego, 1993.

[3] Bedford T., The box dimension of self-affine graphs and repellers, Nonlinearity 1, 53-71, 1989.

[3] Edgar G.A., Measure, Topology, and Fractal Geometry, Undergraduate Texts in Mathematics, Springer- Verlag, 1990.

[4] Falconer K. J., Fractal Geometry-Mathematical Foundations and Applications, John Wiley, 2nd ed. 2003.

[5] Hacısalihoğlu H. H., Fraktal Geometri I, 2006.

[6] Mandelbrot, B., The Fractal Geometry of Nature, 1982.

[7] McMullen C., The Hausdorff dimension of general Sierpinski carpets, Nagoya Math. J. 96,(1984), pp. 1–9.

[8] Yaglom I.M., A simple non-Eucledian geometry and its physical basis: an elementary account of Galilean geometry and the Galilean principle of relativity, New-York: Springer-Verlag, 1979.

205 Ankara University, Faculty of Science, Department of Mathematics, Tandoğan, 06100, Ankara, E- posta: [email protected] 206 Ankara University, Faculty of Science, Department of Mathematics, Tandoğan, 06100, Ankara, E- posta: [email protected] 207 Ankara University, Faculty of Science, Department of Mathematics, Tandoğan, 06100, Ankara, E- posta: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Some Characterizations for Bertrand and Mannheim offsets of null-scrolls

Pınar BALKI OKULLU208 Mehmet ÖNDER209

Abstract

In this study, we consider the notion of Bertrand and Mannheim offsets for null-scrolls in the Minkowski 3-space. First, we define Bertrand and Mannheim offsets of a null scroll. Then we give some characterizations of these offset surfaces. We obtain that the offset distance is constant for Bertrand offsets of a null-scroll while the offset distance is not constant for Mannheim offsets of reference surface.

Key Words: Null-scroll; B-scroll; Bertrand offset; Mannheim offset.

References

3 3 [1] Alias L.J., Ferrandez A., Lucas P., 2-type surfaces in S1 and H1 , Tokyo Journal of Mathematics, Vol. 17(1994), 447-454. [2] Alias L.J., Ferrandez A., Lucas P., Moreno M.A., On the Gauss map of B -scrolls. Tsukuba Journal of Mathematics, Vol. 22(1998), 371-377. [3] Balgetir H., Bektaş B., Inoguchi J., Null Bertrand curves in Minkowski 3-space and their characterizations, Note di Matematica, Vol. 23(1) (2004), 7-13. [4] Balgetir H., Bektas M., Ergüt M., Bertrand curves for nonnull curves in 3-dimensional Lorentzian space. Hadronic Journal, Vol. 27(2004), 229-236. [5] Ferrandez A., Lucas P., On surfaces in the 3-dimensional Lorentz–Minkowski space, Paciﬁc Journal of Mathematics, Vol. 152(1992), 93-100. [6] Ferrandez A., Lucas P., On the Gauss map of -scrolls in 3-dimensional Lorentzian space forms, Chechoslovak Mathematical Journal, Vol. 50(2000), 699-704. [7] Graves L.K., Codimension one isometric immersions between Lorentz spaces, Transactions of the American Mathematical Society, Vol. 252(1979), 367–392. [8] Izumiya S., Takeuchi N., Generic properties of helices and Bertrand curves, Journal of Geometry, Vol. 74(2002), 97-109. [9] Kahraman T., Önder M., Kazaz M., Uğurlu H.H., Some characterizations of Mannheim partner curves in 3 Minkowski 3-space E1 , Proceedings of the Estonian Academy of Sciences, Vol. 60(4) (2011), 210-220. 3 [10] Kasap E., Kuruoğlu N., The Bertrand offsets of ruled surfaces in 1 , Acta Mathematica Vietnamica, Vol. 31(1) (2006), 39-48. [11] Kim D.S., Kim Y.H., -scrolls with non-diagonalizable shape operators, Rocky Mountain Journal of Mathematics., Vol. 33(1) (2003), 175-190. [12] Kim D.S., Kim Y.H., Yoon D.W., Extended -scrolls and their Gauss maps, Indian Journal of Pure and Applied Mathematics, Vol. 33(7) (2002), 1031-1040. [13] Liu H., Characterizations of ruled surfaces with lightlike ruling in Minkowski 3-space. Results in Mathematics, Vol. 56 (2009), 357-368. [14] Liu H, Yuan Y., Pitch functions of ruled surfaces and -scrolls in Minkowski 3-space. Journal of Geometry and Physics, Vol. 62(2012) 47-52. [15] O’Neill B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, 1983. [16] Orbay K., Kasap E., Aydemir İ., Mannheim offsets of ruled surfaces, Mathematical Problems in Engineering, (2009), Article ID 160917.

208 Celal Bayar University, Faculty of Art and science, Department of Mathematics, Muradiye Campus, 45140, Muradiye/Manisa, E-mail: [email protected] 209 Celal Bayar University, Faculty of Art and science, Department of Mathematics, Muradiye Campus, 45140, Muradiye/Manisa, E-mail: [email protected].

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[17] Önder M., Some characterizations of Bertrand offsets of timelike ruled surfaces. Journal of Advanced Research in Dynamical and Control Systems, Vol. 3(3) (2011), 21-35. [18] Önder M., Darboux approach to Bertrand surface offsets, International Journal of Pure and Applied Mathematics, Vol. 74(2) (2012), 221-234. [19] Önder M., Arı Z., Küçük A., On the developable of Bertrand trajectory timelike ruled surface offsets in Minkowski 3-space, International Journal of Pure and Applied Mathematical Sciences, Vol 5(1–2) (2011), 15-26. [20] Önder M., Uğurlu H.H., Frenet frames and invariants of timelike ruled surfaces. Ain Shams Engineering Journal, Vol 4(4) (2013), 507-513. [21] Önder M., Uğurlu H.H., On the developable Mannheim offsets of timelike ruled surfaces. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 84(4), (2014), 541-548. 3 [22] Öztekin H.B., Ergüt M., Null Mannheim curves in the Minkowski 3-space E1 , Turkish Journal of Mathematics, Vol. 35(2011), 107-114. [23] Ravani B., Ku T.S., Bertrand offsets of ruled and developable surfaces, Computer Aided Geometric Design Vol 23(2) (1991), 145-152. [24] Uğurlu H.H., Önder M., On Frenet frames and Frenet invariants of skew spacelike ruled surfaces in Minkowski 3-space, VII. Geometry Symposium, Kırşehir, 2009. [25] Uğurlu H.H., Çalışkan A., Darboux Ani Dönme Vektörleri ile Spacelike ve Timelike Yüzeyler Geometrisi, Celal Bayar Üniversitesi Yayınları, Yayın No: 0006, 2012. [26] Struik D.J., Lectures on Classical Differential Geometry, Dover; 2nd ed. Addison Wesley, 1988. [27] Wang F., Liu H., Mannheim partner curves in 3-Euclidean space, Mathematics in Practice and Theory, Vol. 37(1), (2007), 141-143. [28] Whittemore JK. Bertrand curves and helices. Duke Mathematical Journal, Vol. 6(1) (1940), 235-245.

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Spatial Quaternionic Involute Curve A New View

Süleyman ŞENYURT210, Ceyda CEVAHİR211 , Yasin ALTUN212

Abstract

In this study, the normal vector and the unit Darboux vector of spatial involute curve of the spatial quaternionic curve are taken as the position vector, the curvature and torsion of obtained smarandahce curve were calculeted.

Key Words: : Quaterniyonic Space, Involute Curve, Quaternionic Smarandache Curves

References

[1] Ali, A. T., Special Smarandache Curves in the Euclidean Space, International Journal of Mathematical Combinatorics, 2, 30-36, 2010. [2] Bharathi, K., Nagaraj, M., Quaternion Valued Function of a Real Variable Serret-Frenet Formula, Indian Journal of Pure and Applied Mathematics, 18(6), 507-511, 1987. [3] Çetin, M., Kocayiğit, H., On the Quaternionic Smarandache Curves in Euclidean 3-Space, Int. J. Contemp. Math. Sciences, 8(3),139 - 150, 2013. 4 [4] Çöken, A.C., Tuna A., On The Quaternionic Inclined Curves In The Semi-Euclidean Space E2 , Applied Mathematics and Computation, 155, 373-389, 2004. [5] Demir, S., Özdaş, K., Serret-Frenet Formulas by Real Quaternions (in Turkish), Süleyman Demirel University, Journal of Natural and Applied Sciences, 9(3), 1-7, 2005. [6] Erişir, T., Güngör, M.A., Some Characterizations of Quaternionic Rectifying Curves in the Semi-Euclidean Space , Honam Mathematical J., 36 (1), 67-83, 2014, http://dx.doi.org/10.5831/HMJ.2014.36.1.67. [7] Güngör, M.A., Tosun, M., Some characterizations of quaternionic rectifying curves, Differential Geometry - Dynamical Systems, 13, 89-100 , 2011. [8] Karadağ, M., Sivridağ, A. İ., Quaternion valued functions of a single real variable and inclined curves (in Turkish), Erciyes University, journal of the Institute of Science and Technology,13(1-2),23-36, 1997. [9] Soyfidan T., Quaternionic Involute-Evolute Cauple Curves, Master Thesis, University of Sakarya, 2011. [10] Şenyurt, S., Çalışkan, A.S., An Application According to Spatial Quaternionic Smarandache Curve, Applied Mathematical Sciences, 9(5), 219-228, 2015, http://dx.doi.org/10.12988/ams.2015.411961. [11] Şenyurt, S., Grilli, L., Spherical Indicatrix Curves of Spatial Quaternionic Curves, Applied Mathematical Sciences, 9(90), 4469 - 4477, 2015, http://dx.doi.org/10.12988/ams.2015.53279. [12] Şenyurt, S., Sivas, S., An Application of Smarandache Curve (in Turkish), Ordu Univ. J. Sci. Tech., 3(1), 46-60, 2013. [13] Turgut, M., Yılmaz, S., Smarandache Curves in Minkowski Space-time, International J.Math. Combin., 3, 51-55, 2008.

210Ordu Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, 52200, Altınordu / ORDU, E-mail: [email protected] 211 Ordu Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, 52200, Altınordu / ORDU, E-mail: [email protected] 212 Ordu Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, 52200, Altınordu / ORDU, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On the Darboux Vector Belonging to involute Curve a Different View

Süleyman ŞENYURT213, Yasin ALTUN214 , Ceyda CEVAHİR 215

Abstract

In this study, we investigated special Smarandache curves in terms of Sabban frame drawn on the surface of the sphere by the unit Darboux vector of involute curve. Here was created Sabban frame belonging to this curve. We explained Smarandache curves position vector is composed by Sabban vectors belonging to this curve. Then, we calculated geodesic curvatures of this Smarandache curves. Found results were expressed depending on the main curve. Also, we given example belonging to the results found.

Key Words: Involute Curve, Darboux Vector, Smarandache Curves, Sabban Frame, Geodesic Curvature

References

[1] Ali A.T., Special Smarandache curves in the Euclidian space, International Journal of Mathematical Combinatorics, Vol.2(2010), 30-36.

[2] Çalışkan A. and Şenyurt, S., Smarandache Curves In Terms of Sabban Frame of Fixed Pole Curve, Boletim da Sociedade parananse de Mathematica 3 srie.Vol:34(2016),53-62.

[3] Hacısalihoğlu H.H., Differantial Geometry(in Turkish), Academic Press Inc. Ankara, 1994.

[4] Turgut M. and Yılmaz S., Smarandache Curves in Minkowski Space-time, International Journal of Mathematical Combinatorics, Vol.3(2008), 51-55.

[5] Taşköprü K. and Tosun M., Smarandache Curves on S^2, Boletim da Sociedade Paranaense de Matematica 3 Srie. vol.32(2014), 51-59.

[6] Fenchel, W., On The Differential Geometry of Closed Space Curves, Bulletin of the American Mathematical Society, Vol. 57(1951), 44-54.

[7] Çalışkan A. and Şenyurt, S., Smarandache Curves In Terms of Sabban Frame of Spherical Indicatrix Curves, Gen. Math. Not., Vol.31(2015), 1-15.

213 Ordu University, Faculty of Art and science, Department of Mathematics, Ordu, E-mail: [email protected] 214 Ordu University, Faculty of Art and science, Department of Mathematics, Ordu, E-mail: [email protected] 215 Ordu University, Faculty of Art and science, Department of Mathematics, Ordu, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Surface family with a common natural asymptotic lift

Ergin BAYRAM216, Evren ERGÜN217, Emin KASAP218

Abstract

In the present paper, we find a surface family possessing the natural lift of a given curve as a common asymptotic curve. We express necessary and sufficient conditions for the given curve such that its natural lift is an asymptotic curve on any member of the surface family. We present important results for ruled surfaces. Finally, we illustrate the method with some examples. Key Words: Asymptotic curve, Surface family, Natural lift curve

References

[1] Wang G. J., Tang K. & Tai C. L., Parametric representation of a surface pencil with a common spatial geodesic, Comput. Aided Des., 36(5) (2004), 447-459. [2] Kasap E., Akyıldız F. T. & Orbay K., A generalization of surfaces family with common spatial geodesic, Appl. Math. Comput., 201 (2008), 781-789. [4 ]Bayram E., Güler F. & Kasap E., Parametric representation of a surface pencil with a common asymptotic curve, Comput. Aided Des., 44 (2012), 637-643. [5] do Carmo M. P., Differential geometry of curves and surfaces, Englewood Cliffs (New Jersey): Prentice Hall Inc, 1976. [6] Thorpe J. A., Elementary topics in differential geometry, Springer-Verlag (New York, Heidelberg-Berlin), 1979.

216 Ondokuz Mayıs University, Faculty of Art and Science, Department of Mathematics, 55200, Samsun, E-mail: [email protected] 217 Ondokuz Mayıs University, Çarşamba Chamber of Commerce Vocational School, Çarşamba, Samsun, E-mail:[email protected] 218 Ondokuz Mayıs University, Faculty of Art and Science, Department of Mathematics, 55200, Samsun, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A new method for designing involute trajectory timelike ruled surfaces in Minkowski 3-space

Mustafa BİLİCİ219

Abstract

The aim of this paper is to present a new perspective on the generation of developable trajectory ruled surfaces in Minkowski 3-space. Also, some new results and theorems related to the developability of the involute trajectory timelike ruled surfaces are obtained. Finally, we illustrate these surfaces by presenting one example.

Keywords: trajectory ruled surface, involute-evolute, Frenet frame, Minkowski 3-space

References

1. B. Ravani and T.S. Ku, “Bertrand Offsets of ruled and developable surfaces”, Comput. Aided Geom. Design, 1991, 23, 145-152. 2. E. Kasap and F. T. Akyıldız, “Surfaces with common geodesics in Minkowski 3-space”, Appl. Math. Comput. 2006, 177, 260-270. 3. Y. J. Chen and B. Ravani, “Offset Surface Generation and Contouring in Computer Aided Design”, J. Mech. Des., 1987, 109, 133-142. 4. R.T Farouki, “The Approximation of Non-Degenerate offset Surfaces”, Comput. Aided Geom. Design, 1986, 3, 15-43. 5. A. Turgut and H. H. Hacısalihoğlu, “Timelike Ruled Surfaces in the Minkowski 3-Space”, Far East J. Math. Sci., 1997, 5, 83-90. 6. A. Turgut and H. H. Hacısalihoğlu, “Spacelike Ruled Surfaces in the Minkowski 3-Space” Commun. Fac. Sci. Univ. Ank., Ser. A1, Math. Stat., 1997, 46, 83-91. 7. A. Turgut and H. H. Hacısalihoğlu, “Timelike Ruled Surfaces in the Minkowski 3-Space-II”, Turkish J. Math., 1998, 22, 33-46. 8. A. Turgut and H. H. Hacısalihoğlu, “On the Distribution Parameter of Timelike Ruled Surfaces in the Minkowski 3-Space”, Far East J. Math. Sci., 1997, 5,321-328. 9. Y. Yaylı and S. Saracoğlu, “On Developable Ruled Surfaces in Minkowski Space”, Adv. Appl. Cliﬀord Algebr., 2011, 22, 499-510. 10. I. Van de Woestijne, “Minimal Surfaces of the 3- dimensional Minkowski Space. Geometry and Topology of Submanifolds II”, World Scientific Publ., 1990, Singapur, pp. 344-369. 11. Y. H. Kim and D. W. Yoon, “Classification of ruled surfaces in Minkowski 3-spaces”, J. Geom. Phys., 2004, 49, 89-100. 12. K. Akutagawa and S. Nishikawa, “The Gauss map and space-like surfaces with prescribed mean curvature in Minkowski 3-space”, Tohoku Math. J., 1990, 42, 67-82. 3 13. A. Küçük, “On the developable timelike trajectory ruled surfaces in Lorentz 3-space1 ”, Appl. Math. Comput. 2004, 157, 483-489. 14. R. Aslaner, “Hyperruledsurfaces in Minkowski 4-space”, Iran. J. Sci. Technol. Trans. A Sci., 2005, 29, 341-347. 15. B. O’Neill, Semi-Riemannian Geometry, Academic Press, New York, 1983, p. 69. 16. H. H. Uğurlu, “On The Geometry of Time-like Surfaces”, Commun. Fac. Sci. Univ. Ank., Ser. A1, Math. Stat., 1997, 46, 211-223. 17. J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer-Verlag, New York, 1994, pp. 59-72.

18. M. Bilici and M. Çalışkan, “Some New Notes on the Involutes of the Timelike Curves in Minkowski 3- Space”, Int. J. Contemp. Math. Sci., 2011, 6, 2019-2030.

219 Ondokuz Mayıs University, Department of Mathematics, Education Faculty, 55200 Samsun – Turkey, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Archimedean Polyhedral Metric and Its Isometry Group

Özcan GELİŞGEN 220, Temel ERMİŞ221

Abstract Polyhedra have interesting symmetries. Therefore they have attracted the attention of scientists and artists from past to present. Thus polyhedra are discussed in a lot of scientific and artistic works. There are only five regular convex polyhedra known as the platonic solids. Semi-regular convex polyhedron which are composed of two or more types of regular polygons meeting in identical vertices are called Archimedean solids. The duals of the Archimedean solids are known as the Catalan solids. Platonic solids are very important in the sense that they can be used not only in studies on properties of geometric structures, but also investigations on physical and chemical properties of the system under consideration. Atoms are arranged in the form of regular polyhedrons described earlier by Plato, when they are associated for composing of the crystal structures. The presence of Platonic atomic solids except for Dodecahedron in many studies of crystal structures has been known until 2006. However, dodecahedron crystal alignment has been proved in the crystal structure of gold-palladium atoms [1,2,8]. Also outers protein walls of many virus form a polyhedron. For example, HIV forms dodecahedron. Therefore we encounter polyhedra in the study of medicine. Since other disiplines use polyhedra, it is important that give mathematical equations of polyhedra. Minkowski geometry is a non-Euclidean geometry in a finite number of dimensions that is different from elliptic and hyperbolic geometry. Linear structure of Minkowski geometry which is different from Minkowskian geometry of space-time is the same as the Euclidean one. There is only one difference which distance is not uniform in all directions. This difference cause chancing concepts with respect to distance. For example, instead of the usual sphere in Euclidean space, the unit ball is a general symmetric convex set. Unit ball of Minkowski geometries is a general symmetric convex set[9]. Therefore this show that one can find a relation between symmetries convex set and metrics. In [5], we introduce a family of metrics, and show that the spheres of the 3- dimensional analytical space furnished by these metrics are some well-known some Platonic and Archimedean polyhedra. One of the fundamental problem in geometry for a space with a metric is to determine the group of isometries. In this work, we show that the group of isometries of the 3-dimesional space covered Archimedean polyhedral metric is the semi-direct product of octahedral group Oh and T(3), where T(3) is the group of all translations of the 3-dimensional space. Key Words: Polyhedra, Platonic solids, Archimedean solids, Isometry group, Archimedean polyhedral metric, References

[1] Atiyah M., Sutcliffe P. , Polyhedra in Physics, Chemistry and Geometry, Milan Journal of Mathematics, 71, 33-58, 2003. [2] Carrizales J. M. M. , Lopez J. L. R. , Pal U. , Yoshida M. M. and Yacaman M. J., The Completion of the Platonic Atomic Polyhedra: The Dodecahedron, Small, 2, 3, 351-355, 2006. [3] Ermiş T. and Kaya R., On the Isometries of 3-Dimensional Maximum Space, Konuralp Journal Of Mathematics, 3,1, 103-114, 2015. [4] Gelişgen, Ö., Kaya, R., The Isometry Group of Chinese Checker Space, International Electronic Journal Geometry, Vol.8, No:2, 82-96, 2015. [5] Gelişgen, Ö., Can Z., On The Family of Metrics for Some Platonic and Archimedean Polyhedra, ESOGU Preprint No:001, 2016. [6] Gelişgen, Ö., Kaya, R., The Taxicab Space Group, Acta Mathematica Hungarica, DOI:10.1007/s10474- 008-8006-9, Vol.122, No.1-2, 187-200, 2009. [7] Lopez J. L. R, Carrizales J. M. M. and Yacaman M. J. , Low Dimensional Non - Crystallographic Metallic Nanostructures: Hrtem Simulation, Models and Experimental Results, Modern Physics Letters B. , 20, 13, 725-751, 2006. [8] Thompson A. C., Minkowski Geometry, Cambridge University Press, 1996.

220 Eskişehir Osmangazi University, Faculty of Arts and Sciences, Department of Mathematics-Computer, E-mail: [email protected] 221 Eskişehir Osmangazi University, Faculty of Arts and Sciences, Department of Mathematics-Computer, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

New Results for General Helices in Minkowski 3-space

Kazım İLARSLAN222

Abstract

Without any doubt, helix one of the most fascinating curve in science and nature. Scientist have long held a fascination, sometimes bordering on mystical obsession, for helical structures in nature. In this talk, we discuss the answer of the question whether there exist any general helix whose curvatures satisfying the condition |k1|=|k2| in Minkowski 3-space. Then we show that the answer of the question is related to the casual character of slope axis of given curve.

This talk based on the following papers.

Key Words: General helix, Minkowski 3-space, slope axis, biharmonic curve.

References

[1] Uçum A., Camcı Ç. and İlarslan K., On general helices with spacelike slope axis in Minkowski 3-space, submitted (2015).

[2] Uçum A., Camcı Ç. and İlarslan K., On general helices with timelike slope axis in Minkowski 3-space, accepted to publish in Advances in Applied Clifford Algebras (2015).

[3] Camcı Ç. and İlarslan K. and Uçum A., On general helices with lightlike slope axis in Minkowski 3-space, to appear (2016).

222 Kırıkkale University, Faculty of Art and Science, Department of Mathematics, Kırıkkale. E-mail: [email protected]

132

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On The Semi-Parallel Tensor Product Surfaces In Semi-Euclidean Space E₂⁴

Mehmet YILDIRIM223

Abstract

In this article, the tensor product surfaces are studied that arise from taking the tensor product of a unit circle centered at the origin in Euclidean plane E² and a non-null, unit planar curve in Lorentzian plane E₁². Also we have shown that the tensor product surfaces in 4-dimensional semi-Euclidean space with index 2, E₂⁴, satisfying the semi-parallelity condition R(X,Y).h=0 if and only if the tensor product surface is a totally geodesic surface in E₂⁴.

Key Words: Tensor product immersion, Euclidean circle, Lorentzian curves, semiparallel surface, normal curvature.

References

[1] Arslan K., Bulca B., Kılıc B., Kim Y. H. , Murathan C. and Ozturk G., Tensor Product Surfaces with Pointwise 1-Type Gauss Map, Bull. Korean Math.Soc. 48 (2011), 601-609.

[2] Arslan K. and Murathan C., Tensor product surfaces of pseudo-Euclidean planar curves, Geometry and topology of submanifolds, VII (Leuven, 1994/Brussels, 1994) World Sci. Publ.,

River Edge, NJ (1995), 71-74.

[3] Bulca B. and Arslan K., Semiparallel tensor product surfaces in E⁴, Int. Electron. J. Geom., 7,1,(2014), 36- 43.

[4] İlarslan K. and Nesovic E., Tensor product surfaces of a Euclidean space curve and a Lorentzian plane curve, Differential Geometry - Dynamical Systems 9 (2007),47-57.

[5] Mihai I., and Rouxel B., Tensor Product Surfaces of Euclidean Plane Curves, Results in Mathematics, 27 (1995), no.3-4, 308-315.

[6] Mihai I., Woestyne I. Van de, Verstraelen L. and Walrave J., Tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve. Rend. Sem. Mat. Messina Ser. II 3(18) (1994/95), 147--158.

223 Kırıkkale University, Faculty of Art and Science, Department of Mathematics, Kırıkkale. E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Generalized Pseudo Null Bertrand Curves in semi-Euclidean 4 space

Osman KEÇİLİOĞLU224 and Ali UÇUM225

Abstract

In the present paper, generalized pseudo null Bertrand curves in semi-Euclidean 4-space with index 2 is studied. Because the (1,3)-normal plane of a pseudo null curve is timelike, the (1,3)-Bertrand mate curves of the given curve can be a pseudo null curve, a non-null curve, a Cartan null curve or a partially null curve. However we show that there exists no pseudo null curve such that its (1,3)-Bertrand mate curve is a partially null curve. For other cases, we give the necessary and sufficient conditions for a pseudo null curve to be a (1,3)-Bertrand curve. Also we give the related examples.

Key Words: Generalized Bertrand curve, Semi-Euclidean Space, pseudo null curves.

References

[1] Duggal K. L. and Jin D. H., Null Curves and Hypersurfaces of Semi- Riemannian Manifolds, World Scientic, London, (2007).

[2] Matsuda H. and Yorozu S., Notes on Bertrand curves, Yokohama Math. J., 50 (2003) 41-58.

4 [3] Sakaki M., Null Cartan Curves in R2 , Toyama Mathematical Journal, 32 (2009) 31-39.

[4] Uçum A., Keçilioğlu O. and İlarslan K., Generalized Pseudo Null Bertrand curves in Semi-Euclidean 4- Space with index 2, accepted in Rendiconti del Circolo Matematico di Palermo (2016).

224 Kırıkkale University, Faculty of Art and Science, Department of Statistics, Kırıkkale. E-mail: [email protected] 225 Kırıkkale University, Faculty of Art and Science, Department of Mathematics, Kırıkkale. E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A New Method To Obtain Special Curves In The Three-Dimensional Euclidean Space

Fırat YERLİKAYA226, Savaş KARAAHMETOĞLU227 , İsmail AYDEMİR 228

Abstract

In this paper, we study the problem of obtaining a general equation of all the curves in space a given curvature and torsion. Also, we improve this work with regard to a linear relationship between curvature and torsion. First, we give a main theorem which presents the Frenet apparatus by rotation angle. Second, we examine the situaiton which is fixed the angle of rotation in the main theorem, involved in general equations of LW-curves. In particular, to find a general equation containing curvatures of Bertrand curve is considerable, so we work through this curve. Finally, we characterize the slant helix in cases where a real-valued function of it and we obtain their a natural representation.

Key Words: Lancret Curves, Bertrand Curve, Slant Helix, Frenet Frame

References

[1] Lancret, Michel-Ange. "Memoire sur les courbes ‘a double courbure." Memoires presentes alInstitut 1 (1806): 416-454.

[2] Bertrand, Joseph. "Mémoire sur la théorie des courbes à double courbure."Journal de Mathématiques Pures et Appliquées (1850): 332-350.

[3] Izumiya, Shyuichi, and Nobuko Takeuchi. "Generic properties of helices and Bertrand curves." Journal of Geometry 74.1 (2002): 97-109.

[4] Izumiya, Shyuichi, and Nobuko Takeuchi. "New special curves and developable surfaces." Turkish Journal of Mathematics 28.2 (2004): 153-164.

[5] Ruffa, Anthony A. A Novel Solution to the Frenet-Serret Equations. arXiv preprint arXiv:0709.2855 (2007).

[6] Hacısalihoğlu, H. Hilmi. Diferensiyel geometri. İnönü Üniversitesi, 1983.

[7] Salkowski, E. "Zur transformation von raumkurven." Mathematische Annalen66.4 (1909): 517-557.

226 Ondokuz Mayıs University, Faculty of Art and science, Department of Mathematics, Kurupelit Campus, 55200, Atakum/Samsun, E-mail: [email protected] 227 Ondokuz Mayıs University, Faculty of Art and science, Department of Mathematics, Kurupelit Campus, 55200, Atakum/Samsun, E-mail: [email protected] 228 Ondokuz Mayıs University, Faculty of Art and science, Department of Mathematics, Kurupelit Campus, 55200, Atakum/Samsun, E-mail: [email protected]

135

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

* Some Notes on Integrability Conditions and Tachibana operators on Cotangent Bundle T (M n )

Haşim ÇAYIR229

Abstract

The main aim of this paper is to find integrability conditions by calculating Nijenhuis Tensors

~ ~ ~ 1 N(X H ,Y H ) , N(X H ,Y V ) , N(X V ,Y V ) of almost complex structure 퐹퐶 + 훾(푁퐹) and to show the 2 1 results of Tachibana operators applied 푋퐻 and 푋퐶 according to structure 퐹퐶 + 훾(푁퐹) in cotangent bundle 2 ∗ 푇 (푀푛).

Key Words: Integrability Conditions, Tachibana operators, Horizontal Lift, Vertical Lift, Almost Complex Structure, Cotangent Bundle

References

[1] Çayır H., Köseoğlu G., Lie Derivatives of Almost Contact Structure and Almost Paracontact Structure

With Respect to X C and X V on Tangent Bundle T (M ) . New Trends in Mathematical Sciences, Vol. 4 (1) (2016), 153-159.

[2] Omran T., Sharffuddin A. and Husain S.I., Lift of Structures on Manifolds, Publications de 1’Instıtut Mathematıqe, Nouvelle serie, 360 (50) (1984), 93 – 97.

[3] Salimov A. A., Tensor Operators and Their applications, Nova Science Publ., New York, 2013.

[4] Salimov A. A., Çayır H., Some Notes On Almost Paracontact Structures, Comptes Rendus de 1’Acedemie Bulgare Des Sciences, Vol. 66 (3) (2013), 331-338.

[5] Yano K., Ishihara S., Tangent and Cotangent Bundles, Marcel Dekker Inc, New York, 1973.

[6] Yano K., Patterson E. M. Vertical and complete lifts from a manifold to its cotangent bundle, J. Math. Soc. Japan, Vol. 19 (1967), 91-113.

[7] Yıldırım F, On a Special Class of Semi-Cotangent Bundle, Proceedings of the Institute of Mathematics and Mechanics, Vol. 41 (1) (2015), 25-38.

229 Giresun University, Faculty of Art and Science, Department of Mathematics, 28100, Giresun, Turkey, E-mail:[email protected] & [email protected]

136

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On the parametric representation of the zero constant mean curvature surface family in Minkowski space

Sedat KAHYAOĞLU230, Emin KASAP231

Abstract

We derive a parametric representation to the zero constant mean curvature surface family prescribed by a given curve in Minkowski 3-space. We present some timelike minimal surfaces and spacelike maximal surfaces as examples.

Key Words: Constant mean curvature surface, Spacelike maximal surface, Timelike minimal surface, Frenet frame

References

[1] Alias L.J., Chaves R.M.B., and Mira P., Björling problem for maximal surfaces in Lorentz-Minkowski space, Proc. Cambridge Philos. Soc. Vol. 134(2003), 289-316.

[2] Chaves R.M.B., Dussan M.P., and Magi M., Björling problem for timelike surfaces in the Lorentz- Minkowski space, J. Math. Anal. Appl. Vol. 377(2011), 481--494.

[3] Kahyaoglu S. and Kasap E., Spacelike maximal surface family prescribed by a spacelike curve in 3- dimensional Minkowski space, Int. J. Contemp. Math. Sci. Vol. 11(2016) 131-138.

[4] Kahyaoglu S. and Kasap E., Timelike minimal surface family prescribed by a spacelike curve in Minkowski 3-space, Int. Electron. J. Pure Appl. Math. Vol. 10 (2016) 83-90.

[4] Kim Y.W., Koh S.-E., Shin H., and Yang S.-D., Spacelike maximal surfaces timelike minimal surfaces and björling representation formulae, J. Korean Math. Soc. Vol. 48 (2011), 1083-100.

[5] Kim Y.W. and Yang S.-D., Prescribing singularities of maximal surfaces via a singular björling representation formula, J. Geom. Phys. Vol. 57(2007), 2167-2177.

[6] Kobayashi O., Maximal surfaces in the 3-dimensional Minkowski space , Tokyo J. Math. Vol. 6(1983), 297- 309.

[7] O'Neill B., Semi-riemannian geometry with application to general relativity, Academic Press, 1983.

230 Ondokuz Mayis University, Yeşilyurt Demir.Çelik. V.S., Department of Mechatronics, 55300, Samsun, Turkey. E-mail: [email protected] 231 Ondokuz Mayis University, Art and Science Faculty, Department of Mathematics, 5300, Samsun, Turkey. E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Equivalence Problem for a Riccati Type Pde in Three Dimensions

Tuna BAYRAKDRA232, Abdullah Aziz ERGİN233

Abstract

Jacobi identity for a Poisson structure reduces to a Riccati type equation in three dimensions [1]. In this study we consider the contact equivalence problem for a Riccati type pde, which is a nonlinear partial differential equation with one dependent and three independent variables, in geometric context via Cartan’s method of equivalence. We obtain an invariant co-frame on base manifold and we compute its structure invariants.

Key Words: Riccati equation, Cartan’s equivalence method.

References

[1] E. Abadoğlu, H. Gümral, Bi-Hamiltonian structure in Frenet-Serret Frame, Physica D 238 (2009) 526-530.

[2] P. Olver, Application of Lie groups to differential equations, Second Edition, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1993.

[3] Cartan, E., Les problémes d’équivalence, Séminaire de Mathématiques, exposé du11janvier 1937 (1937), pp. 113û136.

[4] Robert B. Gardner, The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 58, Society for Industrial and Applied

Mathematics (SIAM), Philadelphia, PA, 1989.

[5] Peter J. Olver, Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995.

[6] Morozov O.I., Moving coframes and symmetries of differential equations, J. Phys. A: Math. Gen., V.35, N 12, 2965-2977.

[7] O. I. Morozov, “Contact-equivalence problem for linear hyperbolic equations,” Journal of Mathematical Sciences, vol. 135, no. 1, pp. 2680–2694, 2006.

232 Akdeniz University Faculty of Science, Department of Mathematics, Dumlupınar Boulevard 07058 Campus Antalya, E-mail: [email protected] 233 Akdeniz University Faculty of Science, Department of Mathematics, Dumlupınar Boulevard 07058 Campus Antalya, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Meridian Surfaces of Weingarten Type in 4-dimensional Euclidean Space 피4

Betül BULCA234, Günay ÖZTÜRK235, Bengü Bayram236, Kadri ARSLAN237

Abstract

In the present study we consider meridian surfaces in Euclidean 4-space 피4. This study consists of third parts. In the first part we give some basic concepts of surfaces in Euclidean 4-space 피4. In the second part we introduce meridian surfaces in 피4. Further, we give basic results related with Weingarten type surfaces in Euclidean 4-space 피4. Finally, we classified all meridian surfaces of Weingarten type in 피4.

Key Words: Second fundamental form, Meridian surfaces, Weingarten surfaces

References

[1] K. Arslan and V. Milousheva, Meridian Surfaces of Elliptic or Hyperbolic Type with Pointwise 1-type Gauss map in Minkowski 4-Space, Taiwanese J. Math., 20(2) (2016), 311-322.

[2] B. Bulca, K. Arslan and V. Milousheva, Meridian Surfaces in E⁴ with Pointwise 1-type Gauss Map, Bull. Korean Math. Soc., 51 (2014), 911-922.

[3] B. Y. Chen, Geometry of Submanifolds , Dekker, New York, 1973.

[4] F. Dillen and W. Kühnel, Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Math., 98 (1999), 307-320.

[5] G. Ganchev and V. Milousheva, Invariants and Bonnet-type theorem for surfaces in R⁴, Cent. Eur. J. Math., 8(6) (2010) 993-1008.

[6] G. Ganchev and V. Milousheva, Special Class of Meridian Surfaces in the Four-Dimensional Euclidean Space, arXiv: 1402.5848v1 [math.DG], 24 Feb. 2014.

[7] W. Kühnel and M. Steller, On Closed Weingarten Surfaces, Monatsh. Math., 146 (2005), 113-126.

[8] W. Kühnel, Ruled W-surfaces, Arch. Math., 62 (1994), 475-480.

[9] J. Weingarten, Ueber eine Klasse auf einander abwickelbarer Flaachen, J. Reine Angew. Math. 59 (1861), 382--393.

[10] J. Weingarten, Ueber die Flachen, derer Normalen eine gegebene Flache beruhren, J. Reine Angew. Math. 62 (1863), 61-63.

234 Uludag University, Faculty of Art and Science, Department of Mathematics, Görükle Campus, 16059, Bursa, E-mail: [email protected] 235 Kocaeli University, Faculty of Art and Science, Department of Mathematics, Kocaeli, E- mail:[email protected] 236 Balıkesir University, Faculty of Art and Science, Department of Mathematics, Çağış Campus, Balıkesir, E-mail: [email protected] 237 Uludag University, Faculty of Art and Science, Department of Mathematics, Görükle Campus, 16059, Bursa, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

The Fermi-Walker Derivative and Principal Normal Indicatrix in Minkowski 3-Space

Fatma KARAKUŞ238, Yusuf YAYLI239

Abstract

In this study we explained the Fermi-Walker derivative along the principal normal indicatrix of a timelike curve in Minkowski 3-space. We get a timelike curve in Minkowski 3-space. According to the principal normal indicatrix of the timelike curve Fermi-Walker derivative, Fermi-Walker parallelism, non-rotating frame and Fermi-Walker termed Darboux vector concepts are given. We proved while the curve is a timelike helix Frenet frame is a non-rotating frame along the principal normal indicatrix. And then we proved when the principal normal indicatrix is a timelike slant helix Fermi-Walker termed Darboux vector is Fermi-Walker parallel along the principal normal indicatrix of a timelike curve.

Key Words: Fermi-Walker derivative, Fermi-Walker parallelism, Non-rotating frame Fermi-Walker termed Darboux vector, Principal Normal Indicatrix, Helix, Slant helix

References

[1] Karakuş F. and Yaylı Y., On the Fermi-Walker derivative and Non-rotating frame, Int. Journal of Geometric Methods in Modern Physics, Vol.9, No.8 (2012), 1250066 (11 pp).

[2] Karakuş F. and Yaylı Y., The Fermi-Walker Derivative on the Spherical Indicatrix of Timelike Curve in Minkowski 3-Space, Adv. Appl. Clifford Algebras, Vol.26 (2016), 199-215.

[3] Fermi, E.: Atti Accad. Naz. Lincei Cl. Sci. Fiz. Mat. Nat. 31, 184-306 (1922).

[4] Hawking, S.W.and Ellis, G.F.R., The large scale structure of spacetime, Cambridge Univ. Press (1973).

[5] İlarslan, K. and Nesovic, E., Timelike and Null Normal Curves in Minkowski Space E₁³, Indian J. Pure Appl. Math. 35(7) (2004) 881-888.

[6] O’Neill, B., Semi Riemannian Geometry, With Applications to Relativity. Pure and Applied Mathematics, 103. Academic Press,Inc., New York,1983.

[7] Petrovıc-Torgasev, M. and Sucurovıc, E., Some characterizations of Lorentzian spherical timelike and null curves. Mat. Vesn. 53 (2001), 21–27.

[8] Ilarslan, K. and Nesovic, E., Timelike and null normal curves in Minkowski space, Indian J. Pure Appl. Math. 35(7) (2004), 881–888.

238 Sinop University, Faculty of Art and Science, Department of Mathematics, 57000, Sinop, E-mail: fkarakus @sinop.edu.tr 239 Ankara University, Faculty of Science, Department of Mathematics, Tandoğan Campus, 06100, Tandoğan/Ankara, E-mail: Yusuf.Yayli @science.ankara. edu.tr

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Split Semi-Quaternions and Semi-Euclidean Planar Motions

Murat BEKAR240 and Yusuf YAYLI241

Abstract

In this study, the basic structures of the algebra of split semi-quaternions are given. Furthermore, the planar motions in semi-Euclidean three-space are expressed by split semi-quaternions.

Key Words: Pseudo-rotation, split semi-quaternion, semi-Euclidean planar motion.

References

[1] Ell T. A. and Sangwine S. J., Quaternion involutions and anti-involutions, J. Comput. Math. Appl. Vol. 53(2007), 137-143.

[2] M. Jafari, Split Semi-quaternions Algebra in Semi-Euclidean 4-space, Cumhuriyet Sci. J. Vol. 36(2015), 70- 77.

240 Necmettin Erbakan University, Faculty of Science, Department of Mathematics and Computer Sciences, 42090, Konya/TURKEY, E-mail: [email protected] 241 Ankara University, Faculty of Science, Department of Mathematics, 06100, Ankara/TURKEY, E- mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Suborbital Graphs For a Special Möbius Transformation on The Upper Half Plane ℍ

Murat BEŞENK242

Abstract

Let PSL(2, ℤ) be the modular group and Γ0(푝) denote the subgroup represented by the matrices 푎 푏 Γ (푝): = {( ) ∈ SL(2, ℤ) ∶ 푐 ≡ 0 푚표푑(푝)} where 푝 is a prime. Let ℍ ≔ {푧 ∈ ℂ | Imz > 0} denote the 0 푐 푑 upper half plane which the lines of the model are the open rays orthogonal to the real axis together with the open ∗ ∗ semicircles orthogonal to the real axis. And also ℍ ≔ ℍ ∪ ℚ ∪ {∞}. Then ℍ ⁄Γ0(푝) is compact Riemann surface. In this paper we examine some properties of suborbital graphs for a special Möbius transformation. In addition, we give edge and circuit circumstances for the suborbital graph. And finally we give necessary and sufficient conditions for graphs to have hyperbolic triangles.

Key Words: Suborbital graphs, Congruence subgroup, Orbit, Circuit, Hyperbolic plane

References

[1] Akbaş M., On Suborbital Graphs For The Modular Group, Bulletin of The London

Mathematical Society, Vol. 33(2001), 647-652.

[2] Beşenk M. et al., Circuit Lengths of Graphs For The Picard Group, Journal of Inequalities

and Applications, Vol. 1(2013), 106-114.

[3] Jones G.A., Singerman D., Wicks K., The Modular Group and Generalized Farey Graphs,

Bulletin of The London Mathematical Society, Vol. 160(1991), 316-338.

[4] Schoeneberg B., Elliptic Modular Functions, Springer Verlag, Berlin, 1974.

[5] Beardon A.F., The Geometry of Discrete Groups, Springer Verlag, Cambridge, 1995.

[6] Sims C.C., Graphs and Finite Permutation Groups, Mathematische Zeitschrift, Vol.

95(1967), 76-86.

[7] Güler B.Ö., Beşenk M., Değer A.H., Kader S., Elliptic Elements and Circuits in

Suborbital Graphs, Hacettepe Journal of Math. and Statistics, Vol. 40(2011), 203-210.

[8] Rankin R.A., Modular Forms and Functions, Cambridge University Press, 2008.

242 Karadeniz Technical University, Faculty of Science, Department of Mathematics, Kanuni Campus, 61080, Ortahisar /Trabzon, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Control invariants of non-directional Bezier curve

İdris ÖREN243

Abstract

Let 푀(푛) be the group of all motions of the n-dimensional Euclidean space. This paper presents the definition of a non-directional Beziver curve and the conditions of푀(푛)-equivalence for two non-directional Beziver curves in ℝ푛 of degree m, where 푚 ≥ 1.

Key Words: Bezier curve; invariant; non-directional curve.

References

[1] Bez, H. E., Generalized invariant-geometry conditions for the rational Bezier paths, Int J Comput Math, 87(2010), 793-811.

[2] Chen X., Ma, W., Deng, C., Conditions for the coincidence of two quartic Bezier curves, Appl Math Comput, 225(2013), 731-736.

[3] Chen XD, Yang, C., Ma, W., Coincidence condition of two B´ezier curves of an arbitrary degree, Comput. Graph, 54(2016), 121-126.

[4] Khadjiev, D., Ören, İ, Peksen, Ö., Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry, Turkish J. Math. 37(2013), 80-94.

[5] Ören, İ., The equivalence problem for vectors in the two-dimensional Minkowski spacetime and its application to Bezier curves, J. Math. Comput. Sci, 6 (2016), No. 1, 1-21.

[6] Pekşen, Ö., Khadjiev, D., Ören, İ., Invariant parametrizations and complete systems of global invariants of curves in the pseudo-Euclidean geometry, Turkish J. Math. 36(2012), 147-160.

[7] S´anchez-Reyes, J., On the conditions for the coincidence of two cubic Bezier curves, J. Comput. Appl. Math., 236(2011), 1675-1677.

[8] Wang, WK, Zhang, H, Liu, XM, Paul, JC, Conditions for coincidence of two cubic Bezier curves, J. Comput.Appl. Math.,235(2011), 5198-5202.

243 Karadeniz Technical University, Faculty of Science, Department of Mathematics, Kanuni Campus, 61080, Ortahisar/Trabzon, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On the Generalization of Quaternions

Muttalip ÖZAVŞAR244, E.Mehmet ÖZKAN245

Abstract

In this work, we introduce an algebraic generalization of the algebra of quaternions.

Key Words: quaternions, noncommutative algebras

References

[1] Hamilton W. Rowan, Elements of Quaternions, Vol. 2 (1899-1901) reprinted Chelsea, New York,

1969

244 Yıldız Technical University, Faculty of Art and science, Department of Mathematics, Davutpasa Campus, 34210, Esenler/İstanbul, E-mail: [email protected] 245 Yıldız Technical University, Faculty of Art and science, Department of Mathematics, Davutpasa Campus, 34210, Esenler/İstanbul, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

C-Curves in Minkowski Space

Emre ÖZTÜRK246, Yusuf YAYLI247

Abstract

In this study, we present the characterizations of the W-curves and their kinematic applications using a different approach in Minkowski space. We also examine the relations between W-curves and C-curves. Using kinematic applications, we get equations of the C-curves by utilizing algebraic methods in 3-dimensional Minkowski space. Finally we specify the relations between the curvatures of the curve in Minkowski 3-space.

Key Words:Minkowski Space, Kinematics, W-curve

References

[1] Aminov Y., Differential Geometry and Topology Of Curves, Gordon and Breach Science Publishers imprint, 2000

[2] Chen B.Y., Kim D.S., Kim Y.H., New characterizations of W-Curves Publ. Math. Debrecen, 69 (4) (2006) 457-472

[3] Ferus D., Schirrmacher S., Mathematische Annalen, 260 (1982) 57-62

[4] Kim Y.H., Lee K.E., Surfaces of Euclidean 4-Space Whose geodesics are W-Curves, Nihonkai Math. J. Vol 4 (1993) 221-232

[5] Kim D.S., Kim Y.H., New characterizations of spheres, cylinders and W-curves, Linera Algebra and Its Applications 432 (2010) 3002-3006

[6] O'Neill B., Semi Riemann Geometry, Academic Press New York,1983

[7] Rademacher H. and Toeplitz O., The enjoyment of mathematics, Princeton Science Library, Princeton University Press, 1994

[8] Ünal Z., Kinematics With Algebraic Methods In Lorentzian Spaces, Ankara University, Ph.D. Thesis, Ankara 2007

[9] Walrave J., Curves and Surfaces in Minkowski Space, Doctoraatsverhandeling, 1995

246 Sayıştay Başkanlığı, İnönü Bulvarı (Eskişehir Yolu), No:45 06520, Balgat, Çankaya/ANKARA, E- mail: [email protected] 247 Ankara University, Faculty of science, Department of Mathematics, Dögol Caddesi, 06100, Tandoğan/Ankara, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A Generalization of Cheeger-Gromoll Metric on Tangent Bundle

Murat ALTUNBAŞ248, Aydın GEZER249

Abstract

In this work, the Riemannian metric obtained from multiplying with a positive defined function c to the horizontal part of well-known two parameter Cheeger-Gromoll type Riemannian metric is considered on tangent bundle. The compatible almost complex structure is defined and the conditions are given under which the tangent bundle is almost Kahlerian and Kahlerian. Finally, some curvature properties of the metric are studied.

Key Words: Tangent bundle, almost complex structure, Riemannian manifold, curvature tensor.

References

[1] Gezer A. and Altunbaş M., Some notes concerning Riemannian metrics of Cheeger-Gromoll type, Journal of Mathematical Analysis and Applications, Vol. 396 (2012) 119–132.

[2] Hou Z. and Sun L., Geometry of tangent bundle with Cheeger-Gromoll type metric, Journal of Mathematical Analysis and Applications, Vol. 402 (2013) 493-504.

[3] Munetanu M., Some aspects on the geometry of the tangent bundles and tangent sphere bundles of a Riemannian manifold, Mediterrenan J. Math., Vol. 5 (2008) 43-59.

[4] Benyounes, M., Loubeau, E., and Todjihounde, L., Harmonic maps and Kaluza-Klein metrics on spheres, Rocky Mount. J. Math., Vol. 42 (3) 2012, 791-821.

248 Erzincan University, Faculty of Art and Science, Department of Mathematics, Yalnızbağ Campus, 24100, Erzincan, E-mail: [email protected] 249 Atatürk University, Faculty of Science, Department of Mathematics, 25040, Erzurum, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Osculating Curves in Semi-Euclidean 4 Space

Nihal KILIÇ ASLAN250, Hatice ALTIN ERDEM251

Abstract

Osculating curves in Minkowski space time firstly defined by İlarslan and Nesovic in [3] as a curve whose position vector always lie in osculating space of the curve. In this paper, we define the first kind and the second kind osculating non-null curves with non-null normals in E4_2. We characterize such curves in terms of their curvature functions. We obtain the explicit equations of such osculating curves with constant curvatures. Also we give some examples of non-null Osculating curves in E4_2 .

Key Words: 4-dimensional Semi-Euclidean space with index 2, spacelike and timelike curves, osculating curve, curvature.

References

[1] Kılıç N., Altın Erdem H., İlarslan K., Osculating Curves in 4-Dimensional Semi-Euclidean Space with index 2, Demonstratio Mathematica (Accepted, publish in 2017)

[2] İlarslan K., Nesovic E., Some characterizations of Osculating Curves in the Euclidean spaces, Demonstratio Mathematica, Vol.XLI No:4 (2008), 931-939.

[3] İlarslan K., Nesovic E., The first kind and the second kind Osculating curves in Minkowski Space-time, Compt. Rend. Acad. Bulg. Sci., 62(6) (2009), 677-686.

250 Kırıkkale Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, 71450, Kırıkkale, E-mail: [email protected] 251 Kırıkkale Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, 71450, Kırıkkale, E-mail: [email protected]

147

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On The Complete Arcs in The Left Near Field Projective Plane Of Order 9

Elif ALTINTAŞ252, Ayşe BAYAR253 Ziya AKÇA254, Süheyla EKMEKÇİ255

Abstract

In this work, the complete (k, 2)- arcs with 6≤k≤10 in the left near field projective plane of order 9 were determined and classified by using a computer program.

Key Words: Projective plane, Hall plane, Arcs, Complete arcs.

References

[1] Altıntaş E., 9. mertebeden sol yaklaşık cisim düzleminde Fano düzlemi içeren arklar üzerine, ESOGÜ Fen Bilimleri Enstitüsü Yüksek lisans tezi, (2015). [2] Hall M., The theory of groups , New York: Macmillan (1959). [3] Hall M., Swift Jr, J.D., Killgrove R., On projective planes of order nine, Math. Tables and Other Aids Comp. 13 (1959) 233-246. [4] Hirschfeld J. W.P., Projective geometries over finite fields, Second Edition, Clarendon Press, Oxford, 1998.

[5] Room T.G., Kirkpatrick P.B., Miniquaternion Geometry, London, Cambridge University Press, 177, (1971).

252 İstanbul Aydın University, ABMYO, Department of Automotive Technology, Florya Campus, 34295, Küçükçekmece/İstanbul, E-mail: [email protected] 253 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-Computer, Meşelik Campus, 26480, Odunpazarı/Eskişehir, E-mail: akorkmaz @ogu.edu.tr 254 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-Computer, Meşelik Campus, 26480, Odunpazarı/Eskişehir, E-mail: : zakca @ogu.edu.tr 255 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-Computer, Meşelik Campus, 26480, Odunpazarı/Eskişehir, E-mail sekmekci @ogu.edu.tr

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Bi- f -harmonic immersions

Selcen YÜKSEL PERKTAŞ256, Feyza Esra ERDOĞAN257

Abstract

In the present paper, we study bi- -harmonic maps which generalize not only -harmonic maps but also biharmonic maps. We derive bi- -harmonic equations for curves and hypersurfaces.

Key Words: Harmonic maps, Biharmonic maps, -harmonic maps, bi- -harmonic maps.

References

[5] Jiang G. Y., 2-harmonic maps and their first and second variation formulas, Chinese Ann. Math. Ser. A., 7, (1986), 389-402.

[6] Djaa M., Cherif A. M., Zegga K., Ouakkas S., On the Generalized of Harmonic and Bi-harmonic Maps, Int. Electron. J. Geom., 5(1)(2012), 90-100.

[7] Eells J., Sampson J. H., Harmonic mapping of the Riemannian manifold, American J. Math.,86, (1964), 109-160.

[8] Keleş S., Yüksel Perktaş S., Kılıç E., Biharmonic curves in LP-Sasakian Manifolds, Bull. Malays. Math. Sci. Soc., 33(2), (2010), 325-344.

[9] Lu W.-J., On -biharmonic maps and bi- -harmonic maps between Riemannian manifolds, Sci China Math., 58(7), (2015), 1483-1498.

[10] Ou Y.-L., On -biharmonic maps and -biharmonic submanifolds, Pacific J. Of Math., 271(2), (2014), 461-477.

[11] Yüksel Perktaş S., Kılıç E., Biharmonic maps between doubly warped product manifolds, Balkan J. of Geom. And its Appl., 15(2), (2010), 159-170.

[12] Zegga K., Cherif A. M., Djaa M., On the -biharmonic maps and submanifolds, Kyunpook Math. J. 55(2015), 157-168.

256 Adıyaman University, Faculty of Arts and Sciences, Department of Mathematics, Adıyaman, E-mail: [email protected] 257 Adıyaman University, Faculty of Education, Department of Elementary Education, Adıyaman, E-mail: [email protected]

149

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Characterizations for Timelike Slant Ruled Surfaces in Dual Lorentzian Space

Seda ALTINGÜL258, Mustafa KAZAZ259

Abstract

In this paper, we study timelike slant ruled surfaces in dual Lorentzian space by means of dual Darboux frame. By using E. Study’s mapping, we consider a timelike ruled surface as a dual hyperbolic spherical curve lying on the dual hyperbolic unit sphere, and study the notion of timelike slant ruled surface. We obtain some dual characterizations for dual hyperbolic spherical curves for which the real parts of them give real characterizations for timelike slant ruled surfaces.

Key Words: Dual Darboux Frame, Dual Hyperbolic Spherical Curve, Dual Slant Curve, Timelike Slant Ruled Surface.

References

3 [1] Önder, M., Timelike and spacelike slant ruled surfaces in Minkowski 3-space E1 , arXiv: 1604.03813v1[Math.DG] (2016). [2] Barros, M., General Helices and a Theorem of Lancret, Proc. Amer. Math. Soc. 125(5), 1503–1509, (1997). [3] Izumiya, S., Takeuchi, N., New Special Curves and Developable Surfaces, Turk. J. Math., 28, 153-163, (2004). [4] Kula, L. and Yayli, Y., On Slant Helix and its Spherical Indicatrix, Applied Mathematics and Computation, 169, 600-607, (2005). [5] Kula, L. Ekmekçi, N., and Yayli, Y., İlarslan, K., Characterizations of Slant Helix in Euclidean 3-Space, Turk. J. Math., 33, 1-13, (2009). [6] Ali, A. T., Position Vectors of Slant Helices in Euclidean Space E3 , Journal of Egyptian Mathematical Society, 20(1), 1-6, (2012). [7] Ali, A.T., Turgut, M., Position vector of a time-like slant helix in Minkowski 3-space, J. Math. Anal. Appl. 365, 559–569, (2010) 3 [8] Ali, A.T., Lopez, R., Slant Helices in Minkowski Space E1 , J. Korean Math. Soc. 48(1), 159-167, (2011). [9] Önder, M., Slant Ruled Surfaces in Euclidean 3-Space E3 , arXiv:1311.0627v1 [math.DG]. (2013). [10] Önder, M., Kaya, O. Darboux Slant Ruled Surfaces, Azerbaijan Journal of Mathematics, 5(1), 64–72, (2015). [11] Önder, M., Kaya, O. Characterizations of Slant Ruled Surfaces in the Euclidean 3-Space, Caspian Journal of Mathematical Sciences (CJMS), (2015). (In Press). [12] Study, E. (1903). Geometrie der Dynamen, Leibzig. [13] Veldkamp, G. R. On the Use of Dual Numbers, Vectors and Matrices in Instantaneous Spatial Kinematics, Mechanism and Machine Theory, 11(2), 141-156, (1975). [14] Oral, S., Kazaz, M., Characterizations for Slant Ruled Surfaces in Dual Space, Iranian Journal of Sciences and Technology (In Press)

[15] O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, (1983). [16] Uğurlu, H.H., Çalışkan, A., Darboux Ani Dönme Vektörleri ile Spacelike ve Timelike Yüzeyler Geometrisi, Celal Bayar Üniversitesi Yayınları, Yayın No: 0006, (2012). [17] Beem, J.K., Ehrlich, P.E., Global Lorentzian Geometry, Marcel Dekker, New York, (1981).

258 Celal Bayar University, Faculty of Art and science, Department of Mathematics, Muradiye Campus, 45140, Yunusemre/Manisa, E-mail: [email protected] 259 Celal Bayar University, Faculty of Art and science, Department of Mathematics, Muradiye Campus, 45140, Yunusemre/Manisa, E-mail: [email protected]

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[18] Dimentberg F.M., The Screw Calculus and its Applications in Mechanics. English translation: AD680993, Clearinghouse for Federal and Scientific Technical Information, (Izdat. Nauka, Moscow, USSR), (1965). [19] Blaschke, W. Differential Geometrie and Geometrischke Grundlagen ven Einsteins Relativitasttheorie, New York, Dover, (1945). [20] Uğurlu HH, Çalışkan A, The study mapping for directed spacelike and timelike lines in minkowski 3- space R1 3 . Math Comput Appl 1(2):142–148, (1996). [21] Yaylı Y., Çalışkan A., Uğurlu H. H., “The E. Study Mapping of Circles on Dual Hyperbolic and 2 2 Lorentzian Unit Spheres H0 and S1 ”, Mathematical Proceedings of the Royal Irish Academy, 102 (A (1)) (2002), 37-47. [22] Önder M, Uğurlu, HH Frenet frames and invariants of timelike ruled surfaces. Ain Shams Eng J. (2012). doi:10.1016/j. asej. 2012.10.003 [23] Karger, A., and Novak, J.. Space Kinematics and Lie Groups, Prague, Gordon and Breach Science Publishers, (1978). [24] Önder, M., Uğurlu, H. H., “ Dual Darboux Frame of a Timelike Ruled Surface and Darboux Approach to Mannheim Offsets of Timelike Ruled Surfaces”, Proceedings of a National Academy of Science, India Section A: Physical Science, Vol. 83, No. 2, 163-169, (2013).

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Numerical Computation of Fibered Projective Planes

Mehmet Melik UZUN260, Ziya AKÇA261 , Süheyla EKMEKÇİ 262, Ayşe BAYAR 263

Abstract

In this study, we give fibered projective planes with base projective planes of order 2 and 3 by using Matlab script.

Key Words: Projective Planes, Matlab, Fibred Projective Planes

References

[1] Bayar A., Ekmekçi S., Akça Z., A note on fibered projective plane geometry, Information Sciences, 178 (2008) 1257-1262.

[2] Akca Z., Bayar A., Uzun M. M., A Computer Program to Determine Projective Planes over Galois Fields, Int. Math. Forum, Vol. 11(2016), No. 1, 1 - 9

[3] Bayar A., Ekmekçi S., On the Menelaus and Ceva 6-figures in the fibered projective planes, Abstract and Applied Analysis, (2014) 1-5.

[4] Hirschfeld J.W.P., 1979, Projective Geometries Over Finite Fields, Clarendon Press, 1979

[5] Kuijken L., Van Maldeghem H., Fibered geometries, Discrete Mathematics 255 (2002) 259-274.

[6] Zadeh L., Fuzzy sets, Inform. Control, 8 (1965) 338-358.

260 Central Bank of the Republic of Turkey, Eskişehir Branch, 26010, Odunpazarı/Eskişehir, E-mail: [email protected] 261 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics- Computer, Meşelik Campus, 26480, Odunpazarı/Eskişehir, E-mail: [email protected] 262 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics- Computer, Meşelik Campus, 26480, Odunpazarı/Eskişehir, E-mail: [email protected] 263 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics- Computer, Meşelik Campus, 26480, Odunpazarı/Eskişehir, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On The Polar Taxicab Metric In Three Dimensional Space

Temel ERMİŞ264, Özcan GELİŞGEN265

Abstract

We see that researchers give alternative distance functions of which paths are different from path of Euclidean metric in the distance geometry. Considering distance of air travel or travel over water in terms of Euclidean distance, these travels are made through the interior of spherical Earth which is impossible [5]. In this work, using the idea given in [3], a new alternative metric defined [4] on spherical surfaces due to disadvantage and disharmony of Euclidean distance on earth’s surface. This metric composed of arc length on sphere and length of line segments. Also, this metric which is very much used in navigation and spherical trigonometry will contribute to advancement of logistics and optimal facility location on spherical surfaces.

Key Words: Metric Geometry, Distane Geometry

References

[1] A. Bayar, R. Kaya, On A Taxicab Distance On A Sphere, MJMS,17(1) (2005), 41-51.

[2] H. B. Çolakoğlu and R. Kaya, A Generalization of Some Well-Known Distances and Related Isometries, Math. Commun. Vol. 16 (2011), 21 - 35.

[3] H. G. Park, K. R. Kim, I. S. Ko, B. H. Kim, On Polar Taxicab Geometry In A Plane, J. Appl. Math. & Informatics, 32 (2014), 783-790.

[4] T. Ermiş, Ö. Gelişgen, On An Extension of the Polar Taxicab Distance in Space, ESOGU preprint 2016

[5] J. J. Mwemezi, Y. Haung, Optimal Facilitiy Location On Spherical Surfaces: Algorithm And Application, New York Science Journal, 4(7) (2011), 21-28.

[6] O. Gelisgen,, R. Kaya, Generalization of α-distance to n−dimensional space, KoG. Croat. Soc. Geom. Graph. 10 (2006), 33-35.35-40.

[7] Z. Akca, R. Kaya, On the Distance Formulae In three Dimensional Taxicab Space, Hadronic Journal, 27 (2006), 521-532.

264 Eskişehir Osmangazi University, Faculty of Art and Science, Department of Mathematics and Computer Sciences, Meşelik Campus, 26480, Eskişehir, E-mail: [email protected] 265 Eskişehir Osmangazi University, Faculty of Art and Science, Department of Mathematics and Computer Sciences, Meşelik Campus, 26480, Eskişehir, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Generalized Null Bertrand Curves in semi-Euclidean 4-space

Ali UÇUM266

Abstract

In the present paper, generalized Cartan null Bertrand curves in semi-Euclidean 4-space with index 2 is considered. Because the (1,3)-normal plane of a Cartan null curves is timelike, the (1,3)-Bertrand mate curves of the given curve can be a pseudo null curve, a non-null curve or a Cartan null curve, respectively. Thus, we give the necessary and sufficient conditions for these three cases to be (1,3)-Bertrand curves and we also give the related examples.

Key Words: Generalized Bertrand curve, Semi-Euclidean Space, Cartan null curves.

References

[1] Duggal K. L. and Jin D. H., Null Curves and Hypersurfaces of Semi- Riemannian Manifolds, World Scientic, London, (2007).

[2] Matsuda H. and Yorozu S., Notes on Bertrand curves, Yokohama Math. J., 50 (2003) 41-58.

4 [3] Sakaki M., Null Cartan Curves in R2 , Toyama Mathematical Journal, 32 (2009) 31-39.

[4] Uçum A., Keçilioğlu O. and İlarslan K., Generalized Pseudo Null Bertrand curves in Semi-Euclidean 4- Space with index 2, accepted in Rendiconti del Circolo Matematico di Palermo (2016).

266 Kırıkkale University, Faculty of Art and Science, Department of Mathematics, Kırıkkale. E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Freeness Conditions of Crossed Modules

Tufan Sait KUZPINARI267, Alper ODABAŞ268 , Enver Önder USLU 269

Abstract

Free crossed modules were first defined by Whitehead [1] and in this manner Porter and Arvasi has defined the (totally) free 2-crossed modules of commutative algebras. Some applications for algebraic geometry can be found at [3]. For the algebra case, Arvasi and Porter [2] has used step-by-step constriction which is defined by Andre.

In this study (totally) free 3-crossed modules over commutative algebras of free crossed modules has been defined.

Key Words: Free crossed modules, simplicial objects, category theory

References

[1] J.H.C. Whitehead. Combinatorial Homotopy. Bull. Amer. Math. Soc. 55 (1949), 453-496

[2] Z.Arvasi and T.Porter. Freeness Conditions for 2-Crossed Modules of Commutative Algebras Applied Categorical Structures , 6 , 455-471(1998).

[3] J.G.Ratcliffe. Free and Projective Crossed Modules. J. London Math. Soc. 22 (1980), 66-74.

267 Aksaray University, Faculty of Art and science, Department of Mathematics, 68100,Aksaray, E-mail: [email protected] 268 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics and Computer Science, 26540, Eskişehir, E-mail: [email protected] 269 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics and Computer Science 26540, Eskişehir, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On The Fibered Projective Planes

Süheyla EKMEKÇİ270, Ziya AKÇA271 , Ayşe BAYAR272

Abstract

In this study, the fibered projective plane and the fibered versions of some classical theorems and quadrangle in projective plane are given.

Key Words: Fibered projective plane, quadrangle,

References

[1] Ekmekçi S., Bayar A., A note on fibered quadrangles, Konuralp Journal of Mathematics, 3(2), 185-189.

[2] Bayar A., Ekmekçi S., On the Menelaus and Ceva 6-Figures in the Fibered

Projective Planes, Abstract and Applied Analysis, 2014, 1-5., Doi: 10.1155/2014/803173

[3] Bayar A., Akça Z., Ekmekçi S., A note on fibered projective plane geometry, Information Science, 178, 1257-1262, 2008.

[4] Kuijken L., Van Maldeghem H., Fibered geometries, Discrete Mathematics, 255, 259-274, 2002.

270 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-computer, Meşelik Campus, 26480, Eskişehir, E-mail: [email protected] 271 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-computer, Meşelik Campus, 26480, Eskişehir, E-mail: [email protected] 272 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-computer, Meşelik Campus, 26480, Eskişehir, E-mail: [email protected]

156

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Some Classical Theorems in Intuitionistic Fuzzy Projective Plane

Ayşe BAYAR273, Süheyla EKMEKÇİ274 , Ziya AKÇA275

Abstract

In this work, we introduce that intuitionistic fuzzy versions of some classical configurations in projective plane are valid in intuitionistic fuzzy projective plane with base Desarguesian or Pappian plane.

Key Words: Projective plane, intuitionistic fuzzy projective plane, Desargues and Pappus theorems.

References

[1] Atanassov K. T., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986) 87-96.

[2] Bayar A., Ekmekçi S., Akça Z., A note on fibered projective plane geometry, Information Sciences, 178 (2008) 1257-1262.

[3] Bayar A., Ekmekçi S., On the Menelaus and Ceva 6-figures in the fibered projective planes, Abstract and Applied Analysis, (2014) 1-5.

[4] Çoker D., Demirci M., On intuitionistic fuzzy points, NIFS 1 (1995) 2, 79-84.

[5] Ghassan E. A., Intuitionistic fuzzy projective geometry, J. of Al-Ambar University for Pure Science, 3 (2009) 1-5.

[6] Hughes D. R., Piper F.C., Projective planes, Springer, New York, Heidelberg, Berlin, 1973.

[7] Kuijken L., Van Maldeghem H., Fibered geometries, Discrete Mathematics 255 (2002) 259-274.

[8] Turanlı N., An overview of intuitionistic fuzzy supratopological spaces, Hacettepe Journal of Mathematics and Statistics, 32(2003)-(17-26).

[9] Zadeh L., Fuzzy sets, Inform. Control, 8 (1965) 338-358.

273 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-Computer, Meşelik Campus, 26480, Odunpazarı/Eskişehir, E-mail: akorkmaz @ogu.edu.tr 274 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-Computer, Meşelik Campus, 26480, Odunpazarı/Eskişehir, E-mail: [email protected] 275 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-Computer, Meşelik Campus, 26480, Odunpazarı/Eskişehir, E-mail: zakca @ogu.edu.tr

157

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A Computer Search for some Subplanes of Projective Plane Coordinatized a Left Nearfield

Ziya AKÇA276, Ayşe BAYAR277 , Süheyla EKMEKÇİ278

Abstract

In this work, we introduce some subplanes of the left near field projective plane of order 9 which is coordinatized as homogenous. We give an algorithm for checking subplanes of order 2 of this projective plane and apply the algorithm (implemented in C#) to determine and classify Fano subplanes.

Key Words: Near field, Projective plane, Fano plane

References

[1] Akça Z., Günaltılı İ., Güney Ö., On the Fano subplanes of the left semifield plane of order 9. Hacet. J. Math. Stat. 35 (2006), no. 1, 55--61 [2] Akpınar A., On some projective planes of finite order, G.U. Journal of Science 18 (2) (2005) 315-325. [3] Çalıskan C., Moorhouse C., Eric G., Subplanes of order 3 in Hughes planes, The Electronic Journal of Combinatorics 18 (2011). [4] Çiftçi S., Kaya R., On the Fano Subplanes in the Translation Plane of order 9, Doğa-Tr. J. of Mathematics 14 (1990), 1-7. [5] Hall M., The theory of groups , New York: Macmillan (1959). [6] Hall M., Swift Jr, J.D., Killgrove R., On projective planes of order nine, Math. Tables and Other Aids Comp. 13 (1959) 233-246. [7] Room T.G., Kirkpatrick P.B., Miniquaternion Geometry, London, Cambridge University Press, 177, (1971). [8] Stevenson F.W., Projective Planes, W. H. Freeman and Company, San Francisco, 416 (1972). [9] Veblen O., Wedderburn J.H.M., Non-Desarguesian and non-Pascalian geometries, Trans. Amer. Math. Soc. 8 (1907), 379--388

276 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-Computer, Meşelik Campus, 26480, Odunpazarı/Eskişehir, E-mail: [email protected] 277 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-Computer, Meşelik Campus, 26480, Odunpazarı/Eskişehir, E-mail: [email protected] 278 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-Computer, Meşelik Campus, 26480, Odunpazarı/Eskişehir, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

The Dual Euler Parameters in Dual Lorentzian Space

Buşra AKTAŞ279 and Halit GÜNDOĞAN280

Abstract

In this paper, by using dual Lorentzian matrix multiplication, Cayley formula and Euler parameters of dual lorentz ortohogonal matrix are obtained in Dual Lorentzian space.

Then, dual lorentz rodrigues parameters are obtained by using dual lorentzian matrix multiplication in Dual Lorentzian space.

Key Words: Dual Lorentz Rodrigues Parameter, Cayley Formula, Euler Parameter

References

[1] H. Gundogan and O. Keçilioğlu, Lorentzian Matrix Multiplication and The Motion on Lorentzian Plane, Glasnik Matematicki, Vol. 41(61)

[2] S. Ozkaldı and H. Gundogan, Cayley Formula, Euler parameters and Rotations in 3-Dimensional Lorentzian Space, Advances in Applied Clifford Algebras, 20(2010), 367-377

[3] I. Karakılıc, The Dual Rodrigues Parameters, International Journal of Engineering and Applied Sciences, Vol. 2, Issue 2(2010), 23-32

[4] J. M. McCarthy, An Introduction to Theoretical Kinematics, The MIT Press, Cambridge, Massachusetts, London, England, 1990

[5] A. Dagdeviren, Properties of Lorentz Matrix Multiplication and Dual Matrices, Istanbul, 2013

279 Kırıkkale University, Faculty of Art and science, Department of Mathematics, Yahsihan Campus, 71450, Yahsihan/Kırıkkale, E-mail: [email protected] 280 Kırıkkale University, Faculty of Art and science, Department of Mathematics, Yahsihan Campus, 71450, Yahsihan/Kırıkkale, E-mail: [email protected]

159

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

The Tangent Operator in Lorentzian Space

Olgun DURMAZ281 and Halit GÜNDOĞAN282

Abstract

In this paper, by using Lorentzian matrix multiplication, L-tangent operator is obtained in Lorentzian space. L-tangent operator of L-SO(3), L-H(3), L-H(4) is studied in Lorentzian space. L-tangent operators are related to vectors.

Key Words: L-tangent operator, L-SO(3), L-H(3), L-H(4).

References

[1] B. O' Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, Inc, New York, 1983.

[2] R. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer-Verlag, New York, 1994.

[3] S. Ozkaldı, H. Gundogan, Cayley Formula, Euler Parameters and Rotations in 3-Dimensional Lorentzian Space, Advances in Applied Clifford Algebras 20(2010), 367-377.

[4] H. Gundogan and O. Kecilioglu, Lorentzian Matrix Multiplication and the Motions on Lorentzian Plane, Glasnik Matematikci Vol. 41 (61)(2006), 329-334.

[5] O. Kecilioglu, S. Ozkaldı and H. Gundogan, Rotations and Screw Motion with Timelike Vector in 3- Dimensional Lorentzian Space, Advances in Applied Clifford Algebras 22(2012), 1081-1091.

[6] J.M. McCarthy, An İntroduction to Theoretical Kinematics, The MIT Press, Cambridge, Massachusetts, London, England, 1990.

[7] A. Dagdeviren, Properties of Lorentz Matrix Multiplication and Dual Matrices, Istanbul, 2013.

281 Kırıkkale University, Faculty of Art and science, Department of Mathematics, Yahşihan Campus, 71450, Yahşihan/Kırıkkale, E-mail:[email protected] 282 Kırıkkale University, Faculty of Art and science, Department of Mathematics, Yahşihan Campus, 71450, Yahşihan/Kırıkkale, E-mail: : [email protected]

160

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Surfaces endowed with canonical principal direction in Minkowski 3-space

Alev KELLECİ283, Nurettin Cenk TURGAY284 , Mahmut ERGÜT 285

Abstract

A submanifold M in Minkowski space is said to be a surface endowed with canonical principal direction, if the angle function, 휃 between the fixed direction, k and the unit normal vector of M, N is not constant. In this talk, we will present a short survey on constant angle surfaces and surfaces endowed with canonical principal direction in semi-Euclidean spaces. We give a new classification for space-like surfaces endowed with canonical principal direction in Minkowski 3-space. In this direction, we will classify constant 3 angle surfaces in E 1 whose unit normal vector field makes a constant hyperbolic angle with a fixed timelike vectors. Also, we present some examples of these surfaces. Therefore, we complete the classification of both space-like surfaces endowed with canonical principal direction and constant angle surfaces in Minkowski 3- space.

Key Words: Space-like surface, Constant angle surface, Canonical principal direction, Minkowski space

References

[1] Lopez R., Munteanu M. I, Constant angle surfaces in Minkowski space, Bulletin of Belgian Mathematical Society Simon Stevin 18 (2011), 271-286. [2] Nistor A. I, A note on spacelike surfaces in Minkowski 3-space, Filomat 27:5 (2003), 843-849. [3] Dillen F., Fastenakels J., Veken J. Van der, Surfaces in S2 × R with a canonical principal direction, Annals of Global Analysis and Geometry 35 (2009) 381–396. [4] Dillen F., Munteanu M. I., Nistor A. I., Canonical coordinates and principal directions for surfaces in H2 × R, Taiwanese Journal of Mathematics 15 (2011) 2265–2289. [5] Dillen F., Fastenakels J., Veken J. Van der and Vrancken L.., Constant Angle Surfaces in S2 × R, Monaths. Math., 152(2) (2007), 89–96. [6] Dillen F., Munteanu M. I., Constant Angle Surfaces in H2 × R, Bull Braz Math Soc, New Series 40(1), 85-97 2009, Sociedade Brasileira de Matemática.

283 Firat University, Faculty of Science, Department of Mathematics, 23200, Merkez/Elazig Turkey, E- mail: [email protected] 284 Istanbul Technical University, Faculty of Science and Letters, Department of Mathematics, 34469, Maslak/Istanbul Turkey, E-mail: [email protected] 285 Namik Kemal University, Faculty of Science and Letters, Department of Mathematics, 59030, Merkez/Tekirdag Turkey, E-mail: [email protected]

161

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A Study On The Elastic Curves

Gülşah AYDIN ŞEKERCİ286 , Sibel SEVİNÇ287, A. Ceylan ÇÖKEN 288

Abstract

An elastic curve bound to a surface is reflect the geometry of its environment. The behavior of the elastic curve by the help of the tangent vectors of the curve and the normal vector of the surface can be determine. In this study, we give a generalization of the bending energy for the curve on the surface. For this, we use Frenet and Darboux frames. According to the causal character of the curve and the surface, we research Euler-Lagrange equation describing the equilibrium states of the curve with this energy.

Key Words: Elastic curves, semi-Riemannian manifold, Frenet frame, Darboux frame, Euler Lagrange equation.

References

[1] Duggal K. and Bejancu A., Lightlike submanifolds of semi- Riemannian manifolds and applications, Kluwer Academic Publishers, The Netherlands, 1996.

[2] Duggal K. and Jin D. H., Null curves and hypersurfaces of semi- Riemannian manifolds, World Scientific Publishing Co. Pte. Ltd., Singapore, 2007.

[3] Guven J., Valencia D. M. and Montejo P. V., Environmental bias and elastic curves on surfaces, arXiv: 1405.7387v2 (2014).

[4] Manning G. S., Relaxed elastic line on a curved surface, Quarterly of Applied Mathematics 45 (1987), 515- 527.

[5] Nickerson H. K. and Manning G. S., Intrinsic equations for a relaxed elastic line on a oriented surface, Geometriae Dedicata 27 (1988), 127- 136.

[6] Singer D. A., Lectures on elastic curves and rods, AIP Conf. Proc. 1002 (2008) 3.

286 Süleyman Demirel University, Faculty of Arts and Science, Department of Mathematics, 32000, Isparta, E-mail: [email protected] 287 Cumhuriyet University, Faculty of Science, Department of Mathematics, 58000, Sivas, E-mail: [email protected] 288 Akdeniz University, Faculty of Science, Department of Mathematics, 07000, Antalya, E-mail: [email protected]

162

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Some Results About Harmonic Curves On Lorentzian Manifolds

Sibel SEVİNÇ289, Gülşah AYDIN ŞEKERCİ290 , A. Ceylan ÇÖKEN 291

Abstract

Key Words: Harmonic curves, transversal harmonic curves, harmonic surfaces, Lorentzian manifold.

References

[1] Duggal K. L., Bejancu A., Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers, 346, 1996.

[2] Duggal, K. L., Jin D. H., Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific Publishing, 2007.

[3] Kılıç B., -Harmonic Curves and Surfaces in Euclidean Space, Commun. Fac. Sci. Univ. Ank. Series A1. 54(2) (2005), 13-20.

[5] Matea S., K-Harmonic Curves into a Riemannian Manifold with Constant Sectional Curvature, arXiv: 1005.1393v2 [math.DG]8Jun2010.

289 Cumhuriyet University, Faculty of Science, Department of Mathematics, 58000, Sivas, E-mail: [email protected] 290 Süleyman Demirel University, Faculty of Arts and Science, Department of Mathematics, 32000, Isparta, E-mail: [email protected] 291 Akdeniz University, Faculty of Science, Department of Mathematics, 07000, Antalya, E-mail: [email protected]

163

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

4 On Spherical Indicatries Of Partially Null Curves In R2

Ümit Ziya SAVCI292, Süha YILMAZ293

Abstract

4 In this study, we investigate spherical indicatrix of partially null curves in Semi-Riemann space R2 . First, we calculate Frenet apparatus of tangent, normal, first and second binormal indicatrices. Second, we devote to some special curves of spherical indicatrices. In this situation, we obtained some interesting result.

Key Words: Semi Euclidean space, spherical indicatrices, patially null curves, timelike curves, spacelike curves

References

[1] Hacısalihoğlu H. H., Diferensiyel Geometri, Inönü Unv Fen Edebiyat Fak. Yayınları, 1983.

[2] O’Neill B., Semi-Riemannian Geometry, Adacemic Press, New York, 1983.

[3] Petrovic–Torgasev M., Ilarslan K. And Nesovic E., On partially null and pseudo null curves in the semi- 4 euclidean space R2 , J. Geom., Vol. 84(2005), 106-116.

[4] Yılmaz S., Spherical Indicatrix of Curves and Characterization of Some Special Curves Four Dimensional Lorentzian Space L⁴, PhD, Dokuz Eylül University, İzmir, Turkey, 2001.

[5]Yilmaz S. and Turgut M., On Frenet apparatus of partially null curves in semi-Euclidean space, Scientia Magna, Vol. 4(2008), 39-44.

292 Celal Bayar University, Department of Mathematics Education , 45900, Manisa-Turkey. E-mail: [email protected] 293 Dokuz Eylül University, Buca Educational Faculty, 35150, Buca-Izmir, Turkey. E-mail: [email protected]

164

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

The New Frame Approach For Spatial Curves

Çağla RAMİS294, Yusuf YAYLI295

Abstract

In this study, we revise the orthonormal frame called alternative frame for space curves [3]. Moreover, the new curve characterizations are given for some special space curves and generalized the casual ones.

Key Words: Frenet-Serret formula, Bishop’s frame, Slant helix, C-Slant Helix, Magnetic curve

References

[1] O.Neill B., Elementary Differential Geometry, Academic Press, New York, (1966).

[2] Özdemir Z. B., Gök İ., Yaylı Y., Ekmekci F. N., A new approach for magnetic curves in 3D Riemannian manifolds, Journal of Mathematical Physics (2014), 1-12.

[3] Uzunoğlu B., Ramis Ç., Yaylı Y., On Curves of Nk–Slant Helix and Nk –Constant Precession in Minkowski 3–Space, Journal of Dynamical Systems and Geometric Theories, Vol. 12 (2014), 175-189.

294 Ankara University, Faculty of Art and Science, Department of Mathematics, Tandogan Campus, 06100, Tandogan/Ankara, E-mail: [email protected] 295 Ankara University, Faculty of Art and Science, Department of Mathematics, Tandogan Campus, 06100, Tandogan/Ankara, E-mail: [email protected]

165

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Curvatures of Surfaces via Quaretnions in Minkowski Space

Muhammed Talat SARIAYDIN296, Talat KÖRPINAR297 , Vedat ASİL 298

Abstract

In this paper, we study Gaussian and Mean curvatures of the Bisector Ruled Surface via split quaternions in Minkowski 3-Space. Then, we firstly give derivatives of the bisector surface in E₁3. Then, we obtain curvature of the this surface generated by point-curve via split quaternions in E₁³.

Key Words: Bisector Surface, Minkowski Space, Ruled Surface, Curvatures.

References

[1] Cigliola A., Split Quaternions, Generalized Quaternions and Integer-Valued Polynomials, Universit a Degli Studi Roma Tre, PhD Thesis in Mathematics, 2014.

[2] Elber G., Kim M.S., A Computational Model for Nonrational Bisector Surfaces: Curve-Surface and Surface-Surface Bisectors, (2000), 364-372.

[3] Hanson A.J., Quaternion Gauss Maps and Optimal framings of Curves and Surfaces, Technical Report No:518, Indiana University, 1998.

[4] Jirapong K., Krawczyk R.J., Seashell Architectures, ISAMA, Bridges Conference, 2003.

[5] Körpinar T., Asil V., New Effect for Faraday Tensor for Biharmonic Particles in Heisenberg Spacetime, International Journal of Theoretical Physics, 54(5) (2015), 1545-1552.

[6] O'Neill B., Semi Riemannian Geometry, Academic Press, New York, 1983.

[7] Ozdemir M. and Ergin A.A., The Roots of a Split Quaternion, Applied Mathematics Letters, 22 (2009), 258-263.

[8] Sarıaydın M.T., Characterization of Some Quaternionic Surface in Minkowski 3-Space, Fırat University, PhD Thesis.

296 Muş Alparslan University, Faculty of Art and Science, Department of Mathematics, 49250, Muş/Turkey, E-mail: [email protected] 297 Muş Alparslan University, Faculty of Art and Science, Department of Mathematics, 49250, Muş/Turkey, E-mail: [email protected] 298 Fırat University, Faculty of Science, Department of Mathematics, 23119, Elazığ/Turkey, E-mail: [email protected]

166

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Weierstrass Representation Formula In Bianchi Type-I Spacetime

Talat KÖRPINAR299, Gülden ALTAY 300 , Handan ÖZTEKİN301, Mahmut ERGÜT 302

Abstract

In this paper, we study Weierstrass representation formula with Hubble parameter in Bianchi Type-I Spacetime. Therefore, we construct a new characterization for surfaces in Bianchi Type-I Spacetime.

Key Words: Bianchi Type-I Spacetime., Weierstrass representation, Hubble parameter

References

[1] Einstein A., Relativity: The Special and General Theory, New York: Henry Holt, 1920

[2] Kenmotsu K., Weierstrass Formula for Surfaces of Prescribed Mean Curvature, Math. Ann. 245 (1979), 89-99

[3] O'Neill B., Semi-Riemannian Geometry, Academic Press, New York, 1983

[4] Körpınar T., Turhan E., Bianchi Type-I Cosmological Models for Biharmonic Particles and its Transformations in Spacetime, Int. J. Theor. Phys. 54 (2015), 664-671

[5] Pradhan A., Anisotropic Bianchi Type-I Magnetized String Cosmological Models with Decaying Vacuum Energy Density, Commun. Theor. Phys. 55 (2011), 931-941

299 Muş Alparslan University, Faculty of Art and Science, Department of Mathematics, 49250, Muş/Turkey, E-mail: [email protected] 300 Fırat University, Faculty of Science, Department of Mathematics, 23119, Elazığ/Turkey, E-mail: [email protected] 301 Fırat University, Faculty of Science, Department of Mathematics, 23119, Elazığ/Turkey, E-mail: handanoztekin@@gmail.com 302 Namık Kemal University, Faculty of Art and Science, Department of Mathematics, 59030, Tekirdağ/Turkey, E-mail: [email protected]

167

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Metric n-Hyperplanes of Euclidean and Hyperbolic Geometry

Oğuzhan DEMİREL303

Abstract

The lines of Euclidean and hyperbolic geometries are characterized by W. Benz (Monatsh Math 141:1–10, 2004) as metric lines in the sense of Blumenthal and Menger (Studies in Geometry. San Francisco: Freeman, 1970). Inspired by the work of W.Benz, we extend the notion of metric lines to metric n-hyperplanes and characterize the hyperplanes of Euclidean geometries as metric hyperplanes. Moreover, we see that there do not exist metric n-hyperplanes (n≥ 2) in hyperbolic geometry.

Key Words: Metric spaces, functional equations of metric and their solutions, hyperbolic geometry,

References

[1] Blumenthal, L.M., Menger, K., Studies in Geometry. Freeman, San Francisco (1970).

[2] Benz, W., Metric and periodic lines in real inner product space geometries, Monatsh.

Math. 141(2004), 1–10.

[3] Ungar, A. A., Analytic Hyperbolic Geometry: Mathematical Foundations and Appli-cations, Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., (2005)

[4] Demirel, O., Seyrantepe, E.S, Sönmez, N., Metric and periodic lines in the Poincaré ball model of hyperbolic geometry, Bull Iran. Math. Soc. 38(2012), 805–815.

[5] Demirel, O., Seyrantepe, E.S.: The cogyrolines of Möbius gyrovector spaces are metric but not periodic, Aequationes Math., 85(2013), 185–200.

[6] Demirel, O., A new proof of the nonexistence of isometries between higher dimensional Euclidean and hyperbolic space, Aequationes Math., 89(2015), 1449–1459.

303 Afyon Kocatepe University, Faculty of Science and Literaure, Department of Mathematics, Ahmet Necdet SEZER Campus, 03200 , Afyonkarahisar, E-mail: [email protected]

168

14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

On Golden Riemannian Tangent Bundles with C-G Metric

Ahmet KAZAN304, H.Bayram KARADAĞ305

Abstract

In this study, we define the metallic structure 퐽̃ on the tangent bundle 푇푀 and give a condition for integrability of 퐽̃ by using the Nijenhuis tensor field 푁퐽̃ . We find a condition for the Cheeger-Gromoll (C-G) metric 푔̃ to be a pure metric with respect to the metallic structure 퐽̃. Also, we investigate the condition for (푇푀, 퐽̃, 푔̃) to be locally decomposable golden Riemannian tangent bundle and give a theorem for it.

Key Words: Tangent bundle, Metallic structure, Golden structure, Cheeger-Gromoll, , Pure metric.

References

[1] Cheeger, J. and Gromoll, D., On the structure of complete manifolds of non-negative curvature, Ann. of Math., 96 (1972), 413-443.

[2] Gezer, A., Cengiz, N., and Salimov, A., On integrability of golden riemannian structures, Turk J Math., 37 (2013), 693-703.

[3] Gudmundsson, S. and Kappos, E., On the geometry of tangent bundles, Expo. Math., 20 (2002), 1-41.

[4] Hretcanu, C.-E. and Crasmareanu, M., Metallic structures on riemannian manifolds, Revista De La Union Matematica Argentina, 54 (2013), No. 2, 15-27.

[5] Kowalski, O., Curvature of the induced riemannian metric of the tangent bundle of riemannian manifold, J. Reine Angew. Math., 250 (1971), 124-129.

304 İnönü University, Sürgü School of Higher Education, Department of Computer Technologies, Malatya/Turkey E-mail: [email protected] 305 İnönü University, Faculty of Sciences and Arts, Department of Mathematics, Malatya/Turkey E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Pseudosymmetric Lightlike Hypersurfaces in indefinite Sasakian Space Forms*

Sema KAZAN306, Bayram ŞAHİN307

Abstract

We study pseudosymmetric lightlike hypersurfaces of an indefinite Sasakian space form, tangent to the structure vector field. We obtain sufficient conditions for a lightlike hypersurface to be pseudosymmetric in an indefinite Sasakian space form. Later, we give sufficient conditions for a lightlike hypersurface to be pseudoparallel and Ricci-pseudosymmetric in the indefinite Sasakian space form. We also find certain conditions for a pseudosymmetric lightlike hypersurface of an indefinite Sasakian space form to be totally geodesic and check the effect of Weyl projective pseudosymmetry conditions on the geometry of a lightlike hypersurface of an indefinite Sasakian space form.

Key Words: Pseudosymmetric lightlike hypersurface, pseudoparallel lightlike hypersurface, indefinite Sasakian space form.

References

[1] Adamow A., and Deszcz, R., On totally umbilical submanifolds of some class of Riemannian manifolds, Demonstratio Math, 16 (1983), 39-59.

[2] Arslan, K., Çelik, Y., Deszcz R. and Ezentaş, R., On the equivalence of Ricci-semisymmetry and semisymmetry, Colloquium Mathematicum, 76 (1998) (2), 279-294.

[3] K.L. Duggal and B. Şahin, Differential Geometry of Lightlike Submanifolds, Birkhäuser Verlag AG, 2010.

[4] F. Massamba, Semi-parallel lightlike hypersurfaces of indefinite Sasakian manifolds, Int. J. Contemp. Math. Sciences, 3 (2008) (13), 629-634.

*This talk was published as Kazan, Sema; Şahin, Bayram, Pseudosymmetric Lightlike Hypersurfaces in indefinite Sasakian Space Forms. Journal of Applied Analysis and Computation. Volume 6, Number 3, August 2016, 699-719.

306 İnönü University, Faculty of Art and science, Department of Mathematics, Campus, 44280, Malatya, E-mail: [email protected] 307 İnönü University, Faculty of Art and science, Department of Mathematics, Campus, 44280, Malatya, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Determination of the curves of constant breadth according to Bishop Frame

in Euclidean 3-space by a Galerkin-like method

Şuayip YÜZBAŞI308, Murat KARAÇAYIR309 , Mehmet SEZER310

Abstract

In Euclidean 3-space, curves of constant breadth according to the Bishop Frame are characterized by a first order linear differential equation system with three unknown functions. In this study, by using a scheme reminiscent of the Galerkin method, we obtain approximate solutions of this system. Using a technique known as residual correction, we then estimate the errors of our approximate solutions and use these estimations to improve the accuracy of the already obtained solutions. In order to investigate the efficiency of the proposed scheme, we consider an example problem and present the results.

Keywords: Bishop frame, curves of constant breadth, linear differential equation systems, a Galerkin- like method.

References

[1] Çetin M., Sezer M., Kocayiğit H., Determination of the curves of constant breadth according to Bishop Frame in Euclidean 3-space, New Trends in Mathematical Sciences 2015(3): 18-34.

[2] Köse Ö., On space curves of constant breadth, Doğa Tr. J. Math 1986 10(1) : 11-14.

[3] Çelik, İ., Collocation Method and Residual Correction Using Chebyshev Series, Applied Mathematics and Computation, 2006 174(2): 910-920.

308 Akdeniz University, Faculty of Science, Department of Mathematics, Campus, Tr-07058, Antalya. E-mail: [email protected] 309 Akdeniz University, Faculty of Science, Department of Mathematics, Campus, Tr- 07058, Antalya. E-mail: [email protected] 310 Celal Bayar University, Faculty of Arts and Science, Department of Mathematics, Tr-45000, Manisa. E-mail: [email protected]

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A Laguerre method to determinate the curves of constant breadth according to

Bishop Frame in Euclidean 3-space

Şuayip YÜZBAŞI311, Mehmet SEZER 312 , Esra SEZER313

Abstract

In this study, we consider characterizing curves of constant breadth according to Bishop frame in Euclidean 3-space. This characterizing corresponds to system of differential equations with variable coefficients. Our aim is to give a collocation method based on Laguerre polynomials to determine curves of constant breadth according to Bishop frame in Euclidean 3-space. The method is reduced to orginal problem to a system of algebraic eqautions. We present an error estimation technique by using residual function. To explain he method on the considered problem, we apply to numerical exapmles.

Key Words: Curves of constant breadth, Bishop frame, Laguerre polynomials, collocation points; system of differential Equations.

References

[1] Köse Ö., On space curves of constant breadth, Doğa Tr. J. Math 1986 10(1) :11-14.

[2] Çetin M., Sezer M., Kocayiğit H., Determination of the curves of constant breadth according to Bishop Frame in Euclidean 3-space, New Trends in Mathematical Sciences 2015(3):18-34.

[3] Çelik, İ., Collocation Method and Residual Correction Using Chebyshev Series, Applied Mathematics and Computation, 2006, 174(2), 910-920.

[4] Yüzbaşi Ş., Laguerre approach for Solving pantograph-type Volterra integro-differential equations, Applied Mathematics and Computation, 2014, 232,1183-1199.

311 Akdeniz University, Faculty of Science, Department of Mathematics, Campus, Tr-07058, Antalya. E-mail: [email protected] 312 Celal Bayar University, Faculty of Arts and Science, Department of Mathematics, Tr-45000, Manisa. E-mail: [email protected] 313 Akdeniz University, Faculty of Arts and Science, Department of Mathematics, Tr-07058, Antalya E-mail: [email protected]

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Anet Parallel Surfaces in Heisenberg Group

Gülden ALTAY314, Talat KÖRPINAR 315 , Mahmut ERGÜT 316

Abstract

In this paper, we study Anet parallel surfaces in three dimensional Heisenberg group. We obtain some characterizations of these surfaces.

Key Words: Heisenberg Group, Anet surface, parallel surface.

References

[1] Bobenko A., Either U., Bonnet Surfaces and Painleve Equations, J. Reine Angew Math., 499 (1998), 47- 79. [2] Kanbay F., Bonnet Ruled Surfaces, Acta Mathematica Sinica, English Series, 21 (2005), 623- 630. [3] Körpınar T., Turhan E., Parallel Surfaces to Normal Ruled Surfaces of General Helices in the Sol Space Sol³, Bol. Soc. Paran. Mat., 2 (2013), 245-253. [4] Soyuçok Z., The Problem of Non- Trivial Isometries of Surfaces Preserving Principal Curvatures, Journal of Geometry, 52 (1995), 173- 188. [5] Turhan E., Altay G., Minimal surfaces in three dimensional Lorentzian Heisenberg group, Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 55 (2014),1- 23. [6] Ünlütürk Y., Ekici C., Parallel Surfaces of Spacelike Ruled Weingarten Surfaces in Minkowski 3-space, New Trends in Mathematics, 1 (2013), 85-92.

314 Fırat University, Faculty of Science, Department of Mathematics, 23119, Elazığ/Turkey, E-mail: [email protected] 315 Muş Alparslan University, Faculty of Art and Science, Department of Mathematics, 49250, Muş/Turkey, E-mail: [email protected] 316 Namık Kemal University, Faculty of Art and Science, Department of Mathematics, 59030, Tekirdağ/Turkey, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

New Characterization of Involute Curves in Universal Covering Group

Handan ÖZTEKİN 317, Talat KÖRPINAR318 , Gülden ALTAY 319, Mahmut ERGÜT 320

Abstract

In this paper, we characterize involute curves in the universal covering group of E(2) with Riemannian metric. Finally, we obtain a new parametric equation for this curves in the universal covering group of E(2).

Key Words: Universal covering group, Helices, Involute curves

References

[1] Backes E., Reckziegel H, On symmetric submanifolds of spaces of constant curvature, Math. Ann. 263 (1983), 419-433.

[2] Cook T.A, The curves of life, Constable, London 1914, Reprinted (Dover, London 1979).

[3] Inoguchi J., Van der Veken J., Parallel surfaces in the motion groups E(1,1) and E(2), Bull. Belg. Math. Soc. Simon Stevin 14 (2007), 321--332.

[4] Milnor J., Curvatures of Left-Invariant Metrics on Lie Groups, Advances in Mathematics 21 (1976), 293- 329.

[5] Struik DJ, Lectures on Classical Differential Geometry, New York: Dover, 1988.

317 Fırat University, Faculty of Science, Department of Mathematics, 23119, Elazığ/Turkey, E-mail: handanoztekin@@gmail.com 318 Muş Alparslan University, Faculty of Art and Science, Department of Mathematics, 49250, Muş/Turkey, E-mail: [email protected] 319 Fırat University, Faculty of Science, Department of Mathematics, 23119, Elazığ/Turkey, E-mail: [email protected] 320 Namık Kemal University, Faculty of Art and Science, Department of Mathematics, 59030, Tekirdağ/Turkey, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Normal Section Curves on Semi-Riemannian Manifolds

Feyza Esra ERDOĞAN321, Selcen YÜKSEL PERKTAŞ322

Abstract

Key Words: Semi Riemann Manifold, Null Curve, Normal Section Curve, Curvature, Planar Normal Section.

References

[1 Blomstrom C., Planar geodesic immersions in pseudo-Euclidean Space, Math.Ann. 274(1986),585-589.

[2] Chen B.Y., Geometry of Submanifolds. Pure and Apllied Mathematics, No.22, Marcell Dekker.,Inc., New York, (1973).

[3] Chen B.Y., Submanifolds with planar normal sections, Soochow J. Math. 7(1981),19-24.

[4] Chen B.Y., Differential geometry of submanifolds with planar normal sections, Ann. Mat. Pura Appl.130 (1982), 59-66.

[5] Chen B.Y., S. J. Li, Classification of surfaces with pointwise planar normal sections and its application to Fomenko's conjecture, J.Geom. 26 (1986), 21-34.

[6] Chen B.Y., Classification of surfaces with planar normal sections, J. Of Geometry 20 (1983), 122- 127.

[7] Chen B.Y., P. Verheyen. Submanifolds with geodesic normal sections, Math.Ann.269 (1984) 417-429.

[8] Hong Y., On submanifolds With planar normal Sections, Mich. Math. J. 32 (1985), 203-210.

[9] Kim Y.H., Surfaces in a pseudo-Euclidean space with planar normal sections, J. Geom. 35(1989).

321 Adıyaman University, Faculty of Education, Department of Elementary Education, Adıyaman, E-mail: [email protected] 322 Adıyaman University, Faculty of Arts and Sciences, Department of Mathematics, Adıyaman, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

LS(2,D)− Equivalence Conditions of Dual Control Points in D2

Muhsin İNCESU323

Abstract

In this study we studied the equivalence conditions of compared two different control point systems in planar dual space D2 under the linear similarity transformations LS(2,D) according to the invariant system of these control points system.

Key Words: Linear Similarity, Equivalence conditions, Dual Planar Control Points

References

[1] Dj.Khadjiev, Some Questions in the Theory of Vector Invariants, Math. USSR- Sbornic, 1(3), 383-396 (1967).

[2] Grosshans F., Obsevable Groups and Hilbert’s Problem, American Journal of Math., 95, 229-253 (1973).

[3] H. Weyl, The Classical Groups, Their Invariants and Representations, 2nd ed., with suppl., Princeton, Princeton University Press, 1946.

[4] Dj. Khadjiev , An Application of the Invariant Theory to the Differential Geometry of Curves, Fan, Tashkent, 1988. ( in Russian )

[5] F. Klein, A comperative review of recent researches in geometry (translated by Dr. M.W. Haskell), Bulletin of the New York Mathematical Society, 2, 215-249 (1893).

[6] M.Incesu, The Complete System of Point Invariants in the Similarity Geometry, Phd. Thesis, Karadeniz Technical University, Trabzon, 2008.

[7 ]Muhsin Incesu, Osman Gürsoy, Djavvat Khadjiev, On The First Fundamental Theorem for Dual Orthogonal Group O(2, D), 1st International Eurasian Conference on Mathematical Sciences and Applications (IECMSA), September 03-07, 2012, Prishtine, KOSOVO

[8] Incesu, M. Gürsoy O., On The First Fundamental Theorem for Special Dual Orthogonal Group SO(2, D) and its Application to Dual Bezier Curves, First International Conference on Analysis and Applied Mathematics (ICAAM 2012), October 18-21, 2012 , Gumushane, Turkey.

[9] Incesu, M., Gürsoy, O., On The Orthogonal Invariants of Dual Planar Bezier Curves, 2nd International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2013), 26-29 August 2013,Sarajevo, Bosnia and Herzegovina.

323 Mus Alparslan University Education Faculty Department of Mathematics,49100, Mus, Turkey, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Timelike Directional Tubular Surfaces

Mustafa DEDE324, Hatice TOZAK325 , Cumali EKİCİ 326

Abstract

In this paper, we introduce a new version of the timelike tubular surfaces. We first define a new adapted frame along a spacelike space curve and denote this frame as the q-frame. We then reveal the relationship between the Frenet frame and the q-frame. Finally, we give a parametric representation of a timelike directional tubular surface using the q-frame.

Key Words: Frenet frame, timelike pipe surface, tube, adapted frame.

References

[1] Bishop, R.L., There is more than one way to frame a curve. Am. Math. Mon. 82(1975), 246-251. [2] Klok, F., Two moving coordinate frames for sweeping along a 3D trajectory. Comput. Aided Geom. Des. 3(1986), 217-229. [3] Wang, W., Jüttler, B., Zheng, D., Liu, Y., Computation of rotation minimizing frames. ACM Trans. Graph. 27(1) (2008), 1-18. [4] Guggenheimer, H., Computing frames along a trajectory. Comput. Aided Geom. Des. 6(1989), 77-78. [5] Shin, H., Yoo, S. K., Cho, S. K., Chung, W. H., Directional Offset of a Spatial Curve for Practical Engineering Design, ICCSA, 3(2003), 711-720. [6] Lü, W. and Pottmann, H., Pipe surfaces with rational spine curve are rational, Computer Aided Geometric Design, 13(1996), 621-628. [7] Wang, W. and Joe, B., Robust computation of the rotation minimizing frame for sweep surface modelling. Comput. Aided Des., (29) (1997), 379 391. [8] Xu, Z., Feng, R., Sun, J., Analytic and Algebraic Properties of Canal Surfaces, Journal of Computational and Applied Mathematics, 195(2006), 220-228. [9] Maekawa, T., Patrikalakis, N.M., Sakkalis, T., Yu, G., Analysis and applications of pipe surfaces, Comput. Aided Geom. Design, 15(1988), 437-458. [10] Dogan, F. and Yaylı, Y., Tubes with Darboux Frame, Int. J. Contemp. Math. Sciences, 7(2012), 751-758. [11] Dede, M., Tubular surfaces in Galilean space, Math. Commun., 18(2013), 209-217. [12] Bloomenthal, J., Calculation of reference frames along a space curve, Graphics gems, Academic Press Professional, Inc., San Diego, CA, 1990.

324 Kilis 7 Aralık University, Faculty of Art and science, Department of Mathematics, 79000, Kilis, E-mail: [email protected] 325 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-Computer, Meşelik Campus, 26480, Eskişehir, E-mail: [email protected] 326 Eskişehir Osmangazi University, Faculty of Art and science, Department of Mathematics-Computer, Meşelik Campus, 26480, Eskişehir, E-mail: [email protected]

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[13] Xu, G. Hui, K. Ge, W. Wang, G., Direct manipulation of free-form deformation using curve-pairs. Computer-Aided Design 45 (3) (2013), 605-614. [14] Yılmaz S. and Turgut, M., A new version of Bishop frame and an application to spherical images, J. Math. Anal. Appl., 371(2010), 764-776. [15] Dede M., Ekici C., Tozak H., Directional Tubular Surfaces, International Journal of Algebra, Vol. 9(2015), 527 – 535. [16] Kızıltuğ S., Çakmak A., Kaya S., Timelike tubes around a spacelike curve with Darboux Frame of 3 Weingarten Type 퐸1 , International Journal of Pyhsics and Mathematical Sciences Vol 4(2013), 9-17. [17] Abdel-Aziz H. S. and Saad M. K., Weingarten timelike Tube surfaces around a spacelike curve, Int. Journal of Math. Analysis, Vol 5(2011), 1225-1236. [18] Kızıltuğ S., Çakmak A., Developable ruled surface with Darboux Frame in Minkowski 3-space, Life Science Journal, 10(4) (2013), 1906-1914.

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

A new type of associated curves

Evren ZIPLAR327, Yusuf YAYLI328 , İsmail GÖK329

Abstract

In this study, we enlarge the idea of the principal direction curve to a new idea called as the generalized principal-direction curve in 3-dimensional Euclidean space. Also, we find dealings between such curves and helix curves. Lastly, we investigate circular surfaces linked to generalized principal-direction curves by giving examples.

Key Words: Generalized–direction curve; helix curve.

References

[1] Choi J.H., Kim.Y.H., Associated curves of a Frenet curve and their applications, Applied Mathematics and Computation, 218 (2012) 9116-9124.

[2] Cui.L., Dai.J.S., Kinematic Geometry of Circular Surfaces With a Fixed Radius Based on Euclidean Invariants, Journal of Mechanical Design, October 2009.

[3] Ding.J., Chen.Y., Lv.Y., Song.C., Position-Parameter Selection Criterion for a Helix Curve Meshing Wheel Mechanism Based on Sliding Rates, Journal of Mechanical Engineering, 60(2014)9, 561-570.

[4] Gorjanc.S., Jurkin.E., Circular Surfaces CS(α,p),Filomat 29:4 (2015), 725-737.

[5] Izumiya.S., Takeuchi.N., New special curves and developable surfaces, Turk.J.Math. 28(2004) 153-163.

[6] Klug.A., The Discovery of the DNA Double Helix, J.Mol. Biol.,(2004) 335, 3-26.

[7] Struik.D.J., Lectures on Classical Differential Geometry, Dover,New-York, 1988.

327 Çankırı Karatekin University, Faculty of Science, Department of Mathematics, Çankırı, E-mail: [email protected] 328Ankara University, Faculty of Science, Department of Mathematics, Ankara, E-mail: [email protected] 329Ankara University, Faculty of Science, Department of Mathematics, Ankara, E-mail: [email protected]

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14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016

Spherical Curves with Modified Orthogonal Frame

Bahaddin BÜKCÜ330 , Murat Kemal KARACAN 331

Abstract

In [2-5,8-10], the authors have characterized the spherical curves in different spaces. In this paper, we shall characterize the spherical curves according to modified orthogonal frame in Euclidean 3-space.

Key Words: Spherical curves, Modified orthogonal frame

References

[1] Hacısalihoğlu H. H., Diferansiyel Geometri, Ankara Univ. Fen Fakültesi, Ankara, 1983.

[2] Carmo M.D., Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey,1976.

[3] Bektas M., Ergüt M.,Soylu D.,The Characterization of the Spherical Timelike Curves in 3-Dimensional Lorentzian Space,Bull. Malaysian Math. Soc.,21(1998),117-125.

[4] Petrovic-Torgasev M., Sucurovic E., Some Characterizations of The Lorentzian Spherical Timelike and Null Curves,Mat. Vesnik,53(2001), 21-27.

[5] N.Ayyildiz N., Cöken A.C, Yücesan A., A Characterization of Dual Lorentzian Spherical Curves in The Dual Lorentzian Space,Taiwanese J. Math,11(4)(2007),999-1018.

[6] Milman R.S., Parker G.D., Elements of Differential Geometry, Prentice-Hall Inc., Englewood Clifs, New Jersey,1977.

[7] Sasai T., The Fundamental Theorem of Analytic Space Curves and Apparent Singularities of Fuchsian Differential Equations,Tohoku Math. Journ. 36(1984),17-24.

[8] Pekmen U,S.Pasali, Some characterizations of Lorentzian Spherical Spacelike Curves,Math. Morav., 3(1999), 33-37.

[9] Wong Y.C., A global formulation of the condition for a curve to lie in a sphere, Monatsh.Math., 67 (1963), 363-365.

[10] Wong Y.C., On an Explicit Characterization of Spherical Curves, Proc. Amer. Math. Soc., 34(1972), 239- 242.

330Gazi Osman Pasa University, Faculty of Sciences and Arts, Department of Mathematics,Taslıciftlik Campus, 60250, Tokat-TURKEY,E-mail: [email protected] 331 Usak University, Faculty of Sciences and Arts, Department of Mathematics,1 Eylul Campus, 64200,Usak-TURKEY,E-mail:[email protected]

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Normal Section Curves on Semi-Riemannian Manifolds

Feyza Esra ERDOĞAN332, Selcen YÜKSEL PERKTAŞ333