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流形导论

流形导论

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作者: [法] 图

出版社: 世界图书出版公司

出版时间: 2015-01

版次: 2

ISBN: 9787510084485

定价: 79.00

装帧: 平装

开本: 24开

纸张: 胶版纸

页数: 410页

正文语种: 英语

分类: 自然科学

3 张插图图片

　　Thisisacompletelyrevisededition,withmorethanfiftypagesofnewmaterialscatteredthroughout.Inkeepingwiththeconventionalmeaningofchaptersandsections,Ihavereorgaruzedthebookintotwenty-ninesectionsinsevenchapters.ThemainadditionsareSection200ntheLiederivativeandinteriormultiplication,twointrinsicoperationsonamanifoldtooimportanttoleaveout,newcriteriainSection21fortheboundaryorientation,andanewappendixonquaternionsandthesymplecticgroup.
　　Apartfromcorrectingerrorsandmisprints,Ihavethoughtthrougheveryproofagain,clarifiedmanypassages,andaddednewexamples,exercises,hints,andsolutions.Intheprocess,everysectionhasbeenrewritten,sometimesquitedrastically.Therevisionsaresoextensivethatitisnotpossibletoenumeratethemallhere.Eachchapternowcomeswithanintroductoryessaygivinganoverviewofwhatistocome.Toprovideatimelineforthedevelopmentofideas,Ihaveindicatedwheneverpossi-blethehistoricaloriginoftheconcepts,andhaveaugmentedthebibliographywithhistoricalreferences. PrefacetotheSecondEdition
PrefacetotheFirstEdition
ABriefIntroduction

Chapter1EuclideanSpaces
1SmoothFunctionsonaEuclideanSpace
1.1C∞VersusAnalyticFunctions
1.2Taylor'sTheoremwithRemainder
Problems
2TangentVectorsinRnasDerivations
2.1TheDirectionalDerivative
2.2GermsofFunctions
2.3DerivationsataPoint
2.4VectorFields
2.5VectorFieldsasDerivations
Problems
3TheExteriorAlgebraofMulticovectors
3.1DualSpace
3.2Permutations
3.3MultilinearFunctions
3.4ThePermutationActiononMultilinearFunctions
3.5TheSymmetrizingandAlternatingOperators
3.6TheTensorProduct
3.7TheWedgeProduct
3.8AnticommutativityoftheWedgeProduct
3.9AssociativityoftheWedgeProduct
3.10ABasisfork—Covectors
Problems
4DifferentialFormsonRn
4.1Differential1—FormsandtheDifferentialofaFunction
4.2Differentialk—Forms
4.3DifferentialFormsasMultilinearFunctionsonVectorFields
4.4TheExteriorDerivative
4.5ClosedFormsandExactForms
4.6ApplicationstoVectorCalculus
4.7ConventiononSubscriptsandSuperscripts
Problems

Chapter2Manifolds
5Manifolds
5.1TopologicalManifolds
5.2CompatibleCharts
5.3SmoothManifolds
5.4ExamplesofSmoothManifolds
Problems
6SmoothMapsonaManifold
6.1SmoothFunctionsonaManifold
6.2SmoothMapsBetweenManifolds
6.3Diffeomorphisms
6.4SmoothnessinTermsofComponents
6.5ExamplesofSmoothMaps
6.6PartialDerivatives
6.7TheInverseFunctionTheorem
Problems
7Quotients
7.1TheQuotientTopology
7.2ContinuityofaMaponaQuotient
7.3IdentificationofaSubsettoaPoint
7.4ANecessaryConditionforaHausdorffQuotient
7.5OpenEquivalenceRelations
7.6RealProjectiveSpace
7.7TheStandardC∞AtlasonaRealProjectiveSpace
Problems

Chapter3TheTangentSpace
8TheTangentSpace
8.1TheTangentSpaceataPoint
8.2TheDifferentialofaMap
8.3TheChainRule
8.4BasesfortheTangentSpaceataPoint
8.5ALocalExpressionfortheDifferential
8.6CurvesinaManifold
8.7ComputingtheDifferentialUsingCurves
8.8ImmersionsandSubmersions
8.9Rank，andCriticalandRegularPoints
Problems
9Submanifolds
9.1Submanifolds
9.2LevelSetsofaFunction
9.3TheRegularLevelSetTheorem
9.4ExamplesofRegularSubmanifolds
Problems
10CategoriesandFunctors
10.1Categories
10.2Functors
10.3TheDualFunctorandtheMulticovectorFunctor
Problems
11TheRankofaSmoothMap
11.1ConstantRankTheorem
11.2TheImmersionandSubmersionTheorems
11.3ImagesofSmoothMaps
11.4SmoothMapsintoaSubmanifold
11.5TheTangentPlanetoaSurfaceinR3
Problems
12TheTangentBundle
12.1TheTopologyoftheTangentBundle
12.2TheManifoldStructureontheTangentBundle
12.3VectorBundles
12.4SmoothSections
12.5SmoothFrames
Problems
13BumpFunctionsandPartitionsofUnity
13.1C∞BumpFunctions
13.2PartitionsofUnity
13.3ExistenceofaPartitionofUnity
Problems
14VectorFields
14.1SmoothnessofaVectorField
14.2IntegralCurves
14.3LocalFlows
14.4TheLieBracket
14.5ThePushforwardofVectorFields
14.6RelatedVectorFields
Problems

Chapter4LieGroupsandLieAlgebras
15LieGroups
15.1ExamplesofLieGroups
15.2LieSubgroups
15.3TheMatrixExponential
15.4TheTraceofaMatrix
15.5TheDifferentialofdetattheIdentity
Problems
16LieAlgebras
16.1TangentSpaceattheIdentityofaLieGroup
16.2Left—InvariantVectorFieldsonaLieGroup
16.3TheLieAlgebraofaLieGroup
16.4TheLieBracketongl（n，R）
16.5ThePushforwardofLeft—InvariantVectorFields
16.6TheDifferentialasaLieAlgebraHomomorphism
Problems

Chapter5DifferentialForms
17Differential1—Forms
17.1TheDifferentialofaFunction
17.2LocalExpressionforaDifferential1—Form
17.3TheCotangentBundle
17.4CharacterizationofC∞l—Forms
17.5Pullbackofl—Forms
17.6Restrictionofl—FormstoanImmersedSubmanifold
Problems
18Differentialk—Forms
18.1DifferentialForms
18.2LocalExpressionforak—Form
18.3TheBundlePointofView
18.4Smoothk—Forms
18.5Pullbackofk—Forms
18.6TheWedgeProduct
18.7DifferentialFormsonaCircle
18.8InvariantFormsonaLieGroup
Problems
19TheExteriorDerivative
19.1ExteriorDerivativeonaCoordinateChart
19.2LocalOperators
19.3ExistenceofanExteriorDerivativeonaManifold
19.4UniquenessoftheExteriorDerivative
19.5ExteriorDifferentiationUnderaPullback
19.6Restrictionofk—FormstoaSubmanifold
19.7ANowhere—Vanishing1—FormontheCircle
Problems
20TheLieDerivativeandInteriorMultiplication
20.1FamiliesofVectorFieldsandDifferentialForms
20.2TheLieDerivativeofaVectorField
20.3TheLieDerivativeofaDifferentialForm
20.4InteriorMultiplication
20.5PropertiesoftheLieDerivative
20.6GlobalFormulasfortheLieandExteriorDerivatives
Problems

Chapter6Integration
21Orientations
21.1OrientationsofaVectorSpace
21.2Orientationsandn—Covectors
21.3OrientationsonaManifold
21.4OrientationsandDifferentialForms
21.5OrientationsandAtlases
Problems
22ManifoldswithBoundary
22.1SmoothInvarianceofDomaininRn
22.2ManifoldswithBoundary
22.3TheBoundaryofaManifoldwithBoundary
22.4TangentVectors，DifferentialForms，andOrientations
22.5Outward—PointingVectorFields
22.6BoundaryOrientation
Problems
23IntegrationonManifolds
23.1TheRiemannIntegralofaFunctiononRn
23.2IntegrabilityConditions
23.3TheIntegralofann—FormonRn
23.4IntegralofaDifferentialFormoveraManifold
23.5Stokes'sTheorem
23.6LineIntegralsandGreen'sTheorem
Problems

Chapter7DeRhamTheory
24DeRhamCohomology
24.1DeRharnCohomology
24.2ExamplesofdeRhamCohomology
24.3DiffeomorphismInvariance
24.4TheRingStructureondeRhamCohomology
Problems
25TheLongExactSequenceinCohomology
25.1ExactSequences
25.2CohomologyofCochainComplexes
25.3TheConnectingHomomorphism
25.4TheZig—ZagLemma
Problems
26TheMayer—VietorisSequence
26.1TheMayer—VietorisSequence
26.2TheCohomologyoftheCircle
26.3TheEulerCharacteristic
Problems
27HomotopyInvariance
27.1SmoothHomotopy
27.2HomotopyType
27.3DeformationRetractions
27.4TheHomotopyAxiomfordeRhamCohomology
Problems
28ComputationofdeRhamCohomology
28.1CohomologyVectorSpaceofaTorus
28.2TheCohomologyRingofaTorus
28.3TheCohomologyofaSurfaceofGenusg
Problems
29ProofofHomotopyInvarianee
29.1ReductiontoTwoSections
29.2CochainHomotopies
29.3DifferentialFormsonM×R
29.4ACochainHomotopyBetweeni0*andi1*
29.5VerificationofCochainHomotopy
Problems

Appendices
APoint—SetTopology
A.1TopologicalSpaces
A.2SubspaceTopology
A.3Bases
A.4FirstandSecondCountability
A.5SeparationAxioms
A.6ProductTopology
A.7Continuity
A.8Compactness
A.9BoundednessinRn
A.10Connectedness
A.11ConnectedComponents
A.12Closure
A.13Convergence
Problems
BTheInverseFunctionTheoremonRnandRelatedResults
B.1TheInverseFunctionTheorem
B.2TheImplicitFunctionTheorem
B.3ConstantRankTheorem
Problems
CExistenceofaPartitionofUnityinGeneral
DLinearAlgebra
D.1QuotientVectorSpaces
D.2LinearTransformations
D.3DirectProductandDirectSum
Problems
EQuaternionsandtheSymplecticGroup
E.1RepresentationofLinearMapsbyMatrices
E.2QuaternionicConjugation
E.3QuaternionicInnerProduct
内容简介:
　　Thisisacompletelyrevisededition,withmorethanfiftypagesofnewmaterialscatteredthroughout.Inkeepingwiththeconventionalmeaningofchaptersandsections,Ihavereorgaruzedthebookintotwenty-ninesectionsinsevenchapters.ThemainadditionsareSection200ntheLiederivativeandinteriormultiplication,twointrinsicoperationsonamanifoldtooimportanttoleaveout,newcriteriainSection21fortheboundaryorientation,andanewappendixonquaternionsandthesymplecticgroup.
　　Apartfromcorrectingerrorsandmisprints,Ihavethoughtthrougheveryproofagain,clarifiedmanypassages,andaddednewexamples,exercises,hints,andsolutions.Intheprocess,everysectionhasbeenrewritten,sometimesquitedrastically.Therevisionsaresoextensivethatitisnotpossibletoenumeratethemallhere.Eachchapternowcomeswithanintroductoryessaygivinganoverviewofwhatistocome.Toprovideatimelineforthedevelopmentofideas,Ihaveindicatedwheneverpossi-blethehistoricaloriginoftheconcepts,andhaveaugmentedthebibliographywithhistoricalreferences.
目录:
PrefacetotheSecondEdition
PrefacetotheFirstEdition
ABriefIntroduction

Chapter1EuclideanSpaces
1SmoothFunctionsonaEuclideanSpace
1.1C∞VersusAnalyticFunctions
1.2Taylor'sTheoremwithRemainder
Problems
2TangentVectorsinRnasDerivations
2.1TheDirectionalDerivative
2.2GermsofFunctions
2.3DerivationsataPoint
2.4VectorFields
2.5VectorFieldsasDerivations
Problems
3TheExteriorAlgebraofMulticovectors
3.1DualSpace
3.2Permutations
3.3MultilinearFunctions
3.4ThePermutationActiononMultilinearFunctions
3.5TheSymmetrizingandAlternatingOperators
3.6TheTensorProduct
3.7TheWedgeProduct
3.8AnticommutativityoftheWedgeProduct
3.9AssociativityoftheWedgeProduct
3.10ABasisfork—Covectors
Problems
4DifferentialFormsonRn
4.1Differential1—FormsandtheDifferentialofaFunction
4.2Differentialk—Forms
4.3DifferentialFormsasMultilinearFunctionsonVectorFields
4.4TheExteriorDerivative
4.5ClosedFormsandExactForms
4.6ApplicationstoVectorCalculus
4.7ConventiononSubscriptsandSuperscripts
Problems

Chapter2Manifolds
5Manifolds
5.1TopologicalManifolds
5.2CompatibleCharts
5.3SmoothManifolds
5.4ExamplesofSmoothManifolds
Problems
6SmoothMapsonaManifold
6.1SmoothFunctionsonaManifold
6.2SmoothMapsBetweenManifolds
6.3Diffeomorphisms
6.4SmoothnessinTermsofComponents
6.5ExamplesofSmoothMaps
6.6PartialDerivatives
6.7TheInverseFunctionTheorem
Problems
7Quotients
7.1TheQuotientTopology
7.2ContinuityofaMaponaQuotient
7.3IdentificationofaSubsettoaPoint
7.4ANecessaryConditionforaHausdorffQuotient
7.5OpenEquivalenceRelations
7.6RealProjectiveSpace
7.7TheStandardC∞AtlasonaRealProjectiveSpace
Problems

Chapter3TheTangentSpace
8TheTangentSpace
8.1TheTangentSpaceataPoint
8.2TheDifferentialofaMap
8.3TheChainRule
8.4BasesfortheTangentSpaceataPoint
8.5ALocalExpressionfortheDifferential
8.6CurvesinaManifold
8.7ComputingtheDifferentialUsingCurves
8.8ImmersionsandSubmersions
8.9Rank，andCriticalandRegularPoints
Problems
9Submanifolds
9.1Submanifolds
9.2LevelSetsofaFunction
9.3TheRegularLevelSetTheorem
9.4ExamplesofRegularSubmanifolds
Problems
10CategoriesandFunctors
10.1Categories
10.2Functors
10.3TheDualFunctorandtheMulticovectorFunctor
Problems
11TheRankofaSmoothMap
11.1ConstantRankTheorem
11.2TheImmersionandSubmersionTheorems
11.3ImagesofSmoothMaps
11.4SmoothMapsintoaSubmanifold
11.5TheTangentPlanetoaSurfaceinR3
Problems
12TheTangentBundle
12.1TheTopologyoftheTangentBundle
12.2TheManifoldStructureontheTangentBundle
12.3VectorBundles
12.4SmoothSections
12.5SmoothFrames
Problems
13BumpFunctionsandPartitionsofUnity
13.1C∞BumpFunctions
13.2PartitionsofUnity
13.3ExistenceofaPartitionofUnity
Problems
14VectorFields
14.1SmoothnessofaVectorField
14.2IntegralCurves
14.3LocalFlows
14.4TheLieBracket
14.5ThePushforwardofVectorFields
14.6RelatedVectorFields
Problems

Chapter4LieGroupsandLieAlgebras
15LieGroups
15.1ExamplesofLieGroups
15.2LieSubgroups
15.3TheMatrixExponential
15.4TheTraceofaMatrix
15.5TheDifferentialofdetattheIdentity
Problems
16LieAlgebras
16.1TangentSpaceattheIdentityofaLieGroup
16.2Left—InvariantVectorFieldsonaLieGroup
16.3TheLieAlgebraofaLieGroup
16.4TheLieBracketongl（n，R）
16.5ThePushforwardofLeft—InvariantVectorFields
16.6TheDifferentialasaLieAlgebraHomomorphism
Problems

Chapter5DifferentialForms
17Differential1—Forms
17.1TheDifferentialofaFunction
17.2LocalExpressionforaDifferential1—Form
17.3TheCotangentBundle
17.4CharacterizationofC∞l—Forms
17.5Pullbackofl—Forms
17.6Restrictionofl—FormstoanImmersedSubmanifold
Problems
18Differentialk—Forms
18.1DifferentialForms
18.2LocalExpressionforak—Form
18.3TheBundlePointofView
18.4Smoothk—Forms
18.5Pullbackofk—Forms
18.6TheWedgeProduct
18.7DifferentialFormsonaCircle
18.8InvariantFormsonaLieGroup
Problems
19TheExteriorDerivative
19.1ExteriorDerivativeonaCoordinateChart
19.2LocalOperators
19.3ExistenceofanExteriorDerivativeonaManifold
19.4UniquenessoftheExteriorDerivative
19.5ExteriorDifferentiationUnderaPullback
19.6Restrictionofk—FormstoaSubmanifold
19.7ANowhere—Vanishing1—FormontheCircle
Problems
20TheLieDerivativeandInteriorMultiplication
20.1FamiliesofVectorFieldsandDifferentialForms
20.2TheLieDerivativeofaVectorField
20.3TheLieDerivativeofaDifferentialForm
20.4InteriorMultiplication
20.5PropertiesoftheLieDerivative
20.6GlobalFormulasfortheLieandExteriorDerivatives
Problems

Chapter6Integration
21Orientations
21.1OrientationsofaVectorSpace
21.2Orientationsandn—Covectors
21.3OrientationsonaManifold
21.4OrientationsandDifferentialForms
21.5OrientationsandAtlases
Problems
22ManifoldswithBoundary
22.1SmoothInvarianceofDomaininRn
22.2ManifoldswithBoundary
22.3TheBoundaryofaManifoldwithBoundary
22.4TangentVectors，DifferentialForms，andOrientations
22.5Outward—PointingVectorFields
22.6BoundaryOrientation
Problems
23IntegrationonManifolds
23.1TheRiemannIntegralofaFunctiononRn
23.2IntegrabilityConditions
23.3TheIntegralofann—FormonRn
23.4IntegralofaDifferentialFormoveraManifold
23.5Stokes'sTheorem
23.6LineIntegralsandGreen'sTheorem
Problems

Chapter7DeRhamTheory
24DeRhamCohomology
24.1DeRharnCohomology
24.2ExamplesofdeRhamCohomology
24.3DiffeomorphismInvariance
24.4TheRingStructureondeRhamCohomology
Problems
25TheLongExactSequenceinCohomology
25.1ExactSequences
25.2CohomologyofCochainComplexes
25.3TheConnectingHomomorphism
25.4TheZig—ZagLemma
Problems
26TheMayer—VietorisSequence
26.1TheMayer—VietorisSequence
26.2TheCohomologyoftheCircle
26.3TheEulerCharacteristic
Problems
27HomotopyInvariance
27.1SmoothHomotopy
27.2HomotopyType
27.3DeformationRetractions
27.4TheHomotopyAxiomfordeRhamCohomology
Problems
28ComputationofdeRhamCohomology
28.1CohomologyVectorSpaceofaTorus
28.2TheCohomologyRingofaTorus
28.3TheCohomologyofaSurfaceofGenusg
Problems
29ProofofHomotopyInvarianee
29.1ReductiontoTwoSections
29.2CochainHomotopies
29.3DifferentialFormsonM×R
29.4ACochainHomotopyBetweeni0*andi1*
29.5VerificationofCochainHomotopy
Problems

Appendices
APoint—SetTopology
A.1TopologicalSpaces
A.2SubspaceTopology
A.3Bases
A.4FirstandSecondCountability
A.5SeparationAxioms
A.6ProductTopology
A.7Continuity
A.8Compactness
A.9BoundednessinRn
A.10Connectedness
A.11ConnectedComponents
A.12Closure
A.13Convergence
Problems
BTheInverseFunctionTheoremonRnandRelatedResults
B.1TheInverseFunctionTheorem
B.2TheImplicitFunctionTheorem
B.3ConstantRankTheorem
Problems
CExistenceofaPartitionofUnityinGeneral
DLinearAlgebra
D.1QuotientVectorSpaces
D.2LinearTransformations
D.3DirectProductandDirectSum
Problems
EQuaternionsandtheSymplecticGroup
E.1RepresentationofLinearMapsbyMatrices
E.2QuaternionicConjugation
E.3QuaternionicInnerProduct

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