Differential Geometry: Bundles, Connections, Metrics and CurvatureBundles, 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. |
Contents
1 Smooth manifolds | 1 |
2 Matrices and Lie groups | 14 |
3 Introduction to vector bundles | 25 |
4 Algebra of vector bundles | 39 |
5 Maps and vector bundles | 48 |
6 Vector bundles with Csupn as fiber | 59 |
7 Metrics on vector bundles | 72 |
8 Geodesics | 78 |
13 Flat connections and holonomy | 152 |
14 Curvature polynomials and characteristic classes | 170 |
15 Covariant derivatives and metrics | 205 |
16 The Riemann curvature tensor | 220 |
17 Complex manifolds | 245 |
18 Holomorphic submanifolds holomorphic sections and curvature | 268 |
19 The Hodge star | 282 |
List of lemmas propositions corollaries and theorems | 289 |
Other editions - View all
Differential Geometry: Bundles, Connections, Metrics and Curvature Clifford Henry Taubes Limited preview - 2011 |
Differential Geometry: Bundles, Connections, Metrics and Curvature Clifford Henry Taubes Limited preview - 2011 |
Differential Geometry: Bundles, Connections, Metrics and Curvature Clifford Henry Taubes No preview available - 2011 |
Common terms and phrases
algebra ball bundle isomorphism bundle transition function Chapter Chern classes cohomology compact complex manifold complex structure complex vector bundle components connection coordinate chart corresponding covariant derivative covariantly constant defined definition denote a given diffeomorphism differential element entries equation equivalence class Euclidean example exponential map fiber dimension follows function theorem G-bundle G-equivariant geodesic Geometry given point Gl(n Hermitian identifies identity inner product integral inverse Kähler latter Lemma Let M denote Lie group lie(G linear map matrix Meanwhile metric g n-dimensional n-form neighborhood nonzero Note obeys open set orthogonal orthonormal frame pair principal bundle product bundle proof Proposition pull-back quotient Rham Riemannian metric sends any given smooth manifold smooth map SO(n subbundle subgroup submanifold subspace suppose symmetric tensor Topology vector field vector space viewed write written zero