Differential Geometry (Moscow Lectures, 8)

282 Pages·2022·3.64 MB·english

by  Prasolov

Moscow Lectures 8
Victor V. Prasolov
Differential
Geometry
Moscow Lectures
Volume 8
SeriesEditors
Lev D. Beklemishev, Department of Mathematical Logic, Steklov Mathematical
InstituteoftheRussianAcademyofSciences,Moscow,Russia
Vladimir I. Bogachev, Faculty of Mathematics, National Research University,
HigherSchoolofEconomics,Moscow,Russia
BorisFeigin,FacultyofMathematics,NationalResearchUniversity,HigherSchool
ofEconomics,Moscow,Russia
Valery Gritsenko, Faculty of Mathematics, National Research University, Higher
SchoolofEconomics,Moscow,Russia
YulyS.Ilyashenko,IndependentUniversityofMoscow,Moscow,Russia
Dmitry B. Kaledin, Department of Algebraic Geometry, Steklov Mathematical
InstituteoftheRussianAcademyofSciences,Moscow,Russia
AskoldKhovanskii,IndependentUniversityofMoscow,Moscow,Russia
IgorM.Krichever,CenterforAdvancedStudies,SkolkovoInstituteofScienceand
Technology,Moscow,Russia
AndreiD. Mironov,Laboratoryof Higher EnergyPhysics, P.N. LebedevPhysical
Institute.oftheRussianAcademyofSciences,Moscow,Russia
VictorA.Vassiliev,DepartmentofGeometryandTopology,SteklovMathematical
InstituteoftheRussianAcademyofSciences,Moscow,Russia
ManagingEditor
Alexey L. Gorodentsev, Faculty of Mathematics, National Research University,
HigherSchoolofEconomics,Moscow,Russia
MoscowLecturesisanewbookserieswithtextbooksarisingfromallmathematics
institutionsinMoscow.
Moreinformationaboutthisseriesathttps://link.springer.com/bookseries/15875
Victor V. Prasolov
Differential Geometry
VictorV.Prasolov
IndependentUniversityofMoscow
Moscow,Russia
ISSN2522-0314 ISSN2522-0322 (electronic)
MoscowLectures
ISBN978-3-030-92248-1 ISBN978-3-030-92249-8 (eBook)
https://doi.org/10.1007/978-3-030-92249-8
MathematicsSubjectClassification:53-XX
Translated from the Russian by Olga Sipacheva: Originally published as дифференциальная
геометрияbyMCCMEMoscow,2020
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Preface
Differentialgeometrystudiespropertiesofcurves,surfaces,andsmoothmanifolds
by methods of mathematical analysis. Riemannian geometry is a section of dif-
ferential geometry which studies smooth manifolds with an additional structure,
Riemannian metric. The main part of this book is devoted to exactly Riemannian
geometry. The exception is affine differential geometry, projective differential
geometry, and connections more general than the Levi-Civitá connection, which
originatesfromaRiemannianmetric.
Thebookbeginswiththesimplestobjectofdifferentialgeometry,curvesinthe
plane. The most important characteristic of a curve at a given point is curvature.
The first chapter considers both local properties of curvature (the Frenet–Serret
formula and osculating circles) and global ones (the total curvature of a closed
curveandthe four-vertextheorem).The totalorientedcurvatureof a closedcurve
isinvariantwithrespecttoaregularhomotopy,andviceversa:ifthetotaloriented
curvaturesoftwocurvesareequal,thenthesecurvesareregularlyhomotopic(this
isthe Whitney–Grausteintheorem).To eachcurvecorrespondsits evolute,thatis,
thelocusofcentersofallosculatingcirclesofthiscurve,thatis,theenvelopeofthe
familyofnormals.Withrespecttoitsevolute,thegivencurveisaninvolute.Butthe
inverseoperationofassigninganinvolutetoacurveisambiguous:everycurvehas
awholefamilyofinvolutes(acurveorthogonaltoafamilyofnormalscanbedrawn
througheverypointofanormal).Wegivetwoproofsoftheisoperimetricinequality
betweenthelengthofa closedcurvewithoutself-intersectionsandtheareawhich
it bounds. A part of the differential geometry of plane curves is not related to a
Riemannianmetric,thatis, remainsbeyondthe scope ofRiemanniangeometry.It
includesenvelopingfamiliesofcurves,affineunimodulardifferentialgeometry,and
projective differential geometry. The chapter about plane curves is concluded by
elementsofintegralgeometry:wederiveaformulaexpressingthemeasureofaset
ofstraightlinesintersectingagivencurve.
The second chapter studies curves in spaces, first in three-dimensional space
andtheninmany-dimensionalones.Givenacurveinthree-dimensionalspace,we
definecurvatureandtorsion,derivetheFrenet–Serretformula,anddefineosculating
planesandspheres.We alsodefinethe totalcurvatureofa closedcurveandprove
v
vi Preface
Fenchel’s theorem (that the total curvature of a closed curve is at least 2π) and
the Fáry-Milnor theorem (that the total curvature of a knotted closed curve is at
least4π).Attheendofthechapter,weconsidercurvesinmany-dimensionalspace,
definequantitiesgeneralizingcurvatureandtorsionforsuchcurves,andderivethe
generalizedFrenet–Serretformulas.
Thethirdchapterisdevotedtosurfacesinthree-dimensionalspace.Onasurface
in R3, the first quadratic form is introduced; this is the inner product of tangent
vectors to the surface. With a curve on a surface, we associate a Darboux frame
anduseittodefinethegeodesiccurvature,thenormalcurvature,andthegeodesic
torsionofacurveonasurface.Ageodesiconasurfaceisacurvewithzerogeodesic
curvature; any shortest curve on a surface joining two given points is geodesic.
On a surface in R3, the second quadratic form is also defined. In a basis with
respect to which the matrix of the first quadratic form is the identity matrix and
the matrix of the second quadratic form is diagonal, the diagonal elements of the
second quadratic form are the principal curvatures. The Gaussian curvature of a
surface is the product of principal curvatures. The principal curvatures cannot be
expressedonlyintermsofthefirstquadraticform,buttheGaussiancurvaturecan.
TheGaussiancurvatureofasurfacecanalsobedefinedinadifferentway,interms
of differential forms on the surface. The integral of the Gaussian curvature over
a polygon on the surface can be related to the integral of the geodesic curvature
over the boundary of this polygon and the sum of exterior angles of the polygon
(the Gauss–Bonnet formula). We define the parallel transport of a vector along a
curveandthecovariantdifferentiationofvectorfieldsandintroducetheRiemannian
curvaturetensor.Usinggeodesics,wedefinetheexponentialmapofatangentspace
to a surface. To study properties of geodesics, we derive the first and the second
variationformula.UsingJacobivectorfieldsandconjugatepoints,wefindoutwhen
thelengthofageodesicisnotgloballyminimal.Weprovethetheoremonthelocal
isometryofsurfacesofconstantGaussiancurvature.Attheendofthechapter,we
introduce the Laplace–Beltrami operator, which is a generalization to surfaces of
theLaplaceoperatorintheplane.
Inthefourthchapterwediscusstwotopics,hypersurfacesinmany-dimensional
space and connections on vector bundles. The study of connections on vector
bundlesisbasedoncertainprerequisitesconcerningmanifoldsandvectorbundles
over manifolds. Thus, beginning in Chap. 4, the reader is supposed to have
background knowledge of manifolds, tangent vectors, differential forms, vector
bundles over manifolds, sections of bundles, and the inverse function theorem;
all the necessary information can be found in the books [Pr2] and [Pr3]. For a
hypersurfacein Euclidean space, we define the Weingarten operator and use it to
introduce the second, third, etc. quadratic forms. We define connections first on
hypersurfaces and then on any manifolds and vector bundles over manifolds. In
parallel,weintroducegeodesicswithrespecttoagivenconnection.Wealsodefine
thecurvaturetensorandthetorsiontensorofagivenconnectionandintroducethe
curvaturematrixofaconnection.
ThefifthchapterisconcernedwiththegeneraltheoryofRiemannianmanifolds.
ARiemannianmanifoldisamanifoldwhoseeverytangentspaceisequippedwith
Preface vii
An inner product(Riemannian metric). On a Riemannian manifold, there exists a
unique torsion-free connection compatible with the Riemannian metric (the Levi-
Civitàconnection).TheRiemanntensorofthisconnectionhasseveralsymmetries
(satisfiesseveralidentities).GeodesicsonRiemannianmanifoldshavesomespecific
featuresin comparisonwith geodesicsforarbitraryconnections;in particular,any
such geodesic is locally a shortest curve. A Riemannian manifold is said to be
geodesically complete if all geodesics on this manifold can be extended without
bound.AccordingtotheHopf–Rinowtheorem,geodesiccompletenessisequivalent
to the completeness of the Riemannian manifold as a metric space. The Riemann
tensor can be described by using the sectional curvatures corresponding to two-
dimensionalsubspaces.ForRiemanniansubmanifolds,aswellasforhypersurfaces,
we can introduce the second quadratic form and prove generalizations of Gauss’
and Weingarten’s formulas. An important class of Riemannian submanifolds is
formed by totally geodesic submanifolds (a submanifold M is totally geodesic if
each geodesic on M is also a geodesic on the ambient manifold). In the many-
dimensionalcase,justasinthecaseofsurfaces,weobtainthefirstandthesecond
variation formulas and use them to introduce Jacobi fields and define conjugate
points.Thechapterisconcludedbyadiscussionoftheholonomy(transformations
of the tangent space obtained by the parallel transport of vectors along closed
curves)andaninterpretationofcurvatureasinfinitesimalholonomy.
The sixth chapter discusses the differential geometry of Lie groups, that is,
manifolds endowed with a group structure consistent with the smooth structure.
With each Lie group, its Lie algebra is associated, which is the tangent space at
theidentityelementinwhichthemultiplicationofelementsisdefinedastakingthe
commutatoroftheleft-invariantvectorfieldscorrespondingtotangentvectors.The
Lie algebra of a Lie group is mapped to this Lie group by the exponential map.
For a Lie groupand a Lie algebra,adjointrepresentationsare defined;the adjoint
representationofa Lie algebrais usedtodefinethe Killingform.On aLie group,
thereexistvariousconnectionsandmetricsrelatedtothegroupstructureinvarious
ways.OnacompactLiegroup,invariantintegrationcanbedefined.Someproperties
ofLiegroupsarepossessedbymoregeneralspaces,namely,byhomogeneousand
symmetricones.
The last (seventh)chapteris devotedto some of the applicationsof differential
geometry: comparison theorems, relationship between curvature and topological
propertiesofmanifolds,andtheLaplaceoperatoronRiemannianmanifolds.
BasicNotation
kisthecurvatureofacurveintheplane(p.3)orspace(p.48)
e ,e ,e istheSerret–Frenetframe,p.47
1 2 3
(cid:3) isthetorsionofaspacecurve,p.48
E,F,andGarethecoefficientsofthefirstquadraticform,p.66
g arethecoefficientsofthefirstquadraticform,p.67
ij
viii Preface
gij aretheentriesofthematrixinversetothematrix(g ),p.68
ij
ε ,ε ,ε istheDarbouxframe,p.68
1 2 3
k isgeodesiccurvature,p.69
g
k isnormalcurvature,p.69
n
(cid:3) isgeodesictorsion,p.69
g
L,M,andN arethecoefficientsofthesecondquadraticform,p.72
H ismeancurvature,p.73
K isGaussiancurvature,p.73
(cid:5)k aretheChristoffelsymbols,p.85
ij
S2isthesphere,p.87
∇ V isthecovariantderivativeofavectorfieldV inthedirectionofavectorfield
W
W,p.93
Rl istheRiemanniancurvaturetensor,p.98
ijk
exp (V)istheexponentialmap,p.100
p
Moscow,Russia VictorV.Prasolov
Contents
1 CurvesinthePlane .......................................................... 1
1.1 CurvatureandtheFrenet–SerretFormulas........................... 3
1.2 OsculatingCircles ..................................................... 6
1.3 TheTotalCurvatureofaClosedPlaneCurve........................ 7
1.4 Four-VertexTheorem.................................................. 10
1.5 TheNaturalEquationofaPlaneCurve............................... 13
1.6 Whitney–GrausteinTheorem ......................................... 14
1.7 TubeAreaandSteiner’sFormula..................................... 15
1.8 TheEnvelopeofaFamilyofCurves.................................. 16
1.9 EvoluteandInvolute................................................... 21
1.10 IsoperimetricInequality............................................... 23
1.11 AffineUnimodularDifferentialGeometry ........................... 26
1.12 ProjectiveDifferentialGeometry ..................................... 29
1.13 TheMeasureoftheSetofLinesIntersectingaGivenCurve........ 33
1.14 SolutionsofProblems................................................. 36
2 CurvesinSpace .............................................................. 47
2.1 CurvatureandTorsion:TheFrenet–SerretFormulas ................ 47
2.2 AnOsculatingPlane................................................... 51
2.3 TotalCurvatureofaClosedCurve.................................... 53
2.4 BertrandCurves........................................................ 55
2.5 TheFrenet–SerretFormulasinMany-DimensionalSpace .......... 56
2.6 SolutionsofProblems................................................. 57
3 SurfacesinSpace............................................................. 65
3.1 TheFirstQuadraticForm ............................................. 66
3.2 TheDarbouxFrameofaCurveonaSurface ........................ 68
3.3 Geodesics .............................................................. 70
3.4 TheSecondQuadraticForm .......................................... 72
3.5 GaussianCurvature.................................................... 75
3.6 GaussianCurvatureandDifferentialForms.......................... 77
3.7 TheGauss–BonnetTheorem.......................................... 80
ix