1Introduction
References
2Topology
2.1BasicDefinitions
2.2BaseofTopology,Metric,Norm
2.3Derivatives
2.4Compactness
2.5Connectedness,Homotopy
2.6TopologicalChargesinPhysics
References
3Manifolds
3.1ChartsandAtlases
3.2SmoothManifolds
3.3TangentSpaces
3.4VectorFields
3.5MappingsofManifolds,Submanifolds
3.6Frobenius'Theorem
3.7ExamplesfromPhysics
3.7.1ClassicalPointMechanics
3.7.2ClassicalandQuantumMechanics
3.7.3ClassicalPointMechanicsUnderMomentumConstraints
3.7.4ClassicalMechanicsUnderVelocityConstraints
3.7.5Thermodynamics
References
4TensorFields
4.1TensorAlgebras
4.2ExteriorAlgebras
4.3TensorFieldsandExteriorForms
4.4ExteriorDifferentialCalculus
References
5Integration,HomologyandCohomology
5.1PreludeinEuclideanSpace
5.2ChainsofSimplices
5.3IntegrationofDifferentialForms
5.4DeRhamCohomology
5.5HomologyandHomotopy
5.6HomologyandCohomologyofComplexes
5.7Euler'sCharacteristic
5.8CriticalPoints
5.9ExamplesfromPhysics
References
6LieGroups
6.1LieGroupsandLieAlgebras
6.2LieGroupHomomorphismsandRepresentations
6.3LieSubgroups
6.4SimplyConnectedCoveringGroup
6.5TheExponentialMapping
6.6TheGeneralLinearGroupGl(n,K)
6.7ExamplefromPhysics:TheLorentzGroup
6.8TheAdjointRepresentation
References
7BundlesandConnections
7.1PrincipalFiberBundles
7.2FrameBundles
7.3ConnectionsonPrincipleFiberBundles
7.4ParallelTransportandHolonomy
7.5ExteriorCovariantDerivativeandCurvatureForm
7.6FiberBundles
7.7LinearandAffineConnections
7.8CurvatureandTorsionTensors
7.9ExpressionsinLocalCoordinatesonM
References
8Parallelism,Holonomy,Homotopyand(Co)homology
8.1TheExactHomotopySequence
8.2HomotopyofSections
8.3GaugeFieldsandConnectionsonR4
8.4GaugeFieldsandConnectionsonManifolds
8.5CharacteristicClasses
8.6GeometricPhasesinQuantumPhysics
8.6.1Berry-SimonConnection
8.6.2DegenerateCase
8.6.3ElectricalPolarization
8.6.4OrbitalMagnetism
8.6.5TopologicalInsulators
8.7GaugeFieldTheoryofMolecularPhysics
References
9RiemannianGeometry
9.1RiemannianMetric
9.2HomogeneousManifolds
9.3RiemannianConnection
9.4GeodesicNormalCoordinates
9.5SectionalCurvature
9.6Gravitation
9.7Complex,HermitianandK~ihlerianManifolds
References
Compendium
ListofSymbols
Index
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