国外数学名著系列（续1）（影印版）60：几何6（黎曼几何）

图书标准信息

• 作者 [俄罗斯]波斯特尼科夫 著
• 出版社 科学出版社
• 出版时间 2009-01
• 版次 1
• ISBN 9787030235039
• 定价 96.00元
• 装帧 精装
• 开本 16开
• 纸张 胶版纸
• 页数 503页
• 字数 634千字
• 正文语种 英语

【内容简介】

ThisbooktreatsthatpartofRiemanniangeometryrelatedtomoreclassicaltopicsinaveryoriginal，clearandsolidstyle.BeforegoingtoRiemanniangeometry，theauthorpresentsamoregeneraltheoryofmanifoldswithalinearconnection.HavinginminddifferentgeneralizationsofRiemannianmanifolds，itisclearlystressedwhichnotionsandtheoremsbelongtoRiemanniangeometryandwhichofthemareofamoregeneralnature.Muchattentionispaidtotransformationgroupsofsmoothmanifolds.
Throughoutthebook，differentaspectsofsymmetricspacesaretreated.Theauthorsuccessfullycombinestheco-ordinateandinvariantapproachestodifferentialgeometry，whichgivethereadertoolsforpracticalcalculationsaswellasatheoreticalunderstandingofthesubject.Thebookcontainsaveryusefullargeappendixonfoundationsofdifferentiablemanifoldsandbasicstructuresonthemwhichmakesitself-containedandpracticallyindependentfromothersources.
Theresultsarewellpresentedandusefulforstudentsinmathematicsandtheoreticalphysics，andforexpertsinthesefields.Thebookcanserveasatextbookforstudentsdoinggeometry，aswellasareferencebookforprofessionalmathematiciansandphysicists.

【目录】

Preface
Chapter1.AfneConnections
1.ConnectiononaManifold
2.CovariantDifferentiationandParallelTranslationAlongaCurve
3.Geodesics
4.ExponentialMappingandNormalNeighborhoods
5.WhiteheadTheorem
6.NormalConvexNeighborhoods
7.ExistenceofLerayCoverings

Chapter2.CovariantDifferentiation.Curvature
1.CovariantDifferentiation
2.TheCaseofTensorsofType（r，1）
3.TorsionTensorandSymmetricConnections
4.GeometricMeaningoftheSymmetryofaConnection
5.CommutativityofSecondCovariantDerivatives
6.CurvatureTensorofanAfneConnection
7.SpacewithAbsoluteParallelism
8.BianciIdentities
9.TraceoftheCurvatureTensor
10.RicciTensor

Chapter3.AffineMappings.Submanifolds
1.AfneMappings
2.Affinities
3.AfneCoverings
4.RestrictionofaConnectiontoaSubmanifold
5.InducedConnectiononaNormalizedSubmanifold
6.GaussFormulaandtheSecondFundamentalFormofaNormalizedSubmanifold
7.TotallyGeodesicandAuto-ParallelSubmanifolds
8.NormalConnectionandtheWeingartenFormula
9.VanderWaerden-BortolottiConnection

Chapter4.StructuralEquations.LocalSymmetries
1.TorsionandCurvatureForms
2.CaftanStructuralEquationsinPolarCoordinates
3.ExistenceofAfneLocalMappings
4.LocallySymmetricAfneConnectionSpaces
5.LocalGeodesicSymmetries
6.SemisymmetricSpaces

Chapter5.SymmetricSpaces
1.GloballySymmetricSpaces
2.GermsofSmoothMappings
3.ExtensionsofAffineMappings
4.UniquenessTheorem
5.ReductionofLocallySymmetricSpacestoGloballySymmetricSpaces
6.PropertiesofSymmetriesinGloballySymmetricSpaces
7.SymmetricSpaces
8.ExamplesofSymmetricSpaces
9.CoincidenceofClassesofSymmetricandGloballySymmetricSpaces

Chapter6.ConnectionsonLieGroups
1.InvariantConstructionoftheCanonicalConnection
2.MorphismsofSymmetricSpacesasAffineMappings
3.Left-InvariantConnectionsonaLieGroup
4.CartanConnections
5.LeftCartanConnection
6.Right-InvariantVectorFields
7.RightCartanConnection

Chapter7.LieFunctor
1.Categories
2.Functors
3.LieFunctor
4.KernelandImageofaLieGroupHomomorphism
5.Campbell-HausdorffTheorem
6.DynkinPolynomials
7.LocalLieGroups
8.BijectivityoftheLieFunctor

Chapter8.AffineFieldsandRelatedTopics
1.AffineFields
2.DimensionoftheLieAlgebraofAffineFields
3.CompletenessofAffineFields
4.MappingsofLeftandRightTranslationonaSymmetricSpace
5.DerivationsonManifoldswithMultiplication
6.LieAlgebraofDerivations
7.InvolutiveAutomorphismoftheDerivationAlgebraofaSymmetricSpace
8.SymmetricAlgebrasandLieTernaries
9.LieTernaryofaSymmetricSpace

Chapter9.CartanTheorem
1.Functors
2.ComparisonoftheFunctorswiththeLieFunctor
3.PropertiesoftheFunctors
4.ComputationoftheLieTernaryoftheSpace
5.FundamentalGroupoftheQuotientSpace
6.SymmetricSpacewithaGivenLieTernary
7.Coverings
8.CartanTheorem
9.IdentificationofHomogeneousSpaceswithQuotientSpaces
10.TrauslationsofaSymmetricSpace
11.ProofoftheCartanTheorem

Chapter10.PalaisandKobayashiTheorems
1.Infinite-DimensionalManifoldsandLieGroups
2.VectorFieldsInducedbyaLieGroupAction
3.PalaisTheorem
4.KobayashiTheorem
5.AffineAutomorphismGroup
6.AutomorphismGroupofaSymmetricSpace
7.TranslationGroupofaSymmetricSpace

Chapter11.LagrangiansinRiemannianSpaces
1.RiemannianandPseudo-RiemannianSpaces
2.RiemannianConnections
3.GeodesicsinaRiemannianSpace
4.SimplestProblemoftheCalculusofVariations
5.Euler-LagrangeEquations
6.MinimumCurvesandExtremals
7.RegularLagrangians
8.ExtremalsoftheEnergyLagrangian

Chapter12.MetricPropertiesofGeodesics
1.LengthofaCurveinaRiemannianSpace
2.NaturalParameter
3.RiemannianDistanceandShortestArcs
4.ExtremalsoftheLengthLagrangian
5.RiemannianCoordinates
6.GaussLemma
7.GeodesicsareLocallyShortestArcs
8.SmoothnessofShortestArcs
9.LocalExistenceofShortestArcs
10.IntrinsicMetric
11.Hopf-RinowTheorem

Chapter13.HarmonicFunctionalsandRelatedTopics
1.RiemannianVolumeElement
2.DiscriminantTensor
3.Foss-WeylFormula
4.Casen=2
5.LaplaceOperatoronaRiemannianSpace
6.TheGreenFormulas
7.ExistenceofHarmonicFunctionswithaNonzeroDifferential
8.ConjugateHarmonicFunctions
9.IsothermalCoordinates
10.Semi-CartesianCoordinates
11.CartesianCoordinates

Chapter14.MinimalSurfaces
1.ConformalCoordinates
2.ConformalStructures
3.MinimalSurfaces
4.ExplanationofTheirName
5.PlateauProblem
6.FreeRelativisticStrings
7.SimplestProblemoftheCalculusofVariationsforFunctionsofTwoVariables
8.ExtremalsoftheAreaFunctional
9.Casen=3
10.RepresentationofMinimalSurfacesViaHolomorphicFunctions
11.WeierstrassFormulas
12.AdjoinedMinimalSurfaces

Chapter15.CurvatureinRiemannianSpace
1.RiemannianCurvatureTensor
2.SymmetriesoftheRiemannianTensor
3.RiemannianTensorasaFunctional
4.WalkerIdentityandItsConsequences
5.RecurrentSpaces
6.VirtualCurvatureTensors
7.ReconstructionoftheBianciTensorfromItsValuesonBivectors
……
Chapter16.GaussianCurvature
Chapter17.SomeSpecialTensors
Chapter18.SurfaceswithConformalStructure
Chapter19.MappingsandSubmanifoldsⅠ
Chapter20.SubmanifoldsⅡ
Chapter21.FundamentalFormsofaHypersurface
Chapter22.SpacesofConstantCurvature
Chapter23.SpaceForms
Chapter24.Four-DimensionalManifolds
Chapter25.MetricsonaLieGroupⅠ
Chapter26.MetricsonaLieGroupⅡ
Chapter27.JacobiTheory
Chapter28.SomeAdditionalTheoremsⅠ
Chapter29.SomeAdditionalTheoremsⅡ
Chapter30.SmoothManifolds
Chapter31.TangentVectors
Chapter32.SubmanifoldsofaSmoothManifold
Chapter33.VectorandTensorFieldsDifferentialForms
Chapter34.VectorBundles
Chapter35.ConnectionsonVectorBundles
Chapter36.CurvatureTensor
SuggestedReading
Index