Aspects of Differential Geometry

Preface......Page 13
Acknowledgments......Page 15
4.1 Curves and Surfaces in Rn Given by ODEs......Page 17
4.2 Volume of Geodesic Balls......Page 21
4.3 Holomorphic Geometry......Page 25
4.4 Kähler Geometry......Page 29
5.1 Basic Properties of de Rham Cohomology......Page 33
5.2 Clifford Algebras......Page 42
5.3 The Hodge Decomposition Theorem......Page 47
5.4 Characteristic Classes......Page 57
6 Lie Groups......Page 61
Basic Concepts......Page 62
Lie Algebras......Page 63
The Exponential Function of a Matrix Group......Page 71
The Classical Groups......Page 76
Representations of a Compact Lie Group......Page 78
Bi-invariant pseudo-Riemannian Metrics......Page 85
The Killing Form......Page 87
The Classical Groups in Low Dimensions......Page 92
The Cohomology of Compact Lie Groups......Page 97
The Cohomology of the Unitary Group......Page 100
Smooth Structures on Coset Spaces......Page 103
The Isometry Group......Page 107
The Lie Derivative and Killing Vector Fields......Page 111
Homogeneous Pseudo-Riemannian Manifolds......Page 116
Local Symmetric Spaces......Page 118
The Global Geometry of Symmetric Spaces......Page 120
8.1 Homological Algebra......Page 127
8.2 Simplicial Cohomology......Page 137
8.3 Singular Cohomology......Page 140
8.4 Sheaf Cohomology......Page 144
Bibliography......Page 149
Authors' Biographies......Page 155
Index......Page 157