Compatibility of Gauß Maps with Metrics
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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 ............................................. -
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 -
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. -
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. -
Differential Geometry Peter Petersen
Differential Geometry Peter Petersen CHAPTER 1 Preliminaries 1.1. Vectors-Matrices Given a basis e, f for a two dimensional vector space we expand vectors using matrix multiplication ve v = vee + vf f = ef vf and the matrix representation [L] for a linear⇥ map/transformation⇤ L can be found from L (e) L (f) = ef [L] Le Le ⇥ ⇤ = ⇥ ef⇤ e f Lf Lf e f ⇥ ⇤ Next we relate matrix multiplication and the dot product in R3. We think of vectors as being columns or 3 1 matrices. Keeping that in mind and using transposition ⇥ of matrices we immediately obtain: XtY = X Y, · Xt X Y = X X X Y 2 2 · 2 · 2 t X X X⇥ Y X⇤ = ⇥ 1 ⇤ 1 1 Y ·X 1 · ⇥ t ⇤ X X X Y X Y X Y = 1 2 1 2 , 1 1 2 2 Y ·X Y ·Y 1 · 2 1 · 2 ⇥ ⇤ ⇥ ⇤ X1 X2 X1 Y2 X1 Z2 t · · · X1 Y1 Z1 X2 Y2 Z2 = Y1 X2 Y1 Y2 Y1 Z2 2 Z · X Z · Y Z · Z 3 ⇥ ⇤ ⇥ ⇤ 1 · 2 1 · 2 1 · 2 These formulas can be used to calculate the4 coefficients of a vector with respect5 2 to a general basis. Recall first that if E1,E2 is an orthonormal basis for R , then X =(X E1) E1 +(X E2) E2 · · t = E1 E2 E1 E2 X So the coefficients for X are simply⇥ the dot⇤⇥ products with⇤ the basis vectors. More generally we have Theorem 1.1.1. Let U.V be a basis for R2, then t 1 t X = UV UV UV − UV X ⇥ ⇤ ⇣⇥ ⇤t ⇥ ⇤⌘ 1 ⇥ U X ⇤ = UV UV UV − V · X · ⇥ ⇤ ⇣⇥ ⇤ ⇥ ⇤⌘ 3 1.2. -
Minimal Surfaces and Particles in 3-Manifolds
Minimal surfaces and particles in 3-manifolds Kirill Krasnov ∗and Jean-Marc Schlenker † January 2006 (v3) Abstract We consider 3-dimensional anti-de Sitter manifolds with conical singularities along time-like lines, which is what in the physics literature is known as manifolds with particles. We show that the space of such cone-manifolds is parametrized by the cotangent bundle of Teichm¨uller space, and that moreover such cone-manifolds have a canonical foliation by space-like surfaces. We extend these results to de Sitter and Minkowski cone-manifolds, as well as to some related “quasi-fuchsian” hyperbolic manifolds with conical singularities along infinite lines, in this later case under the condition that they contain a minimal surface with principal curvatures less than 1. In the hyperbolic case the space of such cone-manifolds turns out to be parametrized by an open subset in the cotangent bundle of Teichm¨uller space. For all settings, the symplectic form on the moduli space of 3-manifolds that comes from parameterization by the cotangent bundle of Teichm¨uller space is the same as the 3-dimensional gravity one. The proofs use minimal (or maximal, or CMC) surfaces, along with some results of Mess on AdS manifolds, which are recovered here in different way, using differential-geometric methods and a result of Labourie on some mappings between hyperbolic surfaces, that allows an extension to cone-manifolds. Contents 1 Introductions, results 2 1.1 A mathematical introduction . 2 1.2 Physics introduction . 5 2 Minimal surfaces in quasi-Fuchsian manifolds 9 2.1 Minimal surfaces in germs of manifolds . -
On Parallel Surfaces in Minkowski 3-Space
TWMS J. App. Eng. Math. V.3, N.2, 2013, pp. 214-222. ON PARALLEL SURFACES IN MINKOWSKI 3-SPACE YASIN_ UNL¨ UT¨ URK¨ 1, ERDAL OZ¨ USA¨ GLAM˘ 2 Abstract. In this paper, we study on some properties of parallel surfaces in Minkowski 3-space. The results given in this paper were given in Euclidean space by [7, 8]. By using these two former studies, we show these properties in Minkowski 3-space. Also we give the relation among the fundamental forms of parallel surfaces in Minkowski 3-space. Finally we show that how a curve which is geodesic on M become again a geodesic on parallel surface M r by the normal map in Minkowski 3-space. Keywords: parallel surfaces, fundamental forms, geodesic curve. AMS Subject Classification: 53A05, 53B25, 53B30 1. Introduction Parallel surfaces as a subject of differential geometry have been intriguing for mathe- maticians throughout history and so it has been a research field. In theory of surfaces, there are some special surfaces such as ruled surfaces, minimal surfaces and surfaces of constant curvature in which differential geometers are interested. Among these surfaces, parallel surfaces are also studied in many papers [1, 2, 3, 4, 5, 11, 13]. Craig had studied to find parallel of ellipsoid [2]. Eisenhart gave a chapter for parallel surfaces in his famous A Treatise On the Differential Geometry of Curves and Surfaces [3]. Nizamo˘gluhad stated parallel ruled surface as a curve depending on one-parameter and gave some geometric properties of such a surface [11]. We can explain a parallel surface something like that a surface M r whose points are at a constant distance along the normal from another surface M is said to be parallel to M. -
Download Differential Geometry of Curves and Surfaces Free Ebook
DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES DOWNLOAD FREE BOOK Manfredo P Do Carmo | 512 pages | 27 Jan 2017 | Dover Publications Inc. | 9780486806990 | English | New York, United States Do carmo differential geometry of curves and surfaces The above concepts are essentially all to do with multivariable calculus. It is straightforward to check that the two definitions are equivalent. Parallel transport along geodesics, the "straight lines" of the surface, can also easily be described directly. It would not surprise me if it quickly becomes the market leader. Thomas F. Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating story: in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later Differential Geometry of Curves and Surfaces applied to optics and gears. There is also plenty of figures, examples, exercises and applications which make the differential geometry of curves and surfaces so interesting and intuitive. A ruled surface is one which can be generated by the motion of a straight line in E 3. For readers bound for Differential Geometry of Curves and Surfaces school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems. He is published widely and known to a broad audience as editor and commentator on Abbotts Flatland. The author uses a rich variety of colours and techniques that help to clarify difficult abstract concepts. In Lagrange extended Euler's results on the calculus of variations involving integrals in one variable to two variables. Home Catalog. In general, the eigenvectors and eigenvalues of the shape operator at each point determine the directions in which the surface bends at each point. -
The Mean Curvature of the Second Fundamental Form of a Hypersurface
Beitr¨agezur Algebra und Geometrie Contributions to Algebra and Geometry Volume 51 (2010), No. 1, 155-169. The Mean Curvature of the Second Fundamental Form of a Hypersurface Stefan Haesen∗ Steven Verpoort† Simon Stevin Institute for Geometry Wilhelminaweg 1, nn 2042 Zandvoort, The Netherlands e-mail: [email protected] K. U. Leuven, Departement Wiskunde, Afdeling Meetkunde Celestijnenlaan 200B bus 2400, 3001 Heverlee, Belgium e-mail: [email protected] Abstract. An expression for the first variation of the area functional of the second fundamental form is given for a hypersurface in a semi- Riemannian space. The concept of the “mean curvature of the sec- ond fundamental form” is then introduced for hypersurfaces in semi- Riemannian spaces. Some characterisations of extrinsic hyperspheres in terms of this curvature are given. MSC 2000: 53B25 (primary), 53A10, 53C42 (secondary) Keywords: second fundamental form, mean curvature, extrinsic hyper- sphere ∗S. Haesen was a postdoctoral researcher at the K.U. Leuven while the work was initiated. He was partially supported by the Spanish MEC Grant MTM2007-60731 with FEDER funds and the Junta de Andaluc´ıaRegional Grant P06-FQM-01951. †corresponding author Both authors were partially supported by the Research Foundation – Flanders (project G.0432.07). 0138-4821/93 $ 2.50 c 2010 Heldermann Verlag 156 S. Haesen, S. Verpoort: The Mean Curvature of the Second Fundamental ... 1. Introduction and outline of the article We shall be concerned with hypersurfaces of a semi-Riemannian manifold, for which the real-valued second fundamental form II is a semi-Riemannian metrical tensor. The geometry of such hypersurfaces can be explored with respect to either the first or the second fundamental form. -
Timelike Surfaces in the Lorentz-Minkowski Space with Prescribed Gaussian Curvature and Gauss Map
Timelike Surfaces in the Lorentz-Minkowski Space with Prescribed Gaussian Curvature and Gauss Map Juan A. Aledoa1, Jos´eM. Espinarb and Jos´eA. G´alvezc aDepartamento de Matem´aticas,Universidad de Castilla-La Mancha, EPSA, 02071 Albacete, Spain e-mail: [email protected] b; cDepartamento de Geometr´ıay Topolog´ıa,Universidad de Granada, 18071 Granada, Spain e-mails: [email protected], [email protected] Abstract. In this paper we obtain a Lelieuvre-type representation for timelike surfaces with pre- scribed Gauss map in the 3-dimensional Lorentz-Minkowski space L3. As a main result we classify the complete timelike surfaces with positive constant Gaussian curvature in L3 in terms of harmonic diffeomorphisms between simply connected Lorentz surfaces and the universal covering of the de Sitter Space. 1 Introduction The existence and uniqueness of a surface in R3 (and also in the rest of Riemannian space forms) with prescribed Gauss map and a given conformal structure have been subject of a wide study (see [11], [6], [2] and references therein). This study has also been extended to spacelike surfaces in L3. Specifically, Kobayashi [8] and Akutagawa and Nishikawa [1] obtained Lorentzian versions of the classical Enneper-Weierstrass and Kenmotsu representations for maximal and constant mean curvature surfaces respectively. On the other hand, Mart´ınez,Mil´anand the third author [3] have rencently obtained a representation for spacelike surfaces in L3 using the Gauss map and the conformal structure given by the second fundamental form. Regarding to timelike surfaces in L3, Magid [9] found a Weierstrass representation for any such a surface in terms of its Gauss map and mean curvature, showing that the language of complex analysis used when the surface can be considered as a Riemann surface is a useful device but not essential. -
Arxiv:1804.10425V1 [Math.DG]
INTEGRAL INVARIANTS FROM COVARIANCE ANALYSIS OF EMBEDDED RIEMANNIAN MANIFOLDS JAVIER ÁLVAREZ-VIZOSO, MICHAEL KIRBY, AND CHRIS PETERSON Abstract. Principal Component Analysis can be performed over small domains of an em- bedded Riemannian manifold in order to relate the covariance analysis of the underlying point set with the local extrinsic and intrinsic curvature. We show that the volume of domains on a submanifold of general codimension, determined by the intersection with higher-dimensional cylinders and balls in the ambient space, have asymptotic expansions in terms of the mean and scalar curvatures. Moreover, we propose a generalization of the classical third fundamental form to general submanifolds and prove that the eigenvalue decomposition of the covariance matrices of the domains have asymptotic expansions with scale that contain the curvature information encoded by the traces of this tensor. In the case of hypersurfaces, this covariance analysis recov- ers the principal curvatures and principal directions, which can be used as descriptors at scale to build up estimators of the second fundamental form, and thus the Riemann tensor, of general submanifolds. Contents 1. Introduction ..................................... ................................... 1 2. PCA Integral Invariants of Riemannian Submanifolds. ............................ 3 3. Third Fundamental Form of a Riemannian Submanifold . ........................ 5 4. Cylindrical Covariance Analysis . ................................. 10 5. Spherical Covariance Analysis -
MATH 4030 Differential Geometry Lecture Notes Part 2 After Studying
MATH 4030 Differential Geometry Lecture Notes Part 2 last revised on October 14, 2015 n n After studying curves in R , we now go up one dimension to study the theory of surfaces in R (we are mainly interested in the case n = 3 since the codimension is one). Unlike the theory of curves, where most theorems reduce to theorems about ODE systems, the situation gets drastically more difficult in dimension two since partial derivatives are involved. On the other hand, while there is no intrinsic geometry on a curve (i.e. every curve is locally \isomorphic" isometrically), there is both n intrinsic and extrinsic geometry for a surface in R . The intrinsic and extrinsic geometry are described by two quadratic forms, namely the first and second fundamental forms. Similar to curves, one can define various notions of curvatures to measure the deviation of the surface (both intrinsically and 3 extrinsically) from the standard picture of a “flat” plane sitting inside R . 3 Regular parametrized surfaces in R So what is a \surface"? It should be something two-dimensional in some sense, and just like curves, we want to give a surface some smooth parametrizations in order to do differential geometry. However, we should keep in mind that the parametrization is just a means of \coordinates" which allows us to do calculations but it should be completely artificial. Actual geometric quantities like area and curvatures should be independent of parametrizations. This property of parametrization invariance (think about Einstein's principle of equivalence in relativity) is a guiding principle to define many useful geometric quantities.