04032nmm a2200433 u 4500001001200000003002700012005001700039007002400056008004100080020001800121100002100139245009300160250001700253260004100270300002700311505100200338653002201340653002301362653002601385653002601411653001701437653003701454653002601491653001501517653002301532653002301555653002701578653002801605653005501633653003501688653001301723710003401736041001901770989003801789490002501827856007201852082001101924520166301935EB000637308EBX0100000000000000049039000000000000000.0cr|||||||||||||||||||||140122 ||| eng a97830348924071 aBallmann, Werner00aLectures on Spaces of Nonpositive CurvaturehElektronische Ressourcecby Werner Ballmann a1st ed. 1995 aBaselbBirkhĂ¤user Baselc1995, 1995 a120 pbonline resource0 aI. On the interior geometry of metric spaces -- 1. Preliminaries -- 2. The Hopf-Rinow Theorem -- 3. Spaces with curvature bounded from above -- 4. The Hadamard-Cartan Theorem -- 5. Hadamard spaces -- II. The boundary at infinity -- 1. Closure of X via Busemann functions -- 2. Closure of X via rays -- 3. Classification of isometries -- 4. The cone at infinity and the Tits metric -- III. Weak hyperbolicity -- 1. The duality condition -- 2. Geodesic flows on Hadamard spaces -- 3. The flat half plane condition -- 4. Harmonic functions and random walks on ? -- IV. Rank rigidity -- 1. Preliminaries on geodesic flows -- 2. Jacobi fields and curvature -- 3. Busemann functions and horospheres -- 4. Rank, regular vectors and flats -- 5. An invariant set at infinity -- 6. Proof of the rank rigidity -- Appendix. Ergodicity of geodesic flows -- 1. Introductory remarks -- Measure and ergodic theory preliminaries -- Absolutely continuous foliations -- Anosov flows and the HĂ¶ continuity of invaria aComplex manifolds aAlgebraic Topology aDifferential Geometry aDifferential geometry aGroup theory aGroup Theory and Generalizations aMathematical analysis aLie groups aTopological groups aAlgebraic topology aAnalysis (Mathematics) aManifolds (Mathematics) aManifolds and Cell Complexes (incl. Diff.Topology) aTopological Groups, Lie Groups aAnalysis2 aSpringerLink (Online service)07aeng2ISO 639-2 bSBAaSpringer Book Archives -20040 aOberwolfach Seminars uhttps://doi.org/10.1007/978-3-0348-9240-7?nosfx=yxVerlag3Volltext0 a516.36 aSingular 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