Table Of ContentVictor Andreevich Toponogov
with the editorial assistance of
Vladimir Y. Rovenski
Differential Geometry
of Curves and Surfaces
A Concise Guide
Birkha¨user
Boston • Basel • Berlin
VictorA.Toponogov(deceased) Withtheeditorialassistanceof:
DepartmentofAnalysisandGeometry VladimirY.Rovenski
SobolevInstituteofMathematics DepartmentofMathematics
SiberianBranchoftheRussianAcademy UniversityofHaifa
ofSciences Haifa,Israel
Novosibirsk-90,630090
Russia
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Contents
Preface ....................................................... vii
AbouttheAuthor .............................................. ix
1 TheoryofCurvesinThree-dimensionalEuclideanSpaceandinthe
Plane....................................................... 1
1.1 Preliminaries .............................................. 1
1.2 DefinitionandMethodsofPresentationofCurves ............... 2
1.3 TangentLineandOsculatingPlane............................ 6
1.4 LengthofaCurve .......................................... 11
1.5 Problems:ConvexPlaneCurves .............................. 15
1.6 CurvatureofaCurve........................................ 19
1.7 Problems:CurvatureofPlaneCurves .......................... 24
1.8 TorsionofaCurve.......................................... 45
1.9 TheFrenetFormulasandtheNaturalEquationofaCurve......... 47
1.10 Problems:SpaceCurves..................................... 51
1.11 PhaseLengthofaCurveandtheFenchel–ReshetnyakInequality... 56
1.12 ExercisestoChapter1 ...................................... 61
2 ExtrinsicGeometryofSurfacesinThree-dimensionalEuclidean
Space ...................................................... 65
2.1 DefinitionandMethodsofGeneratingSurfaces ................. 65
2.2 TheTangentPlane.......................................... 70
2.3 FirstFundamentalFormofaSurface .......................... 74
vi Contents
2.4 SecondFundamentalFormofaSurface ........................ 79
2.5 TheThirdFundamentalFormofaSurface...................... 91
2.6 ClassesofSurfaces ......................................... 95
2.7 SomeClassesofCurvesonaSurface .......................... 114
2.8 TheMainEquationsofSurfaceTheory ........................ 127
2.9 Appendix:IndicatrixofaSurfaceofRevolution ................. 139
2.10 ExercisestoChapter2 ...................................... 147
3 IntrinsicGeometryofSurfaces .................................151
3.1 IntroducingNotation........................................ 151
3.2 CovariantDerivativeofaVectorField ......................... 152
3.3 Parallel Translation of a Vector along a
CurveonaSurface ......................................... 153
3.4 Geodesics................................................. 156
3.5 ShortestPathsandGeodesics................................. 161
3.6 SpecialCoordinateSystems.................................. 172
3.7 Gauss–BonnetTheoremandComparisonTheoremfortheAngles
ofaTriangle............................................... 179
3.8 LocalComparisonTheoremsforTriangles ..................... 184
3.9 AleksandrovComparisonTheoremfortheAnglesofaTriangle.... 189
3.10 ProblemstoChapter3....................................... 195
References ..................................................199
Index.......................................................203
Preface
This concise guide to the differential geometry of curves and surfaces can be
recommendedtofirst-yeargraduatestudents,strongseniorstudents,andstudents
specializingingeometry.Thematerialisgivenintwoparallelstreams.
Thefirststreamcontainsthestandardtheoreticalmaterialondifferentialgeom-
etry of curves and surfaces. It contains a small number of exercises and simple
problemsofalocalnature.ItincludesthewholeofChapter1exceptfortheprob-
lems(Sections1.5,1.7,1.10)andSection1.11,aboutthephaselengthofacurve,
andthewholeofChapter2exceptforSection2.6,aboutclassesofsurfaces,The-
orems 2.8.1–2.8.4, the problems (Sections 2.7.4, 2.8.3) and the appendix (Sec-
tion2.9).
Thesecondstreamcontainsmoredifficultandadditionalmaterialandformu-
lationsofsomecomplicatedbutimportanttheorems,forexample,aproofofA.D.
Aleksandrov’s comparison theorem about the angles of a triangle on a convex
surface,1 formulationsofA.V.Pogorelov’stheoremaboutrigidityofconvexsur-
faces, and S.N. Bernstein’s theorem about saddle surfaces. In the last case, the
formulationsarediscussedindetail.
Adistinctivefeatureofthebookisalargecollection(80to90)ofnonstandard
andoriginalproblemsthatintroducethestudentintotherealworldofgeometry.
Most of these problems are new and are not to be found in other textbooks or
booksofproblems.Thesolutionstothemrequireinventivenessandgeometrical
intuition. In this respect, this book is not far from W. Blaschke’s well-known
1AgeneralizationofAleksandrov’sglobalanglecomparisontheoremtoRiemannianspacesofar-
bitrarydimensionisknownasToponogov’stheorem.
viii Preface
manuscript [Bl], but it contains a number of problems more contemporary in
theme. The key to these problems is the notion of curvature: the curvature of
a curve, principal curvatures, and the Gaussian curvature of a surface. Almost
all the problems are given with their solutions, although the hope of the author
is that an honest student will solve them without assistance, and only in excep-
tional cases will look at the text for a solution. Since the problems are given in
increasingorderofdifficulty,eventhemostdifficultofthemshouldbesolvable
byamotivatedreader.Insomecases,onlyshortinstructionsaregiven.Intheau-
thor’sopinion,itisthelargenumberoforiginalproblemsthatmakesthistextbook
interestinganduseful.
Chapter 3, Intrinsic Geometry of a Surface, starts from the main notion of a
covariant derivative of a vector field along a curve. The definition is based on
extrinsicgeometricalpropertiesofasurface.Thenitisproventhatthecovariant
derivativeofavectorfieldisanobjectoftheintrinsicgeometryofasurface,and
thelatertrainingmaterialisnotrelatedtoanextrinsicgeometry.SoChapter3can
beconsideredanintroductionton-dimensionalRiemanniangeometrythatkeeps
thesimplicityandclarityofthe2-dimensionalcase.
Themaintheoremsaboutgeodesicsandshortestpathsareprovenbymethods
thatcanbeeasilyextendedton-dimensionalsituationsalmostwithoutalteration.
TheAleksandrovcomparisontheorem,Theorem3.9.1,fortheanglesofatriangle
isthehighpointinChapter3.
The author is one of the founders of CAT(k)-spaces theory,2 where the com-
parison theorem for the angles of a triangle, or more exactly its generalization
by the author to multidimensional Riemannian manifolds, takes the place of the
basicpropertyofCAT(k)-spaces.
Acknowledgments.Theauthorgratefullythankshisstudentandcolleagueswho
have contributed to this volume. Essential help was given by E.D.Rodionov,
V.V.Slavski,V.Yu.Rovenski,V.V.Ivanov,V.A.Sharafutdinov,andV.K.Ionin.
2TheinitialsareinhonorofE.Cartan,A.D.Aleksandrov,andV.A.Toponogov.
About the Author
Professor Victor Andreevich Toponogov, a well-known Russian geometer, was
bornonMarch6,1930,andgrewupinthecityofTomsk,inRussia.DuringTo-
ponogov’schildhood,hisfatherwassubjectedtoSovietrepression.Afterfinish-
ingschoolin1948,ToponogoventeredtheDepartmentofMechanicsandMath-
ematicsatTomskUniversity,andgraduatedin1953withhonors.
In spite of an active social position and receiving high marks in his studies,
thestampof“sonofanenemyofthepeople”leftToponogovwithlittlehopeof
continuinghiseducationatthepostgraduatelevel.However,afterJosephStalin’s
deathinMarch1953,thesituationintheUSSRchanged,andToponogovbecame
apostgraduatestudentatTomskUniversity.Toponogov’sscientificinterestswere
influenced by his scientific advisor, Professor A.I. Fet (a recognized topologist
and specialistin variational calculus inthe large, apupil ofL.A. Lusternik) and
bytheworksofAcademicianA.D.Aleksandrov.1
In 1956, V.A. Toponogov moved to Novosibirsk, where in April 1957 he be-
came a research scientist at the Institute of Radio-Physics and Electronics, then
directed by the well-known physicist Y.B. Rumer. In December 1958, Topono-
govdefendedhisPh.D.thesisatMoscowStateUniversity.Inhisdissertation,the
AleksandrovconvexityconditionwasextendedtomultidimensionalRiemannian
manifolds. Later, this theorem came to be called the Toponogov (comparison)
theorem.2 In April 1961, Toponogov moved to the Institute of Mathematics and
1AleksandrDanilovichAleksandrov(1912–1999).
2Meyer,W.T.Toponogov’sTheoremandApplications.LectureNotes,CollegeonDifferentialGe-
ometry,Trieste.1989.
x AbouttheAuthor
Computer Center of the Siberian Branch of the Russian Academy of Sciences
at its inception. All his subsequent scientific activity is related to the Institute
of Mathematics. In 1968, at this institute he defended his doctoral thesis on the
theme“ExtremalproblemsforRiemannianspaceswithcurvatureboundedfrom
above.”
From 1980 to 1982, Toponogov was deputy director of the Institute of Math-
ematics, and from 1982 to 2000 he was head of one of the laboratories of the
institute. In 2001 he became Chief Scientist of the Department of Analysis and
Geometry.
ThefirstthirtyyearsofToponogov’sscientificlifeweredevotedtooneofthe
mostimportantdivisionsofmoderngeometry:Riemanniangeometryinthelarge.
Fromsecondary-schoolmathematics,everybodyhaslearnedsomethingabout
synthetic methods in geometry, concerned with triangles, conditions of their
equalityandsimilarity,etc.FromtheArchimedeanera,analyticalmethodshave
come to penetrate geometry: this is expressed most completely in the theory of
surfaces, created by Gauss. Since that time, these methods have played a lead-
ingpartindifferentialgeometry.InthefundamentalworksofA.D.Aleksandrov,
syntheticmethodsareagainused,becausetheobjectsunderstudyarenotsmooth
enoughforapplicationsofthemethodsofclassicalanalysis.Inthecreativework
ofV.A.Toponogov,bothofthesemethods,syntheticandanalytic,areinharmonic
correlation.
TheclassicresultinthisareaistheToponogovtheoremabouttheanglesofa
trianglecomposedofgeodesics.Thisin-depththeoremisthebasisofmodernin-
vestigationsoftherelationsbetweencurvatureproperties,geodesicbehavior,and
thetopologicalstructureofRiemannianspaces.Intheproofofthistheorem,some
ideas of A.D. Aleksandrov are combined with the in-depth analytical technique
relatedtotheJacobidifferentialequation.
ThemethodsdevelopedbyV.A.Toponogovallowedhimtoobtainasequence
offundamentalresultssuchascharacteristicsofthemultidimensionalsphereby
estimates of the Riemannian curvature and diameter, the solution to the Rauch
problem for the even-dimensional case, and the theorem about the structure of
Riemannianspacewithnonnegativecurvaturecontainingastraightline(i.e.,the
shortestpaththatmaybelimitlesslyextendedinbothdirections).Thisandother
theorems of V.A. Toponogov are included in monographs and textbooks written
byanumberofauthors.HismethodshavehadagreatinfluenceonmodernRie-
manniangeometry.
During the last fifteen years of his life, V.A. Toponogov devoted himself to
differentialgeometryoftwo-dimensionalsurfacesinthree-dimensionalEuclidean
space. He made essential progress in a direction related to the Efimov theorem
aboutthenonexistenceofisometricembeddingofacompleteRiemannianmetric
with a separated-from-zero negative curvature into three-dimensional Euclidean
space, and with the Milnor conjecture declaring that an embedding with a sum
ofabsolutevaluesofprincipalcurvaturesuniformlyseparatedfromzerodoesnot
exist.
AbouttheAuthor xi
Toponogov devoted much effort to the training of young mathematicians. He
wasalectureratNovosibirskStateUniversityandNovosibirskStatePedagogical
University for more than forty-five years. More than ten of his pupils defended
theirPh.D.theses,andseventheirdoctoraldegrees.
V.A. Toponogov passed away on November 21, 2004 and is survived by his
wife,LjudmilaPavlovnaGoncharova,andthreesons.
Differential Geometry of
Curves and Surfaces
