Differential Geometry Andrzej Derdzinski - Ohio State Department of

263 Pages·2012·1.37 MB·English

Differential Geometry
Andrzej Derdzinski
Author address:
Dept. of Mathematics, Ohio State University, Columbus, OH 43210
E-mail address: [email protected]

Contents
Preface xi
Chapter 1. Differentiable Manifolds 1
1. Manifolds 1
Problems 2
2. Examples of manifolds 2
Problems 4
3. Differentiable mappings 5
Problems 8
4. Lie groups 12
Problems 14
Chapter 2. Tangent Vectors 17
5. Tangent and cotangent vectors 17
Problems 21
6. Vector fields 22
Problems 24
7. Lie algebras 25
Problems 27
8. The Lie algebra of a Lie group 27
Problems 30
Chapter 3. Immersions and Embeddings 33
9. The rank theorem, immersions, submanifolds 33
Problems 36
10. More on tangent vectors 39
Problems 41
11. Lie subgroups 43
Problems 44
12. Orthogonal and unitary groups 44
Problems 49
13. Orbits of Lie-group actions 51
Problems 51
14. Whitney’s embedding theorem 53
Problems 55
Chapter 4. Vector Bundles 57
15. Real and complex vector bundles 57
Problems 59
16. Vector fields on the 2-sphere 60
v
vi CONTENTS
Problems 65
17. Operations on bundles and vector-bundle morphisms 66
Problems 69
18. Vector bundle isomorphisms and triviality 69
Problems 70
19. Subbundles of vector bundles 71
Problems 73
Chapter 5. Connections and Curvature 75
20. The curvature tensor of a connection 75
Problems 77
21. Connections in the tangent bundle 79
Problems 80
22. Parallel transport and geodesics 81
Problems 83
23. The “comma” notation for connections 84
Problems 86
24. The Ricci-Weitzenb¨ock identity 86
Problems 89
25. Variations of curves and the meaning of flatness 90
Problems 92
26. Bianchi identities 93
Problems 95
27. Further operations on connections 96
Problems 98
Chapter 6. Riemannian Distance Geometry 101
28. Fibre metrics 101
Problems 104
29. Raising and lowering indices 105
Problems 107
30. The Levi-Civita connection 107
Problems 108
31. The lowest dimensions 110
Problems 110
32. Riemannian manifolds as metric spaces 111
Problems 113
33. Completeness 114
Problems 116
34. Convexity 117
Problems 118
35. Myers’s theorem 118
Problems 120
Chapter 7. Integration 121
36. Finite partitions of unity 121
Problems 122
37. Densities and integration 123
Problems 124
38. Divergence operators 125
CONTENTS vii
Problems 127
39. The divergence theorem 127
Problems 128
40. Theorems of Bochner and Lichnerowicz 128
Problems 130
41. Einstein metrics and Schur’s theorem 130
Problems 131
42. Spheres and hyperbolic spaces 132
Problems 133
43. Sectional curvature 134
Problems 134
Chapter 8. Geometry of Submanifolds 137
45. Projected connections 137
Problems 139
46. The second fundamental form 140
47. Hypersurfaces in Euclidean spaces 141
48. Bonnet’s theorem 141
Chapter 9. Differential Forms 143
49. Tensor products 143
Problems 145
50. Exterior and symmetric powers 146
Problems 148
51. Exterior forms 149
52. Cohomology spaces 152
Problems 154
Chapter 10. De Rham Cohomology 157
53. Homotopy invariance of the cohomology functor 157
Problems 159
54. The homotopy type 160
Problems 161
55. The Mayer-Vietoris sequence 161
Problems 163
56. Explicit calculations of Betti numbers 163
Problems 164
57. Stokes’s formula 165
Problems 166
58. The fundamental class and mapping degree 167
Problems 170
59. Degree and preimages 171
Problems 172
Chapter 11. Characteristic Classes 175
60. The first Chern class 175
Problems 176
61. Poincar´e’s index formula for surfaces 176
Problems 177
62. The Gauss-Bonnet theorem 177
viii CONTENTS
63. The Euler class 179
Problems 182
Chapter 12. Elements of Analysis 185
64. Sobolev spaces 185
Problems 185
Problems 187
65. Compact operators 188
Problems 188
66. The Rellich lemma 188
67. The regularity theorem 189
68. Solvability criterion for elliptic equations 190
69. The Hodge-de Rham decomposition theorem 190
Appendix A. Some Linear Algebra 191
69. Affine spaces 191
Problems 191
70. Orientation in real vector spaces 192
Problems 192
71. Complex lines versus real planes 194
72. Indefinite inner products 194
Appendix B. Facts from Topology and Analysis 195
73. Banach’s fixed-point theorem 195
Problems 196
74. The inverse mapping theorem 197
Problems 198
75. The Stone-Weierstrass theorem 199
76. Sard’s theorem 201
Problems 201
Appendix C. Ordinary Differential Equations 203
78. Existence and uniqueness of solutions 203
Problems 205
79. Global solutions to linear differential equations 206
Problems 209
80. Differential equations with parameters 211
Appendix D. Some More Differential Geometry 215
81. Grassmann manifolds 215
Problems 215
82. Affine bundles 216
Problems 217
83. Abundance of cut-off functions 217
Problems 217
84. Partitions of unity 217
Problems 218
85. Flows of vector fields 219
Problems 221
86. Killing fields 222
CONTENTS ix
87. Lie brackets and flows 222
Problems 224
88. Completeness of vector fields 224
Problems 225
Appendix E. Measure and Integration 229
89. The H¨older and Minkowski inequalities 229
90. Convergence theorems 229
Appendix F. More on Lie Groups 231
96. The exponential mapping 231
Problems 233
97. 234
98. 237
99. 240
Bibliography 243
Index 245
x CONTENTS
Preface
The present text evolved from differential geometry courses that I taught at
theUniversityofBonnin1983-1984andattheOhioStateUniversitybetween1987
and 2005.
The reader is expected to be familiar with basic linear algebra and calculus
of several real variables. Additional background in topology, differential equations
and functional analysis, although obviously useful, is not necessary: self-contained
expositions of all needed facts from those areas are included, partly in the main
text, partly in appendices.
This book may serve either as the basis of a course sequence, or for self-study.
It is with the latter use in mind that I included over 600 practice problems, along
with hints for those problems that seem less than completely routine.
The exposition uses the coordinate-free language typical of modern differential
geometry. However, whenever appropriate, traditional local-coordinate expressions
arepresentedaswell,evenincaseswhereacoordinate-freedescriptionwouldsuffice.
Althoughseeminglyredundant,thisfeaturemayteachthereadertorecognizewhen
and how to take advantage of shortcuts in arguments provided by local-coordinate
notation.
I selected the topics so as to include what is needed for a reader who wishes to
pursue further study in geometric analysis or applications of differential geometry
to theoretical physics, including both general relativity and classical gauge theory
of particle interactions.
The text begins with a rapid but thorough presentation of manifolds and dif-
ferentiable mappings, followed by the definition of a Lie group, along with some
examples. A list of all the topics covered can best be glimpsed from the table of
contents.
One topic which I left out, despite its prominent status, is complex differen-
tial geometry (including K¨ahler manifolds). This choice seems necessary due to
limitations of space.
Finally, I need to acknowledge several books from which I first learned differ-
ential geometry and which, consequently, influenced my view of the subject. These
are Riemannsche Geometrie im Großen by Gromoll, Klingenberg and Meyer, Mil-
nor’sMorse Theory,SulankeandWintgen’sDifferentialgeometrie und Faserbu¨ndel,
Kobayashi and Nomizu’s Foundations of Differential Geometry (both volumes), Le
spectre d’une vari´et´e riemannienne by Berger, Gauduchon and Mazet, Warner’s
Foundations of Differentiable Manifolds and Lie Groups, and Spivak’s A Compre-
hensive Introduction to Differential Geometry.
Andrzej Derdzinski
xi
xii PREFACE