Page 9 - Curvature and Homology

........................ v
PREFACE TO THE ENLARGED EDITION
PREFACE ........................................ vii
...
NOTATION INDEX
................................... x111
INTRODUCTION
..................................... xv
Chapter I
RIEMANNIAN MANIFOLDS .................. I
1.1 Differentiable manifolds ..................... 1
1.2 Tensors ............................. 5
1.3 Tensor bundles ......................... 9
1.4 Differential forms ........................ 12
1.5 Submanifolds .......................... 17
1.6 Integration of differential forms .................. 19
1.7 Affine connections ........................ 23
1.8 Bundle of frames ........................ 27
1.9 Riemannian geometry ...................... 30
1.10 Sectional curvature ....................... 35
1.11 Geodesic coordinates ...................... 40
Exercises ............................. 41

Chapter I1
TOPOLOGY OF DIFFERENTIABLE MANI-
FOLDS ............................. 56
2.1 Complexes ........................... 56
2.2 Singular homology ....................... 60
2.3 Stokes' theorem ......................... 62
2.4 De Rham cohomology ...................... 63
2.5 Periods ............................ 64
2.6 Decomposition theorem for compact Riemann surfaces ........ 65
2.7 The star isomorphism ...................... 68
2.8 Harmonic forms . The operators 6 and A .............. 71
2.9 Orthogonality relations ...................... 73
2.10 Decomposition theorem for compact Riemannian manifolds ..... 75
2.1 1 Fundamental theorem ..................... 76
2.12 Explicit expressions for d, 6 and A ................ 77
Exercises ............................. 78
Chapter Ill
CURVATURE AND HOMOLOGY OF RlE- v
MANNIAN MANIFOLDS .................. 82
3.1 Some contributions of S . Bochner ................. 82
3.2 Curvature and betti numbers ................... 85
3.3 Derivations in a graded algebra .................. 95