Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition

529 Pages·2016·2.59 MB·English

by  Carmo

DIFFERENTIAL
GEOMETRY
OF
CURVES &
SURFACES

DIFFERENTIAL
GEOMETRY
OF
CURVES &
SURFACES
Revised & Updated
SECOND EDITION
Manfredo P. do Carmo
Instituto Nacional de Matemática
Pura e Aplicada (IMPA)
Rio de Janeiro, Brazil
DOVER PUBLICATIONS, INC.
Mineola, New York
Copyright
Copyright © 1976, 2016 by Manfredo P. do Carmo
All rights reserved.
Bibliographical 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. The author has also provided a
new Preface for this edition.
International Standard Book Number
ISBN-13: 978-0-486-80699-0
ISBN-10: 0-486-80699-5
Manufactured in the United States by LSC Communications
80699501 2016
www.doverpublications.com
To Leny,
for her indispensable assistance
in all the stages of this book

Contents
PrefacetotheSecondEdition xi
Preface xiii
SomeRemarksonUsingthisBook xv
1. Curves 1
1-1 Introduction 1
1-2 ParametrizedCurves 2
1-3 RegularCurves;ArcLength 6
1-4 TheVectorProductinR3 12
1-5 TheLocalTheoryofCurvesParametrizedbyArcLength 17
1-6 TheLocalCanonicalForm 28
1-7 GlobalPropertiesofPlaneCurves 31
2. RegularSurfaces 53
2-1 Introduction 53
2-2 RegularSurfaces;InverseImagesofRegularValues 54
2-3 ChangeofParameters;DifferentiableFunctionsonSurface 72
2-4 TheTangentPlane;TheDifferentialofaMap 85
2-5 TheFirstFundamentalForm;Area 94
2-6 OrientationofSurfaces 105
2-7 ACharacterizationofCompactOrientableSurfaces 112
2-8 AGeometricDefinitionofArea 116
Appendix:ABriefReviewofContinuityandDifferentiability 120
ix
x Contents
3. TheGeometryoftheGaussMap 136
3-1 Introduction 136
3-2 TheDefinitionoftheGaussMapandItsFundamental
Properties 137
3-3 TheGaussMapinLocalCoordinates 155
3-4 VectorFields 178
3-5 RuledSurfacesandMinimalSurfaces 191
Appendix:Self-AdjointLinearMapsandQuadraticForms 217
4. TheIntrinsicGeometryofSurfaces 220
4-1 Introduction 220
4-2 Isometries;ConformalMaps 221
4-3 TheGaussTheoremandtheEquationsofCompatibility 235
4-4 ParallelTransport.Geodesics. 241
4-5 TheGauss-BonnetTheoremandItsApplications 267
4-6 TheExponentialMap.GeodesicPolarCoordinates 287
4-7 FurtherPropertiesofGeodesics;ConvexNeighborhoods 302
Appendix:ProofsoftheFundamentalTheoremsoftheLocalTheoryof
CurvesandSurfaces 315
5. GlobalDifferentialGeometry 321
5-1 Introduction 321
5-2 TheRigidityoftheSphere 323
5-3 CompleteSurfaces.TheoremofHopf-Rinow 331
5-4 FirstandSecondVariationsofArcLength;Bonnet’sTheorem 344
5-5 JacobiFieldsandConjugatePoints 363
5-6 CoveringSpaces;TheTheoremsofHadamard 377
5-7 GlobalTheoremsforCurves:TheFary-MilnorTheorem 396
5-8 SurfacesofZeroGaussianCurvature 414
5-9 Jacobi’sTheorems 421
5-10 AbstractSurfaces;FurtherGeneralizations 430
5-11 Hilbert’sTheorem 451
Appendix:Point-SetTopologyofEuclideanSpaces 460
BibliographyandComments 475
HintsandAnswers 478
Index 503