Lectures on Spaces of Nonpositive Curvature by Werner Ballmann

Lectures on Spaces of Nonpositive Curvature by Werner Ballmann

Singular spaces with upper curvature bounds and, in particular, spaces of nonpositive curvature, have been of interest in many fields, including geometric (and combinatorial) group theory, topology, dynamical systems and probability theory. In the first two chapters of the book, a concise introduction into these spaces is given, culminating in the Hadamard-Cartan theorem and the discussion of the ideal boundary at infinity for simply connected complete spaces of nonpositive curvature. In the third chapter, qualitative properties of the geodesic flow on geodesically complete spaces of nonpositive curvature are discussed, as are random walks on groups of isometries of nonpositively curved spaces. The main class of spaces considered should be precisely complementary to symmetric spaces of higher rank and Euclidean buildings of dimension at least two (Rank Rigidity conjecture). In the smooth case, this is known and is the content of the Rank Rigidity theorem. An updated version of the proof of the latter theorem (in the smooth case) is presented in Chapter IV of the book. This chapter contains also a short introduction into the geometry of the unit tangent bundle of a Riemannian manifold and the basic facts about the geodesic flow. In an appendix by Misha Brin, a self-contained and short proof of the ergodicity of the geodesic flow of a compact Riemannian manifold of negative curvature is given. The proof is elementary and should be accessible to the non-specialist. Some of the essential features and problems of the ergodic theory of smooth dynamical systems are discussed, and the appendix can serve as an introduction into this theory.

Werner Ballmann is Professor of Differential Geometry at the University of Bonn and Director at the Max Planck Institute for Mathematics in Bonn.

SKU	Unavailable
ISBN 13	9783764352424
ISBN 10	3764352426
Title	Lectures on Spaces of Nonpositive Curvature
Author	Werner Ballmann
Series	Oberwolfach Seminars
Condition	Unavailable
Binding Type	Paperback
Publisher	Birkhauser Verlag AG
Year published	1995-09-01
Number of pages	120
Cover note	Book picture is for illustrative purposes only, actual binding, cover or edition may vary.