微分几何中的度量结构

图书标准信息

• 作者 Gerard Walschap 著
• 出版社 世界图书出版公司
• 出版时间 2015-01
• 版次 1
• ISBN 9787510086335
• 定价 49.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 226页
• 字数 100千字
• 正文语种 英语

【内容简介】

　　Thistextisanelementaryintroductiontodifferentialgeometry.Althoughitwaswrittenforagraduate-levelaudience,theonlyrequisiteisasolidback-groundincalculus,linearalgebra,andbasicpoint-settopology.
　　Thefirstchaptercoversthefundamentalsofdifferentiablemanifoldsthatarethebreadandbutterofdifferentialgeometry.Alltheusualtopicsarecovered,culnunatinginStokes'theoremtogetherwithsomeapplications.Thestudents'firstcontactwiththesubjectcanbeoverwhelmingbecauseofthewealthofabstractdefinitionsinvolved,soexampleshavebeenstressedthroughout.Oneconcept,forinstance,thatstudentsoftenfindconfusingisthedefinitionoftangentvectors.Theyarefirsttoldthatthesearederivationsoncertainequiv-alenceclassesoffunctions,butlaterthatthetangentspaceofRlis"thesame"asRn.WehavetriedtokeepthesespacesseparateandtocarefullyexplainhowavectorspaceEiscanonicallyisomorphictoitstangentspaceatapoint.Thissubtledistinctionbecomesessentialwhenlaterdiscussingtheverticalbundleofagivenvectorbundle.

【目录】

Preface
Chapter1.DifferentiableManifolds
1.BasicDefinitions
2.DifferentiableMaps
3.TangentVectors
4.TheDerivative
5.TheInverseandImplicitFunctionTheorems
6.Submanifolds
7.VectorFields
8.TheLieBracket
9.DistributionsandFrobeniusTheorem
10.MultilinearAlgebraandTensors
11.TensorFieldsandDifferentialForms
12.IntegrationonChains
13.TheLocalVersionofStokes'Theorem
14.OrientationandtheGlobalVersionofStokes'Theorem
15.SomeApplicationsofStokes'Theorem

Chapter2.FiberBundles
1.BasicDefinitionsandExamples
2.PrincipalandAssociatedBundles
3.TheTangentBundleofSn
4.Cross—SectionsofBundles
5.PullbackandNormalBundles
6.FibrationsandtheHomotopyLifting／CoveringProperties
7.GrassmanniansandUniversalBundles

Chapter3.HomotopyGroupsandBundlesOverSpheres
1.DifferentiableApproximations
2.HomotopyGroups
3.TheHomotopySequenceofaFibration
4.BundlesOverSpheres
5.TheVectorBundlesOverLow—DimensionalSpheres

Chapter4.ConnectionsandCurvature
1.ConnectionsonVectorBundles
2.CovariantDerivatives
3.TheCurvatureTensorofaConnection
4.ConnectionsonManifolds
5.ConnectionsonPrincipalBundles

Chapter5.MetricStructures
1.EuclideanBundlesandRiemannianManifolds
2.RiemannianConnections
3.CurvatureQuantifiers
4.IsometricImmersions
5.RiemannianSubmersions
6.TheGaussLemma
7.Length—MinimizingPropertiesofGeodesics
8.FirstandSecondVariationofArc—Length
9.CurvatureandTopology
10.ActionsofCompactLieGroups

Chapter6.CharacteristicClasses
1.TheWeilHomomorphism
2.PontrjaginClasses
3.TheEulerClass
4.TheWhitneySumFormulaforPontrjaginandEulerClasses
5.SomeExamples
6.TheUnitSphereBundleandtheEulerClass
7.TheGeneralizedGauss—BonnetTheorem
8.ComplexandSymplecticVectorSpaces
9.ChernClasses
Bibliography
Index