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. 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