Differential Geometry: Bundles, Connections, Metrics and Curvature (Oxford Graduate Texts in Mathematics (23))

Bundles, connections, metrics and curvature are the 'lingua franca' of modern differential geometry and theoretical physics. This book will supply a graduate student in mathematics or theoretical physics with the fundamentals of these objects. Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the definition of characteristic classes, and also an introduction to complex and Kahler geometry.Differential Geometry uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life. Helpfully, proofs are offered for almost all assertions throughout. All of the introductory material is presented in full and this is the only such source with the classical examples presented in detail. Cover......Page 1 Contents......Page 8 1.1 Smooth manifolds......Page 16 1.2 The inverse function theorem and implicit function theorem......Page 18 1.3 Submanifolds of R[sup(m)]......Page 19 1.4 Submanifolds of manifolds......Page 22 1.5 More constructions of manifolds......Page 23 1.6 More smooth manifolds: The Grassmannians......Page 24 Appendix 1.1 How to prove the inverse function and implicit function theorems......Page 26 Additional reading......Page 28 2.1 The general linear group......Page 29 2.2 Lie groups......Page 30 2.3 Examples of Lie groups......Page 31 2.4 Some complex Lie groups......Page 32 2.5 The groups Sl(n; C), U(n) and SU(n)......Page 34 2.6 Notation with regards to matrices and differentials......Page 36 Appendix 2.1 The transition functions for the Grassmannians......Page 37 Additional reading......Page 39 3.1 The definition......Page 40 3.2 The standard definition......Page 42 3.3 The first examples of vector bundles......Page 43 3.4 The tangent bundle......Page 44 3.5 Tangent bundle examples......Page 46 3.6 The cotangent bundle......Page 48 3.7 Bundle homomorphisms......Page 49 3.8 Sections of vector bundles......Page 50 3.9 Sections of TM and T*M......Page 51 Additional reading......Page 53 4.1 Subbundles......Page 54 4.2 Quotient bundles......Page 55 4.3 The dual bundle......Page 56 4.4 Bundles of homomorphisms......Page 57 4.6 The direct sum......Page 58 4.7 Tensor powers......Page 59 Additional reading......Page 61 5.1 The pull-back construction......Page 63 5.2 Pull-backs and Grassmannians......Page 64 5.3 Pull-back of differential forms and push-forward of vector fields......Page 65 5.4 Invariant forms and vector fields on Lie groups......Page 67 5.5 The exponential map on a matrix group......Page 68 5.6 The exponential map and right/left invariance on Gl(n; C) and its subgroups......Page 70 5.7 Immersion, submersion and transversality......Page 72 Additional reading......Page 73 6.1 Definitions......Page 74 6.2 Comparing definitions......Page 75 6.3 Examples: The complexification......Page 77 6.4 Complex bundles over surfaces in R[sup(3)]......Page 78 6.6 Bundles over 4-dimensional submanifolds in R[sup(5)]......Page 79 6.8 Complex Grassmannians......Page 80 6.9 The exterior product construction......Page 83 6.10 Algebraic operations......Page 84 6.11 Pull-back......Page 85 Additional reading......Page 86 7 Metrics on vector bundles......Page 87 7.1 Metrics and transition functions for real vector bundles......Page 88 7.3 Metrics, algebra and maps......Page 90 Additional reading......Page 92 8.1 Riemannian metrics and distance......Page 93 8.2 Length minimizing curves......Page 94 8.3 The existence of geodesics......Page 96 8.4 First examples......Page 97 8.5 Geodesics on SO(n)......Page 100 8.6 Geodesics on U(n) and SU(n)......Page 104 8.7 Geodesics and matrix groups......Page 107 Appendix 8.1 The proof of the vector field theorem......Page 108 Additional reading......Page 109 9.2 The exponential map......Page 111 9.3 Gaussian coordinates......Page 113 9.4 The proof of the geodesic theorem......Page 115 Additional reading......Page 118 10.1 The definition......Page 119 10.2 A cocycle definition......Page 120 10.3 Principal bundles constructed from vector bundles......Page 121 10.4 Quotients of Lie groups by subgroups......Page 123 10.5 Examples of Lie group quotients......Page 125 10.6 Cocycle construction examples......Page 128 10.7 Pull-backs of principal bundles......Page 131 10.8 Reducible principal bundles......Page 133 10.9 Associated vector bundles......Page 134 Appendix 10.1 Proof of Proposition 10.1......Page 136 Additional reading......Page 139 11.1 Covariant derivatives......Page 140 11.2 The space of covariant derivatives......Page 141 11.3 Another construction of covariant derivatives......Page 142 11.4 Principal bundles and connections......Page 143 11.5 Connections and covariant derivatives......Page 149 11.6 Horizontal lifts......Page 150 11.7 An application to the classification of principal G-bundles up to isomorphism......Page 151 11.8 Connections, covariant derivatives and pull-back bundles......Page 152 Additional reading......Page 153 12.1 Exterior derivative......Page 154 12.2 Closed forms, exact forms, diffeomorphisms and De Rham cohomology......Page 156 12.3 Lie derivative......Page 158 12.4 Curvature and covariant derivatives......Page 159 12.5 An example......Page 161 12.7 Connections and curvature......Page 163 12.8 The horizontal subbundle revisited......Page 165 Additional reading......Page 166 13.1 Flat connections......Page 167 13.2 Flat connections on bundles over the circle......Page 168 13.3 Foliations......Page 170 13.4 Automorphisms of a principal bundle......Page 171 13.5 The fundamental group......Page 172 13.7 The universal covering space......Page 174 13.8 Holonomy and curvature......Page 175 13.9 Proof of the classification theorem for flat connections......Page 177 Appendix 13.1 Smoothing maps......Page 179 Appendix 13.2 The proof of the Frobenius theorem......Page 181 Additional reading......Page 184 14.1 The Bianchi Identity......Page 185 14.2 Characteristic forms......Page 186 14.3 Characteristic classes: Part 1......Page 189 14.4 Characteristic classes: Part 2......Page 190 14.5 Characteristic classes for complex vector bundles and the Chern classes......Page 192 14.6 Characteristic classes for real vector bundles and the Pontryagin classes......Page 194 14.7 Examples of bundles with nonzero Chern classes......Page 195 14.8 The degree of the map g → g[sup(m)] from SU(2) to itself......Page 204 Appendix 14.1 The ad-invariant functions on M(n; C)......Page 205 Appendix 14.2 Integration on manifolds......Page 207 Appendix 14.3 The degree of a map......Page 212 Additional reading......Page 219 15.1 Metric compatible covariant derivatives......Page 220 15.2 Torsion free covariant derivatives on T*M......Page 223 15.3 The Levi-Civita connection/covariant derivative......Page 225 15.4 A formula for the Levi-Civita connection......Page 226 15.5 Covariantly constant sections......Page 227 15.6 An example of the Levi-Civita connection......Page 229 15.7 The curvature of the Levi-Civita connection......Page 231 Additional reading......Page 233 16.1 Spherical metrics, flat metrics and hyperbolic metrics......Page 235 16.2 The Schwarzchild metric......Page 238 16.3 Curvature conditions......Page 239 16.4 Manifolds of dimension 2: The Gauss–Bonnet formula......Page 242 16.5 Metrics on manifolds of dimension 2......Page 244 16.6 Conformal changes......Page 245 16.7 Sectional curvatures and universal covering spaces......Page 247 16.8 The Jacobi field equation......Page 248 16.9 Constant sectional curvature and the Jacobi field equation......Page 251 16.10 Manifolds of dimension 3......Page 253 16.11 The Riemannian curvature of a compact matrix group......Page 254 Additional reading......Page 259 17 Complex manifolds......Page 260 17.1 Some basics concerning holomorphic functions on C[sup(n)]......Page 261 17.2 The definition of a complex manifold......Page 262 17.3 First examples of complex manifolds......Page 263 17.4 The Newlander–Nirenberg theorem......Page 266 17.6 The almost Kähler 2-form......Page 270 17.7 Symplectic forms......Page 271 17.8 Kähler manifolds......Page 272 17.9 Complex manifolds with closed almost Kähler form......Page 273 17.10 Examples of Kähler manifolds......Page 274 Appendix 17.1 Compatible almost complex structures......Page 276 Additional reading......Page 282 18.1 Holomorphic submanifolds of a complex manifold......Page 283 18.2 Holomorphic submanifolds of projective spaces......Page 284 18.3 Proof of Proposition 18.2, about holomorphic submanifolds in CP[sup(n)]......Page 286 18.4 The curvature of a Kähler metric......Page 287 18.5 Curvature with no (0, 2) part......Page 290 18.6 Holomorphic sections......Page 292 18.7 Example on CP[sup(n)]......Page 294 Additional reading......Page 296 19.1 Definition of the Hodge star......Page 297 19.2 Representatives of De Rham cohomology......Page 298 19.3 A fairy tale......Page 299 19.4 The Hodge theorem......Page 300 19.5 Self-duality......Page 301 Additional reading......Page 302 List of lemmas, propositions, corollaries and theorems......Page 304 List of symbols......Page 306 D......Page 310 J......Page 311 R......Page 312 Z......Page 313 Bundles, connections, metrics and curvature are the'lingua franca'of modern differential geometry and theoretical physics. This book will supply a graduate student in mathematics or theoretical physics with the fundamentals of these objects. Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the definition of characteristic classes, and also an introduction to complex and Kähler geometry. Differential Geometry uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life. Helpfully, proofs are offered for almost all assertions throughout. All of the introductory material is presented in full and this is the only such source with the classical examples presented in detail. 1. Smooth manifolds 2. Matrices and lie groups 3. Introduction to vector bundles 4. Algebra of vector bundles 5. Maps and vector bundles 6. Vector bundles with C[superscript n] as fiber 7. Metrics on vector bundles 8. Geodesics 9. Properties of geodesics 10. Principal bundles 11. Covariant derivatives and connections 12. Covariant derivatives, connections and curvature 13. Flat connections and holonomy 14. Curvature polynomials and characteristic classes 15. Covariant derivatives and metrics 16. The Riemann curvature tensor 17. Complex manifolds 18. Holomorphic submanifolds, holomorphic sections and curvature 19. The Hodge star. Bundles, Connections, Metrics And Curvature Are The Lingua Franca Of Modern Differential Geometry And Theoretical Physics. Supplying Graduate Students In Mathematics Or Theoretical Physics With The Fundamentals Of These Objects, This Book Would Suit A One-semester Course On The Subject Of Bundles And The Associated Geometry. Clifford Henry Taubes. Includes Bibliographical References And Index. Bundles, connections, metrics & curvature are the lingua franca of modern differential geometry & theoretical physics. Supplying graduate students in mathematics or theoretical physics with the fundamentals of these objects, & providing numerous examples, the book would suit a one-semester course on the subject of bundles & the associated geometry