Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems
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Toponogov.V.A.Differential.Geometry
Victor Andreevich Toponogov with the editorial assistance of Vladimir Y. Rovenski Differential Geometry of Curves and Surfaces A Concise Guide Birkhauser¨ Boston • Basel • Berlin Victor A. Toponogov (deceased) With the editorial assistance of: Department of Analysis and Geometry Vladimir Y. Rovenski Sobolev Institute of Mathematics Department of Mathematics Siberian Branch of the Russian Academy University of Haifa of Sciences Haifa, Israel Novosibirsk-90, 630090 Russia Cover design by Alex Gerasev. AMS Subject Classification: 53-01, 53Axx, 53A04, 53A05, 53A55, 53B20, 53B21, 53C20, 53C21 Library of Congress Control Number: 2005048111 ISBN-10 0-8176-4384-2 eISBN 0-8176-4402-4 ISBN-13 978-0-8176-4384-3 Printed on acid-free paper. c 2006 Birkhauser¨ Boston All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Birkhauser¨ Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA) and the author, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and re- trieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (TXQ/EB) 987654321 www.birkhauser.com Contents Preface ....................................................... vii About the Author ............................................. -
Extended Rectifying Curves As New Kind of Modified Darboux Vectors
TWMS J. Pure Appl. Math., V.9, N.1, 2018, pp.18-31 EXTENDED RECTIFYING CURVES AS NEW KIND OF MODIFIED DARBOUX VECTORS Y. YAYLI1, I. GOK¨ 1, H.H. HACISALIHOGLU˘ 1 Abstract. Rectifying curves are defined as curves whose position vectors always lie in recti- fying plane. The centrode of a unit speed curve in E3 with nonzero constant curvature and non-constant torsion (or nonzero constant torsion and non-constant curvature) is a rectifying curve. In this paper, we give some relations between non-helical extended rectifying curves and their Darboux vector fields using any orthonormal frame along the curves. Furthermore, we give some special types of ruled surface. These surfaces are formed by choosing the base curve as one of the integral curves of Frenet vector fields and the director curve δ as the extended modified Darboux vector fields. Keywords: rectifying curve, centrodes, Darboux vector, conical geodesic curvature. AMS Subject Classification: 53A04, 53A05, 53C40, 53C42 1. Introduction From elementary differential geometry it is well known that at each point of a curve α, its planes spanned by fT;Ng, fT;Bg and fN; Bg are known as the osculating plane, the rectifying plane and the normal plane, respectively. A curve called twisted curve has non-zero curvature functions in the Euclidean 3-space. Rectifying curves are introduced by B. Y. Chen in [3] as space curves whose position vector always lies in its rectifying plane, spanned by the tangent and the binormal vector fields T and B of the curve. Accordingly, the position vector with respect to some chosen origin, of a rectifying curve α in E3, satisfies the equation α(s) = λ(s)T (s) + µ(s)B(s) for some functions λ(s) and µ(s): He proved that a twisted curve is congruent to a rectifying τ curve if and only if the ratio { is a non-constant linear function of arclength s: Subsequently Ilarslan and Nesovic generalized the rectifying curves in Euclidean 3-space to Euclidean 4-space [10]. -
Extension of the Darboux Frame Into Euclidean 4-Space and Its Invariants
Turkish Journal of Mathematics Turk J Math (2017) 41: 1628 { 1639 http://journals.tubitak.gov.tr/math/ ⃝c TUB¨ ITAK_ Research Article doi:10.3906/mat-1604-56 Extension of the Darboux frame into Euclidean 4-space and its invariants Mustafa DULD¨ UL¨ 1;∗, Bahar UYAR DULD¨ UL¨ 2, Nuri KURUOGLU˘ 3, Ertu˘grul OZDAMAR¨ 4 1Department of Mathematics, Faculty of Science and Arts, Yıldız Technical University, Istanbul,_ Turkey 2Department of Mathematics Education, Faculty of Education, Yıldız Technical University, Istanbul,_ Turkey 3Department of Civil Engineering, Faculty of Engineering and Architecture, Geli¸simUniversity, Istanbul,_ Turkey 4Department of Mathematics, Faculty of Science and Arts, Uluda˘gUniversity, Bursa, Turkey Received: 13.04.2016 • Accepted/Published Online: 14.02.2017 • Final Version: 23.11.2017 Abstract: In this paper, by considering a Frenet curve lying on an oriented hypersurface, we extend the Darboux frame field into Euclidean 4-space E4 . Depending on the linear independency of the curvature vector with the hypersurface's normal, we obtain two cases for this extension. For each case, we obtain some geometrical meanings of new invariants along the curve on the hypersurface. We also give the relationships between the Frenet frame curvatures and Darboux frame curvatures in E4 . Finally, we compute the expressions of the new invariants of a Frenet curve lying on an implicit hypersurface. Key words: Curves on hypersurface, Darboux frame field, curvatures 1. Introduction In differential geometry, frame fields constitute an important tool while studying curves and surfaces. The most familiar frame fields are the Frenet{Serret frame along a space curve, and the Darboux frame along a surface curve. -
On the Patterns of Principal Curvature Lines Around a Curve of Umbilic Points
Anais da Academia Brasileira de Ciências (2005) 77(1): 13–24 (Annals of the Brazilian Academy of Sciences) ISSN 0001-3765 www.scielo.br/aabc On the Patterns of Principal Curvature Lines around a Curve of Umbilic Points RONALDO GARCIA1 and JORGE SOTOMAYOR2 1Instituto de Matemática e Estatística, Universidade Federal de Goiás Caixa Postal 131 – 74001-970 Goiânia, GO, Brasil 2Instituto de Matemática e Estatística, Universidade de São Paulo Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brasil Manuscript received on June 15, 2004; accepted for publication on October 10, 2004; contributed by Jorge Sotomayor* ABSTRACT In this paper is studied the behavior of principal curvature lines near a curve of umbilic points of a smooth surface. Key words: Umbilic point, principal curvature lines, principal cycles. 1 INTRODUCTION The study of umbilic points on surfaces and the patterns of principal curvature lines around them has attracted the attention of generation of mathematicians among whom can be named Monge, Darboux and Carathéodory. One aspect – concerning isolated umbilics – of the contributions of these authors, departing from Darboux (Darboux 1896), has been elaborated and extended in several directions by Garcia, Sotomayor and Gutierrez, among others. See (Gutierrez and Sotomayor, 1982, 1991, 1998), (Garcia and Sotomayor, 1997, 2000) and (Garcia et al. 2000, 2004) where additional references can be found. In (Carathéodory 1935) Carathéodory mentioned the interest of non isolated umbilics in generic surfaces pertinent to Geometric Optics. In a remarkably concise study he established that any local analytic regular arc of curve in R3 is a curve of umbilic points of a piece of analytic surface. -
Riemannian Submanifolds: a Survey
RIEMANNIAN SUBMANIFOLDS: A SURVEY BANG-YEN CHEN Contents Chapter 1. Introduction .............................. ...................6 Chapter 2. Nash’s embedding theorem and some related results .........9 2.1. Cartan-Janet’s theorem .......................... ...............10 2.2. Nash’s embedding theorem ......................... .............11 2.3. Isometric immersions with the smallest possible codimension . 8 2.4. Isometric immersions with prescribed Gaussian or Gauss-Kronecker curvature .......................................... ..................12 2.5. Isometric immersions with prescribed mean curvature. ...........13 Chapter 3. Fundamental theorems, basic notions and results ...........14 3.1. Fundamental equations ........................... ..............14 3.2. Fundamental theorems ............................ ..............15 3.3. Basic notions ................................... ................16 3.4. A general inequality ............................. ...............17 3.5. Product immersions .............................. .............. 19 3.6. A relationship between k-Ricci tensor and shape operator . 20 3.7. Completeness of curvature surfaces . ..............22 Chapter 4. Rigidity and reduction theorems . ..............24 4.1. Rigidity ....................................... .................24 4.2. A reduction theorem .............................. ..............25 Chapter 5. Minimal submanifolds ....................... ...............26 arXiv:1307.1875v1 [math.DG] 7 Jul 2013 5.1. First and second variational formulas -
On Jacobi Fields and Canonical Connection in Sub-Riemannian
On Jacobi fields and a canonical connection in sub-Riemannian geometry Davide Barilari♭ and Luca Rizzi♯ Abstract. In sub-Riemannian geometry the coefficients of the Jacobi equa- tion define curvature-like invariants. We show that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first introduced in [15]. We show why this connection is naturally nonlinear, and we discuss some of its properties. Contents 1. Introduction 1 2. Jacobi equation revisited 3 3. The Riemannian case 4 4. The sub-Riemannian case 4 5. Ehresmanncurvatureandcurvatureoperator 9 Appendix A. Normal condition for the canonical frame 12 1. Introduction A key tool for comparison theorems in Riemannian geometry is the Jacobi equation, i.e. the differential equation satisfied by Jacobi fields. Assume γε is a one-parameter family of geodesics on a Riemannian manifold (M,g) satisfying k k i j (1)γ ¨ε +Γij (γε)γ ˙ εγ˙ε =0. ∂ The corresponding Jacobi field J = ∂ε ε=0 γε is a vector field defined along γ = γ0, and satisfies the equation ∂Γk (2) J¨k + 2Γk J˙iγ˙ j + ij J ℓγ˙ iγ˙ j =0. ij ∂xℓ The Riemannian curvature is hidden in the coefficients of this equation. To make arXiv:1506.01827v3 [math.DG] 28 Mar 2017 it appear explicitly, however, one has to write (2) in terms of a parallel transported n frame X1(t),...,Xn(t) along γ(t). Letting J(t) = i=1 Ji(t)Xi(t) one gets the following normal form: P (3) J¨i + Rij (t)Jj =0. -
A New Method for Finding the Shape Operator of a Hypersurface in Euclidean 4-Space
Filomat 32:17 (2018), 5827–5836 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL1817827U University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat A New Method for Finding the Shape Operator of a Hypersurface in Euclidean 4-Space Bahar Uyar D ¨uld¨ula aYıldız Technical University, Education Faculty, Department of Mathematics and Science Education, Istanbul,˙ Turkey Abstract. In this paper, by taking a Frenet curve lying on a parametric or implicit hypersurface and using the extended Darboux frame field of this curve, we give a new method for calculating the shape operator’s matrix of the hypersurface depending on the extended Darboux frame curvatures. This new method enables us to obtain the Gaussian and mean curvatures of the hypersurface depending on the geodesic torsions of the curve and the normal curvatures of the hypersurface. 1. Introduction In differential geometry, the shape of a geometric object (curve, surface, etc.) has always been a point of interest. In Euclidean space, along a space curve we have only one direction to move. This direction is determined by the tangent vector of the curve. The curvature of the curve measures the rate of change of its tangent vector, and together with the torsion of the curve they give us some information to its shape. However, along a surface we have different directions to move. These directions constitute a plane at a point, called the tangent plane of the surface at that point. When we move along a given direction, the tangent plane and thereby the normal vector of this plane, that is, the unit normal vector of the surface will change its direction except for some special cases. -
Modern Developments in the Theory and Applications of Moving Frames
London Mathematical Society Impact150 Stories 1 (2015) 14{50 C 2015 Author(s) doi:10.1112/l150lms/l.0001 e Modern Developments in the Theory and Applications of Moving Frames Peter J. Olver Abstract This article surveys recent advances in the equivariant approach to the method of moving frames, concentrating on finite-dimensional Lie group actions. A sampling from the many current applications | to geometry, invariant theory, and image processing | will be presented. 1. Introduction. According to Akivis, [2], the method of rep`eresmobiles, which was translated into English as moving framesy, can be traced back to the moving trihedrons introduced by the Estonian mathematician Martin Bartels (1769{1836), a teacher of both Gauß and Lobachevsky. The apotheosis of the classical development can be found in the seminal advances of Elie´ Cartan, [25, 26], who forged earlier contributions by Frenet, Serret, Darboux, Cotton, and others into a powerful tool for analyzing the geometric properties of submanifolds and their invariants under the action of transformation groups. An excellent English language treatment of the Cartan approach can be found in the book by Guggenheimer, [49]. The 1970's saw the first attempts, cf. [29, 45, 46, 64], to place Cartan's constructions on a firm theoretical foundation. However, the method remained constrained within classical geometries and homogeneous spaces, e.g. Euclidean, equi-affine, or projective, [35]. In the late 1990's, I began to investigate how moving frames and all their remarkable consequences might be adapted to more general, non-geometrically-based group actions that arise in a broad range of applications. -
Differential Geometry Curves – Surfaces – Manifolds Third Edition
STUDENT MATHEMATICAL LIBRARY Volume 77 Differential Geometry Curves – Surfaces – Manifolds Third Edition Wolfgang Kühnel Differential Geometry Curves–Surfaces– Manifolds http://dx.doi.org/10.1090/stml/077 STUDENT MATHEMATICAL LIBRARY Volume 77 Differential Geometry Curves–Surfaces– Manifolds Third Edition Wolfgang Kühnel Translated by Bruce Hunt American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov Translation from German language edition: Differentialgeometrie by Wolfgang K¨uhnel, c 2013 Springer Vieweg | Springer Fachmedien Wies- baden GmbH JAHR (formerly Vieweg+Teubner). Springer Fachmedien is part of Springer Science+Business Media. All rights reserved. Translated by Bruce Hunt, with corrections and additions by the author. Front and back cover image by Mario B. Schulz. 2010 Mathematics Subject Classification. Primary 53-01. For additional information and updates on this book, visit www.ams.org/bookpages/stml-77 Page 403 constitutes an extension of this copyright page. Library of Congress Cataloging-in-Publication Data K¨uhnel, Wolfgang, 1950– [Differentialgeometrie. English] Differential geometry : curves, surfaces, manifolds / Wolfgang K¨uhnel ; trans- lated by Bruce Hunt.— Third edition. pages cm. — (Student mathematical library ; volume 77) Includes bibliographical references and index. ISBN 978-1-4704-2320-9 (alk. paper) 1. Geometry, Differential. 2. Curves. 3. Surfaces. 4. Manifolds (Mathematics) I. Hunt, Bruce, 1958– II. Title. QA641.K9613 2015 516.36—dc23 2015018451 c 2015 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. -
Environmental Bias and Elastic Curves on Surfaces Jemal Guven,1∗Dulce María Valencia2†And Pablo Vázquez-Montejo3‡
Environmental bias and elastic curves on surfaces Jemal Guven,1∗Dulce María Valencia2yand Pablo Vázquez-Montejo3z 1 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México Apdo. Postal 70-543, 04510 México D.F., MEXICO 2 Departamento de Física, Cinvestav, Instituto Politécnico Nacional 2508, 07360, México D.F., MEXICO 3 Department of Physics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA Abstract The behavior of an elastic curve bound to a surface will reflect the geometry of its environment. This may occur in an obvious way: the curve may deform freely along directions tangent to the surface, but not along the surface normal. However, even if the energy itself is symmetric in the curve’s geodesic and normal curvatures, which control these modes, very distinct roles are played by the two. If the elastic curve binds preferentially on one side, or is itself assembled on the surface, not only would one expect the bending moduli associated with the two modes to differ, binding along specific directions, reflected in spontaneous values of these curvatures, may be favored. The shape equations describing the equilibrium states of a surface curve described by an elastic energy accommodating environmental factors will be identified by adapting the method of Lagrange multipliers to the Darboux frame associated with the curve. The forces transmitted to the surface along the surface normal will be determined. Features associated with a number of different energies, both of physical relevance and of mathematical interest, are described. The conservation laws associated with trajectories on surface geometries exhibiting continuous symmetries are also examined. -
Spherical Product Surfaces in the Galilean Space
Konuralp Journal of Mathematics Volume 4 No. 2 pp. 290{298 (2016) c KJM SPHERICAL PRODUCT SURFACES IN THE GALILEAN SPACE MUHITTIN EVREN AYDIN AND ALPER OSMAN OGRENMIS Abstract. In the present paper, we consider the spherical product surfaces in a Galilean 3-space G3. We derive a classification result for such surfaces of constant curvature in G3. Moreover, we analyze some special curves on these surfaces in G3. 1. Introduction The tight embeddings of product spaces were investigated by N.H. Kuiper (see [17]) and he introduced a different tight embedding in the (n1 + n2 − 1) −dimensional Euclidean space Rn1+n2−1 as follows: Let m n1 c1 : M −! R ; c1 (u1; :::; um) = (f1 (u1; :::; um) ; :::; fn1 (u1; :::; um)) be a tight embedding of a m−dimensional manifold M m satisfying Morse equality and n2−1 n2 c2 : S −! R ; c1 (v1; :::; vn2−1) = (g1 (v1; :::; vn2−1) ; :::; gn2 (v1; :::; vn2−1)) n2 the standard embedding of (n2 − 1) −sphere in R , where u = (u1; :::; um) and m n2−1 v = (v1; :::; vn2−1) are the local coordinate systems on M and S , respectively. Then a new tight embedding is given by m n2−1 n1+n2−1 x = c1 ⊗ c2 : M × S −! R ; (u; v) 7−! (f1 (u) ; :::; fn1−1 (u) ; fn1 (u) g1 (v) ; :::; fn1 (u) gn2 (v)) : n1 n1−1 n1+n2−1 Such embeddings are obtained from c1 by rotating R about R in R (cf. [4]). B. Bulca et al. [6, 7] called such embeddings rotational embeddings and consid- ered the spherical product surfaces in Euclidean spaces, which are a special type 2000 Mathematics Subject Classification. -
Algebraic Geometry and Local Differential Geometry
ANNALES SCIENTIFIQUES DE L’É.N.S. PHILLIP GRIFFITHS JOSEPH HARRIS Algebraic geometry and local differential geometry Annales scientifiques de l’É.N.S. 4e série, tome 12, no 3 (1979), p. 355-452. <http://www.numdam.org/item?id=ASENS_1979_4_12_3_355_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1979, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systéma- tique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. scient. EC. Norm. Sup. 46 serie, t. 12, 1979, p. 355 a 432. ALGEBRAIC GEOMETRY AND LOCAL DIFFERENTIAL GEOMETRY BY PHILLIP GRIFFITHS (1) AND JOSEPH HARRIS (1) CONTENTS Introduction. ................................................................. 356 1. Differential-geometric preliminaries ................................................ 360 (a) Structure equations for the frame manifolds. ........................................ 360 (b) The 2nd fundamental form. .................................................... 363 (c) Examples ................................................................. 368 (d) The higher fundamental forms .................................................. 372 (e)