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Differential Geometry Andrzej Derdzinski

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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'sindex 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¨olderand 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 the University of Bonn in 1983-1984 and at the Ohio State University between 1987 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 are presented as well, even in cases where a coordinate-free description would suffice. Although seemingly redundant, this feature may teach the reader to recognize when 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¨ahlermanifolds). 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's Morse Theory, Sulanke and Wintgen's Differentialgeometrie und Faserb¨undel, Kobayashi and Nomizu's Foundations of Differential Geometry (both volumes), Le spectre d'une vari´et´eriemannienne 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 CHAPTER 1 Differentiable Manifolds 1. Manifolds Topics: Coordinate systems; compatibility; atlases; topology; convergence; maximal atlases; the Hausdorff axiom; manifolds; vector spaces as manifolds. Let r be a natural number (r = 0; 1; 2;::: ), or infinity (r = 1), or an addi- tional symbol ! (r = !). We order these values so that 0 < 1 < : : : < 1 < !.A mapping F between open subsets of Euclidean spaces is said to be of class Cr if r = 0 and F is continuous, or 0 < r < 1 and F has continuous partial derivatives up to order r, or r = 1 and F has continuous partial derivatives of all orders or, finally, r = ! and F is real-analytic. (Each of the regularity properties just named for F means the corresponding property for every real-valued component function of F .) An n-dimensional coordinate system (or chart) in a set M is a pair (U; '), where U (the chart's domain) is a nonempty subset of M and ' : U ! '(U) is a one-to-one mapping of U onto an open subset '(U) of Rn. r Two n-dimensional coordinate systems (U; '), (U;e 'e) in M are C compatible, 0 ≤ r ≤ !, if n a. The images '(U \ Ue), 'e(U \ Ue) are both open in R , −1 b. The (bijective) composite mapping 'e ◦ ' : '(U \ Ue) ! 'e(U \ Ue), and its inverse, are of class Cr. −1 We call 'e ◦ ' the transition mapping between (U; ') and (U;e 'e). An n-dimensional Cr atlas A on a set M is a collection of n-dimensional coordinate systems in M which are mutually Cr compatible and whose domains cover M. When A is fixed, a set Y ⊂ M is said to be open if '(U \ Y ) is open in Rn for each (U; ') 2 A. The family of all open sets is called the topology in M determined by the Cr atlas A. By a neighborhood of a point x 2 M we mean any open set containing x. A sequence xk, k = 1; 2;::: of points in M then is said to converge to a limit x 2 M if every neighborhood of x contains the xk for all but finitely many k. An n-dimensional Cr atlas A on M is called maximal if it is contained in no other n-dimensional Cr atlas. Every n-dimensional Cr atlas A on M is r contained in a unique maximal C atlas Amax.
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