Diferentiable Manifolds - Section C Course 2003 (Lecture Notes)

Differentiable Manifolds Section C Course 2003......Page 1
1 Introduction......Page 2
2.1 Coordinate charts......Page 4
2.2 The definition of a manifold......Page 7
2.3 Further examples of manifolds......Page 9
2.4 Maps between manifolds......Page 11
3.1 Existence of smooth functions......Page 12
3.2 The derivative of a function......Page 14
3.3 Derivatives of smooth maps......Page 18
4.1 The tangent bundle......Page 20
4.2 Vector elds as derivations......Page 24
4.3 One-parameter groups of di eomorphisms......Page 26
4.4 The Lie bracket revisited......Page 30
5 Tensor products......Page 31
5.1 The exterior algebra......Page 32
6.1 The bundle of p-forms......Page 36
6.2 Partitions of unity......Page 37
6.3 Working with di erential forms......Page 39
6.4 The exterior derivative......Page 41
6.5 The Lie derivative of a di erential form......Page 45
6.6 de Rham cohomology......Page 48
7.1 Orientation......Page 53
7.2 Stokes’ theorem......Page 56
8.1 de Rham cohomology in the top dimension......Page 62
9.1 The metric tensor......Page 70
9.2 The geodesic ow......Page 73
9.3 Harmonic forms......Page 78