By Werner Ballmann

ISBN-10: 3034892403

ISBN-13: 9783034892407

ISBN-10: 3764352426

ISBN-13: 9783764352424

Singular areas with top curvature bounds and, particularly, areas of nonpositive curvature, were of curiosity in lots of fields, together with geometric (and combinatorial) staff conception, topology, dynamical structures and chance concept. within the first chapters of the publication, a concise advent into those areas is given, culminating within the Hadamard-Cartan theorem and the dialogue of the best boundary at infinity for easily attached entire areas of nonpositive curvature. within the 3rd bankruptcy, qualitative homes of the geodesic stream on geodesically entire 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 measurement no less than (Rank stress conjecture). within the tender case, this can be identified and is the content material of the Rank stress theorem. An up to date model of the evidence of the latter theorem (in the graceful case) is gifted in bankruptcy IV of the e-book. This bankruptcy includes additionally a brief advent into the geometry of the unit tangent package deal of a Riemannian manifold and the elemental proof in regards to the geodesic stream. In an appendix via Misha Brin, a self-contained and brief facts of the ergodicity of the geodesic circulate of a compact Riemannian manifold of damaging curvature is given. The evidence is common and will be available to the non-specialist. a number of the crucial beneficial properties and difficulties of the ergodic concept of soft dynamical platforms are mentioned, and the appendix can function an advent into this theory.

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**Additional resources for Lectures on Spaces of Nonpositive Curvature**

**Example text**

Harmonic functions and random walks on r Let X be a locally compact Hadamard space containing a unit speed geodesic lR -+ X which does not bound a flat half plane and r a countable group of isometries of X satisfying the duality condition. Assume furthermore that X (00) contains at least three points. Suppose J1 is a probability measure on r whose support generates r as a semigroup. 1) h(cp) L = h(cp'l/J)J1Cl/J) for all cp E r. pH Our objective is to show that r admits many J1-harmonic functions.

Proof. Let 'Y(t) = L

Hence the assertion follows from a comparison with Euclidean geometry. D The above lemma together with our previous considerations shows that for any x E X and any ~ E X (00) there is a unique unit speed ray a x,t; : [0,00) ---7 X with ax,t;(O) = x and ax,t;(oo) = ~. For y E X, y -I- x, we denote by ax,y : [O,d(x,y)l- X the unique unit speed geodesic from x to y. On X = XUX(oo) we introduce a topology by using as a basis the open sets of X together with the sets U(x,~,R,c) = {z E Xlz rt. B(x,R),d(ax,z(R),ax,t;(R)) < c}, where x E X,~ E X(oo).

### Lectures on Spaces of Nonpositive Curvature by Werner Ballmann

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