![]() Finally, in Chapter 5, we use the first and second variations of arc length to derive some global properties of surfaces. Chapter 4 unifies the intrinsic geometry of surfaces around the concept of covariant derivative again, our purpose was to prepare the reader for the basic notion of connection in Riemannian geometry. Chapter 3 is built on the Gauss normal map and contains a large amount of the local geometry of surfaces in R 3. Thus, Chapter 2 develops around the concept of a regular surface in R 3 when this concept is properly developed, it is probably the best model for differentiable manifolds. ![]() We have tried to build each chapter of the book around some simple and fundamental idea. ![]() The presentation differs from the traditional ones by a more extensive use of elementary linear algebra and by a certain emphasis placed on basic geometrical facts, rather than on machinery or random details. This book is an introduction to the differential geometry of curves and surfaces, both in its local and global aspects. Finally, I would like to thank my son, Manfredo Jr., for helping me with several figures in this edition. Cavalcante, participated in the project as if it was a work of her own and I might say that without her this volume would not exist. Thanks are also due to John Grafton, Senior Acquisitions Editor at Dover Publications, who believed that the book was still valuable and included in the text all of the changes I had in mind, and to the editor, James Miller, for his patience with my frequent requests. Here I would like to express my deep appreciation and thank them all. For several reasons it is impossible to mention the names of all the people who generously donated their time doing that. In this edition, I have included many of the corrections and suggestions kindly sent to me by those who have used the book. Theorem of Hopf-Rinow 331 First and Second Variations of Arc Length Bonnet’s Theorem 344 Jacobi Fields and Conjugate Points 363 Covering Spaces The Theorems of Hadamard 377 Global Theorems for Curves: The Fary-Milnor Theorem 396 Surfaces of Zero Gaussian Curvature 414 Jacobi’s Theorems 421 Abstract Surfaces Further Generalizations 430 Hilbert’s Theorem 451 Appendix: Point-Set Topology of Euclidean Spaces 460īibliography and Comments 475 Hints and Answers 478 Index 503 Introduction 321 The Rigidity of the Sphere 323 Complete Surfaces. ![]() Geodesic Polar Coordinates 287 4-7 Further Properties of Geodesics Convex Neighborhoods 302 Appendix: Proofs of the Fundamental Theorems of the Local Theory of Curves and Surfaces 315ĥ. 241 4-5 The Gauss-Bonnet Theorem and Its Applications 267 4-6 The Exponential Map. The Intrinsic Geometry of Surfaces 220 4-1 Introduction 220 4-2 Isometries Conformal Maps 221 4-3 The Gauss Theorem and the Equations of Compatibility 235 4-4 Parallel Transport. The Geometry of the Gauss Map 136 3-1 Introduction 136 3-2 The Definition of the Gauss Map and Its Fundamental Properties 137 3-3 The Gauss Map in Local Coordinates 155 3-4 Vector Fields 178 3-5 Ruled Surfaces and Minimal Surfaces 191 Appendix: Self-Adjoint Linear Maps and Quadratic Forms 217Ĥ. Introduction 53 Regular Surfaces Inverse Images of Regular Values 54 Change of Parameters Differentiable Functions on Surface 72 The Tangent Plane The Differential of a Map 85 The First Fundamental Form Area 94 Orientation of Surfaces 105 A Characterization of Compact Orientable Surfaces 112 A Geometric Definition of Area 116 Appendix: A Brief Review of Continuity and Differentiability 120ģ. Introduction 1 Parametrized Curves 2 Regular Curves Arc Length 6 The Vector Product in R3 12 The Local Theory of Curves Parametrized by Arc Length 17 The Local Canonical Form 28 Global Properties of Plane Curves 31Ģ. Preface to the Second Edition xi Preface xiii Some Remarks on Using this Book xv 1. International Standard Book Number ISBN-13: 978-9-0 ISBN-10: 9-5 Manufactured in the United States by LSC Communications 80699501 2016 To Leny, for her indispensable assistance in all the stages of this book The author has also provided a new Preface for this edition. do Carmo All rights reserved.īibliographical Note Differential Geometry of Curves and Surfaces: Revised & Updated Second Edition is a revised, corrected, and updated second edition of the work originally published in 1976 by Prentice-Hall, Inc., Englewood Cliffs, New Jersey. Mineola, New YorkĬopyright Copyright © 1976, 2016 by Manfredo P. do Carmo Instituto Nacional de Matemática Pura e Aplicada (IMPA) Rio de Janeiro, BrazilĭOVER PUBLICATIONS, INC.
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