Nonpositive Curvature: Geometric and Analytic Aspects

1. Front Matter

   Pages i-viii

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2. Introduction

   • Jürgen Jost

   Pages 1-31

3. Spaces of nonpositive curvature

   • Jürgen Jost

   Pages 33-59

4. Convex functions and centers of mass

   • Jürgen Jost

   Pages 61-68

5. Generalized harmonic maps

   • Jürgen Jost

   Pages 69-83

6. Bochner-Matsushima type identities for harmonic maps and rigidity theorems

   • Jürgen Jost

   Pages 85-98

7. Back Matter

   Pages 99-111

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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.

• calculus
• calculus of variation
• curvature
• geometry
• manifold

"Recollects some basic properties as well as some fairly advanced results [which] is done with a spirit that allows one to understand that, even though the study of such manifolds has important differences from the flat case, some techniques come from the very elementary Euclidean geometry."

--Mathematical Reviews

• Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Deutschland

  Jürgen Jost

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