Mathematics - Differential Geometry - Analysis and Physics.pdf

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Di‹erentialGeometry,AnalysisandPhysics
Je‹reyM.Lee
c 2000Je‹reyMarclee
ii
Contents
0.1Preface ................................viii
1PreliminariesandLocalTheory 1
1.1Calculus................................ 2
1.2ChainRule,ProductruleandTaylor’sTheorem......... 11
1.3Localtheoryofmaps......................... 11
2Di‹erentiableManifolds 15
2.1RoughIdeasI............................. 15
2.2TopologicalManifolds........................ 16
2.3Di‹erentiableManifoldsandDi‹erentiableMaps......... 17
2.4Pseudo-GroupsandModelsSpaces................. 22
2.5SmoothMapsandDi‹eomorphisms................ 27
2.6CoveringsandDiscretegroups ................... 30
2.6.1Coveringspacesandthefundamentalgroup........ 30
2.6.2DiscreteGroupActions................... 36
2.7Grassmannianmanifolds....................... 39
2.8PartitionsofUnity.......................... 40
2.9Manifoldswithboundary. ...................... 43
3TheTangentStructure 47
3.1RoughIdeasII............................ 47
3.2TangentVectors ........................... 48
3.3Interpretations............................ 53
3.4TheTangentMap.......................... 54
3.5TheTangentandCotangentBundles................ 55
3.5.1TangentBundle........................ 55
3.5.2TheCotangentBundle.................... 57
3.6ImportantSpecialSituations..................... 59
4Submanifold,ImmersionandSubmersion. 63
4.1Submanifolds............................. 63
4.2Submanifoldsof R n .......................... 65
4.3RegularandCriticalPointsandValues............... 66
4.4Immersions.............................. 70
iii
iv CONTENTS
4.5ImmersedSubmanifoldsandInitialSubmanifolds......... 71
4.6Submersions.............................. 75
4.7MorseFunctions ........................... 77
4.8Problemset.............................. 79
5LieGroupsI 81
5.1DefinitionsandExamples...................... 81
5.2LieGroupHomomorphisms..................... 84
6FiberBundlesandVectorBundlesI 87
6.1TransitionsMapsandStructure................... 94
6.2Usefulwaystothinkaboutvectorbundles............. 94
6.3SectionsofaVectorBundle..................... 97
6.4Sheaves,GermsandJets....................... 98
6.5JetsandJetbundles.........................102
7VectorFieldsand1-Forms 105
7.1Definitionofvectorfieldsand1-forms...............105
7.2Pullbackandpushforwardoffunctionsand1-forms.......106
7.3FrameFields.............................107
7.4LieBracket..............................108
7.5Localization..............................110
7.6Actionbypullbackandpush-forward................112
7.7FlowsandVectorFields.......................114
7.8LieDerivative.............................117
7.9TimeDependentFields .......................123
8LieGroupsII 125
8.1Spinorsandrotation.........................133
9MultilinearBundlesandTensorsFields 137
9.1MultilinearAlgebra..........................137
9.1.1Contractionoftensors....................141
9.1.2AlternatingMultilinearAlgebra...............142
9.1.3Orientationonvectorspaces ................146
9.2MultilinearBundles .........................147
9.3TensorFields.............................147
9.4TensorDerivations..........................149
10Di‹erentialforms 153
10.1Pullbackofadi‹erentialform. ...................155
10.2ExteriorDerivative..........................156
10.3Maxwell’sequations. ........................159
10.4Liederivative,interiorproductandexteriorderivative.......161
10.5TimeDependentFields(PartII)..................163
10.6Vectorvaluedandalgebravaluedforms...............163
CONTENTS
v
10.7GlobalOrientation..........................165
10.8Orientationofmanifoldswithboundary..............167
10.9IntegrationofDi‹erentialForms...................168
10.10Stokes’Theorem...........................170
10.11VectorBundleValuedForms.....................172
11DistributionsandFrobenius’Theorem 175
11.1Definitions...............................175
11.2IntegrabilityofRegularDistributions ...............175
11.3ThelocalversionFrobenius’theorem................177
11.4Foliations...............................182
11.5TheGlobalFrobeniusTheorem...................183
11.6SingularDistributions........................185
12ConnectionsonVectorBundles 189
12.1Definitions...............................189
12.2LocalFrameFieldsandConnectionForms.............191
12.3ParallelTransport ..........................193
12.4Curvature...............................198
13Riemannianandsemi-RiemannianManifolds 201
13.1TheLinearTheory..........................201
13.1.1ScalarProducts .......................201
13.1.2NaturalExtensionsandtheStarOperator.........203
13.2SurfaceTheory............................208
13.3Riemannianandsemi-RiemannianMetrics.............214
13.4TheRiemanniancase(positivedefinitemetric)..........220
13.5Levi-CivitaConnection........................221
13.6Covariantdi‹erentiationofvectorfieldsalongmaps........228
13.7Covariantdi‹erentiationoftensorfields..............229
13.8ComparingtheDi‹erentialOperators ...............230
14FormalismsforCalculation 233
14.1TensorCalculus............................233
14.2CovariantExteriorCalculus,Bundle-ValuedForms........234
15Topology 235
15.1AttachingSpacesandQuotientTopology.............235
15.2TopologicalSum...........................239
15.3Homotopy...............................239
15.4CellComplexes............................241
16AlgebraicTopology 245
16.1AxiomsforaHomologyTheory...................245
16.2SimplicialHomology.........................246
16.3SingularHomology..........................246

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