微分流形导论

图书标准信息

• 作者 [美]朗 著
• 出版社 世界图书出版公司
• 出版时间 2010-09
• 版次 2
• ISBN 9787510027468
• 定价 35.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 250页
• 正文语种 英语

【内容简介】

thisbookisanoutgrowthofmyintroductiontodifferentiablemanifolds（1962）anddifferentialmanifolds（1972）.bothiandmypublishersfeltitworthwhiletokeepavailableabriefintroductiontodifferentialmanifolds.
thebookgivesanintroductiontothebasicconceptswhichareusedindifferentialtopology，differentialgeometry，anddifferentialequations.indifferentialtopology，onestudiesforinstancehomotopyclassesofmapsandthepossibilityoffindingsuitabledifferentiablemapsinthem（immersions，embeddings，isomorphisms，etc.）.onemayalsousedifferentiablestructuresontopologicalmanifoldstodeterminethetopologicalstructureofthemanifold（forexample，alasmale[sm67]）.indifferentialgeometry，oneputsanadditionalstructureonthedifferentiablemanifold（avectorfield，aspray，a2-form，ariemannianmetric，adlib.）andstudiespropertiesconnectedespeciallywiththeseobjects.formally，onemaysaythatonestudiespropertiesinvariantunderthegroupof.differentiableautomorphismswhichpreservetheadditionalstructure.indifferentialequations，onestudiesvectorfieldsandtheirintegralcurves，singularpoints，stableandunstablemanifolds，etc.acertainnumberofconceptsareessentialforallthree，andaresobasicandelementarythatitisworthwhiletocollectthemtogethersothatmoreadvancedexpositionscanbegivenwithouthavingtostartfromtheverybeginnings.theconceptsareconcernedwiththegeneralbasictheoryofdifferentialmanifolds.myfundamentalsofdifferentialgeometry（1999）canthenbeviewedasacontinuationofthepresentbook.

【目录】

Foreword
Acknowledgments
CHAPTERI
DifferentialCalculus
1.Categories
2.FiniteDimensionalVectorSpaces
3.DerivativesandCompositionofMaps
4.IntegrationandTayiorsFormula
5.TheInverseMappingTheorem

CHAPTERII
Manifolds
1.Atlases，Charts，Morphisms
2.Submanifolds，Immersions，Submersions
3.PartitionsofUnity
4.ManifoldswithBoundary

CHAPTERIII
VectorBundles
l.Definition，PullBacks
2.TheTangentBundle
3.ExactSequencesofBundles
4.OperationsonVectorBundles
5.SplittingofVectorBundles

CHAPTERIV
VectorFieldsandDifferentialEquations
1.ExistenceTheoremforDifferentialEquations
2.VectorFields，Curves，andFlows
3.Sprays
4.TheFlowofaSprayandtheExponentialMap
5.ExistenceofTubularNeighborhoods
6.UniquenessofTubularNeighborhoods

CHAPTERV
OiretionsonVectorFieldsendDifferentialForms
1.VectorFields，DifferentialOperators，Brackets
2.LieDerivative
3.ExteriorDerivative
4.ThePoincareLemma
5.ContractionsandLieDerivative
6.VectorFieldsandl-FormsUnderSelfDuality
7.TheCanonical2-Form
8.DarbouxsTheorem

CHAPTERVI
TheTheoremofFrobenius
1.StatementoftheTheorem
2.DifferentialEquationsDependingonaParameter
3.ProofoftheTheorem
4.TheGlobalFormulation
5.LieGroupsandSubgroups

CHAPTERVII
Metrics
1.DefinitionandFunctoriality
2.TheMetricGroup
3.ReductiontotheMetricGroup
4.MetricTubularNeighborhoods
5.TheMorseLemma
6.TheRiemannianDistance
7.TheCanonicalSpray

CHAPTERVIII
IntegretionofDifferentialForms
1.SetsofMeasure0
2.ChangeofVariablesFormula
3.Orientation
4.TheMeasureAssociatedwithaDifferentialForm

CHAPTERIX
StokesTheorem
1.StokesTheoremforaRectangularSimplex
2.StokesTheoremonaManifold
3.StokesTheoremwithSingularities

CHAPTERX
ApplicationsofStokesTheorem
1.TheMaximaldeRhamCohomology
2.VolumeformsandtheDivergence
3.TheDivergenceTheorem
4.CauchysTheorem
5.TheResidueTheorem
Bibliography
Index