微分几何基础

《微分几何基础(英文版)》介绍了微分拓扑、微分几何以及微分方程的基本概念。《微分几何基础(英文版)》的基本思想源于作者早期的《微分和黎曼流形》，但重点却从流形的一般理论转移到微分几何，增加了不少新的章节。这些新的知识为Banach和Hilbert空间上的无限维流形做准备，但一点都不觉得多余，而优美的证明也让读者受益不浅。在有限维的例子中，讨论了高维微分形式，继而介绍了Stokes定理和一些在微分和黎曼情形下的应用。给出了Laplacian基本公式，展示了其在浸入和浸没中的特征。书中讲述了该领域的一些主要基本理论，如：微分方程的存在定理、唯一性、光滑定理和向量域流，包括子流形管状邻域的存在性的向量丛基本理论，微积分形式，包括经典2-形式的辛流形基本观点，黎曼和伪黎曼流形协变导数以及其在指数映射中的应用，Cartan-Hadamard定理和变分微积分第一基本定理。目次：（第一部分）一般微分方程；微积分；流形；向量丛；向量域和微分方程；向量域和微分形式运算；Frobenius定理；（第二部分）矩阵、协变导数和黎曼几何：矩阵；协变导数和测地线；曲率；二重切线丛的张量分裂；曲率和变分公式；半负曲率例子；自同构和对称；浸入和浸没；（第三部分）体积形式和积分：体积形式；微分形式的积分；Stokes定理；Stokes定理的应用；谱理论。

Foreword
Acknowledgments
PARTⅠ
GeneralDifferentialTheory，
CHAPTERⅠ
OifferenlialCalculus
Categories
TopologicalVectorSpaces
DerivativesandCompositionofMaps
IntegrationandTaylorsFormula
TheInverseMappingTheorem

CHAPTERⅡ
Manifolds
Atlases，Charts，Morphisms
Submanifolds，Immersions，Submersions
PartitionsofUnity
ManifoldswithBoundary

CHAPTERⅢ
VectorBundles
Definition，PullBacks
TheTangentBundle
ExactSequencesofBundles
OperationsonVectorBundles
SplittingofVectorBundles

CHAPTERⅣ
VectorFieldsandDifferentialEquations
ExistenceTheoremforDifferentialEquations.
VectorFields，Curves，andFlows
Sprays
TheFlowofaSprayandtheExponentialMap
ExistenceofTubularNeighborhoods
UniquenessofTubularNeighborhoods

CHAPTERⅤ
OperationsonVectorFieldsandDifferentialForms
VectorFields，DifferentialOperators，Brackets
LieDerivative
ExteriorDerivative
ThePoincarLemma
ContractionsandLieDerivative
VectorFieldsandl-FormsUnderSelfDuality
TheCanonical2-Form
DarbouxsTheorem

CHAPTERⅥ
TheTheoremofFrobenius
StatementoftheTheorem
DifferentialEquationsDependingonaParameter
ProofoftheTheorem
TheGlobalFormulation
LieGroupsandSubgroups

PARTⅡ
Metrics，CovariantDerivatives，andRiemannianGeometry

CHAPTERⅦ
Metrics
DefinitionandFunctoriality
TheHilbertGroup
ReductiontotheHilbertGroup
HilbertianTubularNeighborhoods
TheMorse-PalaisLemma
TheRiemannianDistance
TheCanonicalSpray

CHAPTERⅧ
CovariantDerivativesandGeodesics.
BasicProperties
SpraysandCovariantDerivatives
DerivativeAlongaCurveandParallelism
TheMetricDerivative
MoreLocalResultsontheExponentialMap
RiemannianGeodesicLengthandCompleteness

CHAPTERⅨ
Curvature
TheRiemannTensor
JacobiLifts
ApplicationofJacobiLiftstoTexpx
ConvexityTheorems
TaylorExpansions

CHAPTERⅩ
JacobiLiftsandTensorialSplittingoftheDoubleTangentBundle
ConvexityofJacobiLifts
GlobalTubularNeighborhoodofaTotallyGeodesicSubmanifold.
MoreConvexityandComparisonResults
SplittingoftheDoubleTangentBundle
TensorialDerivativeofaCurveinTXandoftheExponentialMap
TheFlowandtheTensorialDerivative

CHAPTERXI
CurvatureandtheVariationFormula
TheIndexForm，Variations，andtheSecondVariationFormula
GrowthofaJacobiLift
TheSemiParallelogramLawandNegativeCurvature
TotallyGeodesicSubmanifolds
RauchComparisonTheorem
CHAPTERXII
AnExampleofSeminegativeCurvature
Pos，，(R)asaRiemannianManifold
TheMetricIncreasingPropertyoftheExponentialMap
TotallyGeodesicandSymmetricSubmanifolds

CHAPTERXIII
AutomorphismsandSymmetries.，
TheTensorialSecondDerivative
AlternativeDefinitionsofKillingFields
MetricKillingFields
LieAlgebraPropertiesofKillingFields
SymmetricSpaces
ParallelismandtheRiemannTensor
CHAPTERXlV
ImmersionsandSubmersions.
TheCovariantDerivativeonaSubmanifoid
TheHessianandLaplacianonaSubmanifold
TheCovariantDerivativeonaRiemhnnianSubmersion.
TheHessianandLaplacianonaRiemannianSubmersion
TheRiemannTensoronSubmanifolds
TheRiemannTensoronaRiemannianSubmersion

PARTIII
VolumeFormsandIntegration
CHAPTERXV
VolumeForms
VolumeFormsandtheDivergence
CovariantDerivatives
TheJacobianDeterminantoftheExponentialMap
TheHodgeStaronForms
HodgeDecompositionofDifferentialForms
VolumeFormsinaSubmersion
VolumeFormsonLieGroupsandHomogeneousSpaces
HomogeneouslyFiberedSubmersions

CHAPTERXVI
IntegrationofDifferentialForms
SetsofMeasure0
ChangeofVariablesFormula
Orientation
TheMeasureAssociatedwithaDifferentialForm
HomogeneousSpaces

CHAPTERXVII
StokesTheorem
StokesTheoremforaRectangularSimplex
StokesTheoremonaManifold
StokesTheoremwithSingularities

CHAPTERXVIII
ApplicationsofStokesTheorem
TheMaximaldeRhamCohomology
MosersTheorem
TheDivergenceTheorem
TheAdjointofdforHigherDegreeForms
CauchysTheorem
TheResidueTheorem

APPENDIX
TheSpectralTheorem，
HilbertSpace
FunctionalsandOperators
HermitianOperators
Bibliography
Index