Lectures on the differential geometry of curves and surfaces

237 Pages·2005·2 MB·English

by  Blaga P.

ff
Lectures on the Di erential Geometry
of Curves and Surfaces
Paul A. Blaga

ToCristina,withlove

Contents
Foreword 9
I Curves 11
1 Spacecurves 13
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Parameterizedcurves(paths) . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Thedefinitionofthecurve . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4 Analyticalrepresentationsofcurves . . . . . . . . . . . . . . . . . . . 26
1.4.1 Planecurves . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4.2 Spacecurves . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5 Thetangentandthenormalplane . . . . . . . . . . . . . . . . . . . . . 32
1.5.1 The equations of the tangent line and normal plane (line) for
differentrepresentationsofcurves . . . . . . . . . . . . . . . . 35
1.6 Theosculatingplane . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.7 Thecurvatureofacurve . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.7.1 Thegeometricalmeaningofcurvature . . . . . . . . . . . . . . 44
1.8 TheFrenetframe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.8.1 ThebehaviouroftheFrenetframeataparameterchange . . . . 47
1.9 Orientedcurves. TheFrenetframeofanorientedcurve . . . . . . . . . 48
1.10 TheFrenetformulae. Thetorsion . . . . . . . . . . . . . . . . . . . . . 50
6 Contents
1.10.1 Thegeometricalmeaningofthetorsion . . . . . . . . . . . . . 53
1.10.2 SomefurtherapplicationsoftheFrenetformulae . . . . . . . . 54
1.10.3 Generalhelices. Lancret’stheorem . . . . . . . . . . . . . . . 56
1.10.4 Bertrandcurves . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.11 Thelocalbehaviourofaparameterizedcurvearoundabiregularpoint . 62
1.12 Thecontactbetweenaspacecurveandaplane . . . . . . . . . . . . . 64
1.13 Thecontactbetweenaspacecurveandasphere. Theosculatingsphere 66
1.14 Existenceanduniquenesstheoremsforparameterizedcurves . . . . . . 68
1.14.1 ThebehaviouroftheFrenetframeunderarigidmotion . . . . . 68
1.14.2 Theuniquenesstheorem . . . . . . . . . . . . . . . . . . . . . 70
1.14.3 Theexistencetheorem . . . . . . . . . . . . . . . . . . . . . . 72
2 Planecurves 75
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.2 Envelopesofplanecurves . . . . . . . . . . . . . . . . . . . . . . . . 75
2.2.1 Curvesgiventhroughanimplicitequation . . . . . . . . . . . . 77
2.2.2 Familiesofcurvesdependingontwoparameters . . . . . . . . 79
2.2.3 Applications: theevoluteofaplanecurve . . . . . . . . . . . . 79
2.3 Thecurvatureofaplanecurve . . . . . . . . . . . . . . . . . . . . . . 81
2.3.1 Thegeometricalinterpretationofthesignedcurvature . . . . . 84
2.4 Thecurvaturecenter. Theevoluteandtheinvoluteofaplanecurve . . . 86
2.5 Theosculatingcircleofacurve . . . . . . . . . . . . . . . . . . . . . . 91
2.6 Theexistenceanduniquenesstheoremforplanecurves . . . . . . . . . 92
3 Theintegrationofthenaturalequationsofacurve 95
3.1 TheRiccatiequationassociatedtothenaturalequationsofacurve . . . 95
3.2 Examplesfortheintegrationofthenaturalequationofaplanecurve . . 96
Problems 103
II Surfaces 113
4 Generaltheoryofsurfaces 115
4.1 Parameterizedsurfaces(patches) . . . . . . . . . . . . . . . . . . . . . 115
4.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2.1 Representationsofsurfaces. . . . . . . . . . . . . . . . . . . . 116
4.3 Theequivalenceoflocalparameterizations . . . . . . . . . . . . . . . . 119
4.4 Curvesonasurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Contents 7
4.5 Thetangentvectorspace,thetangentplaneandthenormaltoasurface . 123
4.6 Theorientationofsurfaces . . . . . . . . . . . . . . . . . . . . . . . . 127
4.7 Differentiablemapsonasurface . . . . . . . . . . . . . . . . . . . . . 130
4.8 Thedifferentialofasmoothmapbetweensurfaces . . . . . . . . . . . 134
4.9 Thesphericalmapandtheshapeoperatorofanorientedsurface . . . . 136
4.10 Thefirstfundamentalformofasurface . . . . . . . . . . . . . . . . . 139
4.10.1 Firstapplications . . . . . . . . . . . . . . . . . . . . . . . . . 140
Thelengthofasegmentofcurveonasurface . . . . . . . . . . 140
Theangleoftwocurvesonasurface . . . . . . . . . . . . . . . 141
Theareaofaparameterizedsurface . . . . . . . . . . . . . . . 142
4.11 Thematrixoftheshapeoperator . . . . . . . . . . . . . . . . . . . . . 144
4.12 Thesecondfundamentalformofanorientedsurface . . . . . . . . . . 146
4.13 Thenormalcurvature. TheMeusnier’stheorem . . . . . . . . . . . . . 148
4.14 Asymptoticdirectionsandasymptoticlinesonasurface . . . . . . . . . 150
4.15 Theclassificationofpointsonasurface . . . . . . . . . . . . . . . . . 152
4.16 Principaldirectionsandcurvatures . . . . . . . . . . . . . . . . . . . . 156
4.16.1 Thedeterminationofthelinesofcurvature . . . . . . . . . . . 159
4.16.2 Thecomputationofthecurvaturesofasurface . . . . . . . . . 161
4.17 Thefundamentalequationsofasurface . . . . . . . . . . . . . . . . . 162
4.17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.17.2 Thedifferentiationrules. Christoffel’scoefficients . . . . . . . . 162
Christoffel’sandWeingarten’scoefficientsincurvaturecoordinates164
4.17.3 TheGauss’andCodazzi-Mainardi’sequationsforasurface . . 165
4.17.4 Thefundamentaltheoremofsurfacetheory . . . . . . . . . . . 167
4.18 TheGauss’egregiumtheorem . . . . . . . . . . . . . . . . . . . . . . 172
4.19 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.19.2 TheDarbouxframe. Thegeodesiccurvatureandgeodesictorsion 175
4.19.3 Geodesiclines . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Examplesofgeodesics . . . . . . . . . . . . . . . . . . . . . . 181
4.19.4 Liouvillesurfaces . . . . . . . . . . . . . . . . . . . . . . . . . 182
5 Specialclassesofsurfaces 185
5.1 Ruledsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.1.1 Generalruledsurfaces . . . . . . . . . . . . . . . . . . . . . . 185
Theparameterizationofaruledsurface . . . . . . . . . . . . . 186
Thetangentplaneandthefirstfundamentalformofaruledsurface187
5.1.2 TheGaussiancurvatureofaruledsurface . . . . . . . . . . . . 189
8 Contents
5.1.3 Envelopeofafamilyofsurfaces . . . . . . . . . . . . . . . . . 190
5.1.4 Developablesurfaces . . . . . . . . . . . . . . . . . . . . . . . 191
Developablesurfacesasenvelopesofaone-parameterfamilyof
planes. Theregressionedgeofadevelopablesurface . 193
5.1.5 Developable surfaces associated to the Frenet frame of a space
curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Theenvelopeofthefamilyofosculatingplanes . . . . . . . . . 197
Theenvelopeofthefamilyofnormalplanes(thepolarsurface) 198
Theenvelopeofthefamilyofrectifyingplanesofaspacecurves 199
5.2 Minimalsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
5.2.1 Definitionandgeneralproperties . . . . . . . . . . . . . . . . . 200
5.2.2 Minimalsurfacesofrevolution . . . . . . . . . . . . . . . . . . 206
5.2.3 Ruledminimalsurfaces. . . . . . . . . . . . . . . . . . . . . . 207
5.3 Surfacesofconstantcurvature . . . . . . . . . . . . . . . . . . . . . . 213
Problems 217
Bibliography 229
Index 233
Foreword
Thisbookisbasedonthelecturenotesofseveralcoursesonthedifferentialgeometryof
curvesandsurfacesthatIgaveduringthelasteightyears. Thesecourseswereaddressed
to different audience and, as such, the lecture notes have been revised again and again
andoncealmostentirelyrewritten. Itriedtoexposethematerialinamodernlanguage,
without loosing the contact with the intuition. There is a common believe that mathe-
matics is eternal and it never changes, unlike other sciences. I think that nothing could
befartherfromthetruth. Mathematicsdoeschangecontinuously,eitherfortheneedof
more rigor, either for the need of more structure. The mathematics we are doing and
teaching today is essentially different (and not only as language!) from the mathemat-
ics of the nineteenth century. This variation also applies, of course, to the teaching of
mathematics.
Thereareseveralreasonwhyweneedtorenewthetextbooksfromtimetotime:
• themathematicsevolvesandweneedtointroducenewresultsandnotions;
• somepartsofspecificcoursesbecomeobsoleteandshouldberemoved;
• thecurriculumasawholeevolvesandwehavetokeeppacewiththedevelopments
oftheneighboringfields;
• some courses move from the undergraduate level to the graduate level or (more
often)theotherwayaroundandwehavetomodifythecontentsandthelanguage
accordingly.
TheseareonlysomeofthereasonsIdecidedtowritethisbook.
10 Foreword
The material included is fairly classical. I preferred to discuss in more details the
foundations rather than introduce more and more topics. In particular, unlike most of
books of this kind, I decided not to discuss at all problems of global differential geom-
etry. Ibelieveitisnotveryusefultodiscusssuchtopics,ifthereisnotenoughroomto
givethemalltheattentiontheydeserve. Also,theLevi-Civitaparallelismandthecovari-
ant differentiation are not discussed, because I think they would feel a lot better in the
courseofRiemanniangeometry. Thenotionofdifferentiablemanifoldisnotintroduced
butitisonlyonestepaway. Infact,Idecidedtoadaptthelanguageofthecourseinsuch
awaytoqualifyforaprerequisiteforacourseinsmoothmanifolds.
Isnoteasytobeoriginalwhenyouteachsomethingthathavebeentaughtagainand
againformorethantwocenturiesand,ontheotherhand, shouldwebealwaysoriginal
atanycost?! Assuch,itgoeswithoutsayingthatthisbookowesmuchtomanyexcellent
classical or more modern textbooks. All of them are mentioned in the bibliography. I
havetomention,however,thebooksofGray,doCarmoandFedenko.
IhavetoconfessthatIamnotaself-educatedperson(notentirely,anyway),andmy
ideasregardingdifferentialgeometryhavebeeninfluencedbymyteachers(thelateProf.
MarianT¸arina˘ andProf. DorinAndrica),aswellbythecolleagueswithwhomIhadthe
pleasuretoworkandtosharetheideas,Prof. PavelEnghis¸,CsabaVarga,CornelPintea,
DanielVa˘ca˘re¸tuandLianaT¸opan.
This book would have never existed without my students and their enthusiasm (or
lackofenthusiasm)determinedmanychangesduringtheyears.
A book is not something that you leave at the office and this one was no exception
and it would never have been finished without the infinite patience of my wife who
toleratedthe(undeterministic)chaoscreatedbybooksandmanuscripts.
Cluj-Napoca,April2005.