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.
About the Author
Clifford Henry Taubes is the William Petschek Professor of Mathematics at Harvard University. He is a member of the National Academy of Sciences and also the American Academy of Sciences. He was awarded the American Mathematical Society's Oswald Veblen Prize in 1991 for his work in differential geometry and topology. He was also the recipient of the French Academy of Sciences Elie Cartan Prize in 1993, the Clay Research Award in 2008, the National Academy of Sciences' Mathematics Award in 2008, and the Shaw Prize in Mathematics in 2009.
Table of Contents
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 fiber Cn
7. Metrics on vector bundles
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
Indexed list of propositions by subject