The Map of My Life (Universitext)

This is a textbook for Riemannian Geometry and Geometric Analysis, introducing techniques from nonlinear analysis at an early stage. Such techniques have recently become an indispensible tool in research in geometry and they are treated in a textbook for the first time. Subjects treated are: Differentiable and Riemannian manifolds, metric properties, tensor calculus, vector bundles, the Hodge theorem for de Rham cohomology, connections and curvature, the Yang-Mills functional, geodesics and Jacobi fields, Rauch comparison theorem and applications, Morse theory (including an introduction to algebraic topology), applications to the existence of closed geodesics, symmetric spaces and Kahler manifolds, the Palais-Smale condition and closed geodesics, harmonic maps, definition and basic properties, existence and uniqueness theorems, applications, minimal surfaces, regularity results. In an appendix Sobolev spaces and regularity theory for linear elliptic equations are discussed in detail. The book gives both invariant global notation and tensor notation and should thus prove to be of interest to physicists as well.