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The present book contains the lecture notes from a "Nachdiplomvorlesung", a topics course adressed to Ph. D. students, at the ETH ZUrich during the winter term 95/96. Consequently, these notes are arranged according to the requirements of organizing the material for oral exposition, and the level of difficulty and the exposition were adjusted to the audience in Zurich. The aim of the course was to introduce some geometric and analytic concepts that have been found useful in advancing our understanding of spaces of nonpos itive curvature. In particular in recent years, it has been realized that often it is useful for a systematic understanding not to restrict the attention to Riemannian manifolds only, but to consider more general classes of metric spaces of generalized nonpositive curvature. The basic idea is to isolate a property that on one hand can be formulated solely in terms of the distance function and on the other hand is characteristic of nonpositive sectional curvature on a Riemannian manifold, and then to take this property as an axiom for defining a metric space of nonposi tive curvature. Such constructions have been put forward by Wald, Alexandrov, Busemann, and others, and they will be systematically explored in Chapter 2. Our focus and treatment will often be different from the existing literature. In the first Chapter, we consider several classes of examples of Riemannian manifolds of nonpositive curvature, and we explain how conditions about nonpos itivity or negativity of curvature can be exploited in various geometric contexts.
1 Introduction.- 1.1 Examples of Riemannian manifolds of negative or nonpositive sectional curvature.- Appendix to § 1.1: Symmetric spaces of noncompact type.- 1.2 Mordell and Shafarevitch type problems.- 1.3 Geometric superrigidity.- 2 Spaces of nonpositive curvature.- 2.1 Local properties of Riemannian manifolds of nonpositive sectional curvature.- 2.2 Nonpositive curvature in the sense of Busemann.- 2.3 Nonpositive curvature in the sense of Alexandrov.- 3 Convex functions and centers of mass.- 3.1 Minimizers of convex functions.- 3.2 Centers of mass.- 3.3 Convex hulls.- 4 Generalized harmonic maps.- 4.1 The definition of generalized harmonic maps.- 4.2 Minimizers of generalized energy functional.- 5 Bochner-Matsushima type identities for harmonic maps and rigidity theorems.- 5.1 The Bochner formula for harmonic one-forms and harmonic maps.- 5.2 A Matsushima type formula for harmonic maps.- 5.3 Geometrie superrigidity.