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[俄罗斯]波斯特尼科夫 著 / 科学出版社 / 2009-01 / 精装
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国外数学名著系列(续1)(影印版)60:几何6(黎曼几何)
ThisbooktreatsthatpartofRiemanniangeometryrelatedtomoreclassicaltopicsinaveryoriginal,clearandsolidstyle.BeforegoingtoRiemanniangeometry,theauthorpresentsamoregeneraltheoryofmanifoldswithalinearconnection.HavinginminddifferentgeneralizationsofRiemannianmanifolds,itisclearlystressedwhichnotionsandtheoremsbelongtoRiemanniangeometryandwhichofthemareofamoregeneralnature.Muchattentionispaidtotransformationgroupsofsmoothmanifolds.
Throughoutthebook,differentaspectsofsymmetricspacesaretreated.Theauthorsuccessfullycombinestheco-ordinateandinvariantapproachestodifferentialgeometry,whichgivethereadertoolsforpracticalcalculationsaswellasatheoreticalunderstandingofthesubject.Thebookcontainsaveryusefullargeappendixonfoundationsofdifferentiablemanifoldsandbasicstructuresonthemwhichmakesitself-containedandpracticallyindependentfromothersources.
Theresultsarewellpresentedandusefulforstudentsinmathematicsandtheoreticalphysics,andforexpertsinthesefields.Thebookcanserveasatextbookforstudentsdoinggeometry,aswellasareferencebookforprofessionalmathematiciansandphysicists.
PrefaceChapter1.AfneConnections1.ConnectiononaManifold2.CovariantDifferentiationandParallelTranslationAlongaCurve3.Geodesics4.ExponentialMappingandNormalNeighborhoods5.WhiteheadTheorem6.NormalConvexNeighborhoods7.ExistenceofLerayCoveringsChapter2.CovariantDifferentiation.Curvature1.CovariantDifferentiation2.TheCaseofTensorsofType(r,1)3.TorsionTensorandSymmetricConnections4.GeometricMeaningoftheSymmetryofaConnection5.CommutativityofSecondCovariantDerivatives6.CurvatureTensorofanAfneConnection7.SpacewithAbsoluteParallelism8.BianciIdentities9.TraceoftheCurvatureTensor10.RicciTensorChapter3.AffineMappings.Submanifolds1.AfneMappings2.Affinities3.AfneCoverings4.RestrictionofaConnectiontoaSubmanifold5.InducedConnectiononaNormalizedSubmanifold6.GaussFormulaandtheSecondFundamentalFormofaNormalizedSubmanifold7.TotallyGeodesicandAuto-ParallelSubmanifolds8.NormalConnectionandtheWeingartenFormula9.VanderWaerden-BortolottiConnectionChapter4.StructuralEquations.LocalSymmetries1.TorsionandCurvatureForms2.CaftanStructuralEquationsinPolarCoordinates3.ExistenceofAfneLocalMappings4.LocallySymmetricAfneConnectionSpaces5.LocalGeodesicSymmetries6.SemisymmetricSpacesChapter5.SymmetricSpaces1.GloballySymmetricSpaces2.GermsofSmoothMappings3.ExtensionsofAffineMappings4.UniquenessTheorem5.ReductionofLocallySymmetricSpacestoGloballySymmetricSpaces6.PropertiesofSymmetriesinGloballySymmetricSpaces7.SymmetricSpaces8.ExamplesofSymmetricSpaces9.CoincidenceofClassesofSymmetricandGloballySymmetricSpacesChapter6.ConnectionsonLieGroups1.InvariantConstructionoftheCanonicalConnection2.MorphismsofSymmetricSpacesasAffineMappings3.Left-InvariantConnectionsonaLieGroup4.CartanConnections5.LeftCartanConnection6.Right-InvariantVectorFields7.RightCartanConnectionChapter7.LieFunctor1.Categories2.Functors3.LieFunctor4.KernelandImageofaLieGroupHomomorphism5.Campbell-HausdorffTheorem6.DynkinPolynomials7.LocalLieGroups8.BijectivityoftheLieFunctorChapter8.AffineFieldsandRelatedTopics1.AffineFields2.DimensionoftheLieAlgebraofAffineFields3.CompletenessofAffineFields4.MappingsofLeftandRightTranslationonaSymmetricSpace5.DerivationsonManifoldswithMultiplication6.LieAlgebraofDerivations7.InvolutiveAutomorphismoftheDerivationAlgebraofaSymmetricSpace8.SymmetricAlgebrasandLieTernaries9.LieTernaryofaSymmetricSpaceChapter9.CartanTheorem1.Functors2.ComparisonoftheFunctorswiththeLieFunctor3.PropertiesoftheFunctors4.ComputationoftheLieTernaryoftheSpace5.FundamentalGroupoftheQuotientSpace6.SymmetricSpacewithaGivenLieTernary7.Coverings8.CartanTheorem9.IdentificationofHomogeneousSpaceswithQuotientSpaces10.TrauslationsofaSymmetricSpace11.ProofoftheCartanTheoremChapter10.PalaisandKobayashiTheorems1.Infinite-DimensionalManifoldsandLieGroups2.VectorFieldsInducedbyaLieGroupAction3.PalaisTheorem4.KobayashiTheorem5.AffineAutomorphismGroup6.AutomorphismGroupofaSymmetricSpace7.TranslationGroupofaSymmetricSpaceChapter11.LagrangiansinRiemannianSpaces1.RiemannianandPseudo-RiemannianSpaces2.RiemannianConnections3.GeodesicsinaRiemannianSpace4.SimplestProblemoftheCalculusofVariations5.Euler-LagrangeEquations6.MinimumCurvesandExtremals7.RegularLagrangians8.ExtremalsoftheEnergyLagrangianChapter12.MetricPropertiesofGeodesics1.LengthofaCurveinaRiemannianSpace2.NaturalParameter3.RiemannianDistanceandShortestArcs4.ExtremalsoftheLengthLagrangian5.RiemannianCoordinates6.GaussLemma7.GeodesicsareLocallyShortestArcs8.SmoothnessofShortestArcs9.LocalExistenceofShortestArcs10.IntrinsicMetric11.Hopf-RinowTheoremChapter13.HarmonicFunctionalsandRelatedTopics1.RiemannianVolumeElement2.DiscriminantTensor3.Foss-WeylFormula4.Casen=25.LaplaceOperatoronaRiemannianSpace6.TheGreenFormulas7.ExistenceofHarmonicFunctionswithaNonzeroDifferential8.ConjugateHarmonicFunctions9.IsothermalCoordinates10.Semi-CartesianCoordinates11.CartesianCoordinatesChapter14.MinimalSurfaces1.ConformalCoordinates2.ConformalStructures3.MinimalSurfaces4.ExplanationofTheirName5.PlateauProblem6.FreeRelativisticStrings7.SimplestProblemoftheCalculusofVariationsforFunctionsofTwoVariables8.ExtremalsoftheAreaFunctional9.Casen=310.RepresentationofMinimalSurfacesViaHolomorphicFunctions11.WeierstrassFormulas12.AdjoinedMinimalSurfacesChapter15.CurvatureinRiemannianSpace1.RiemannianCurvatureTensor2.SymmetriesoftheRiemannianTensor3.RiemannianTensorasaFunctional4.WalkerIdentityandItsConsequences5.RecurrentSpaces6.VirtualCurvatureTensors7.ReconstructionoftheBianciTensorfromItsValuesonBivectors……Chapter16.GaussianCurvatureChapter17.SomeSpecialTensorsChapter18.SurfaceswithConformalStructureChapter19.MappingsandSubmanifoldsⅠChapter20.SubmanifoldsⅡChapter21.FundamentalFormsofaHypersurfaceChapter22.SpacesofConstantCurvatureChapter23.SpaceFormsChapter24.Four-DimensionalManifoldsChapter25.MetricsonaLieGroupⅠChapter26.MetricsonaLieGroupⅡChapter27.JacobiTheoryChapter28.SomeAdditionalTheoremsⅠChapter29.SomeAdditionalTheoremsⅡChapter30.SmoothManifoldsChapter31.TangentVectorsChapter32.SubmanifoldsofaSmoothManifoldChapter33.VectorandTensorFieldsDifferentialFormsChapter34.VectorBundlesChapter35.ConnectionsonVectorBundlesChapter36.CurvatureTensorSuggestedReadingIndex
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