By Werner Ballmann
Singular areas with top curvature bounds and, specifically, areas of nonpositive curvature, were of curiosity in lots of fields, together with geometric (and combinatorial) crew conception, topology, dynamical platforms and likelihood thought. within the first chapters of the e-book, a concise advent into those areas is given, culminating within the Hadamard-Cartan theorem and the dialogue of the right boundary at infinity for easily attached whole areas of nonpositive curvature. within the 3rd bankruptcy, qualitative homes of the geodesic circulate on geodesically whole areas of nonpositive curvature are mentioned, as are random walks on teams of isometries of nonpositively curved areas. the most classification of areas thought of can be accurately complementary to symmetric areas of upper rank and Euclidean structures of size not less than (Rank tension conjecture). within the tender case, this can be recognized and is the content material of the Rank tension theorem. An up-to-date model of the evidence of the latter theorem (in the sleek case) is gifted in bankruptcy IV of the publication. This bankruptcy includes additionally a quick advent into the geometry of the unit tangent package of a Riemannian manifold and the elemental proof in regards to the geodesic circulate. In an appendix via Misha Brin, a self-contained and brief evidence of the ergodicity of the geodesic circulate of a compact Riemannian manifold of adverse curvature is given. The evidence is uncomplicated and may be available to the non-specialist. a number of the crucial positive factors and difficulties of the ergodic conception of tender dynamical structures are mentioned, and the appendix can function an advent into this idea.
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