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Manifolds of nonpositive curvature and their buildings. (English) Zbl 0643.53037
Let $$\tilde M$$ denote a complete, simply connected manifold of bounded nonpositive sectional curvature that is the Riemannian covering of a complete manifold M of finite volume. Let $$S\tilde M$$ denote the unit tangent bundle of $$\tilde M.$$ For each vector $$v\in S\tilde M$$ let $$\gamma_ v$$ denote the geodesic determined by v and let r(v) denote the dimension of the vector space of all parallel Jacobi vector fields on $$\gamma_ v$$. The rank of $$\tilde M,$$ denoted rank $$(\tilde M)$$, is defined to be the minimum value of the integers r(v), $$v\in S\tilde M$$. In this paper the authors prove the following result obtained independently with another method by W. Ballmann in Ann. Math., II. Ser. 122, 597-609 (1985; Zbl 0585.53031). Theorem. Let $$\tilde M$$ be as above and assume that $$\tilde M$$ is irreducible with rank$$(\tilde M)=k\geq 2$$. Then $$\tilde M$$ is a symmetric space of noncompact type and rank k.
The preliminary stages of the proof use or modify results from W. Ballmann, M. Brin and the reviewer [Ann. Math., II. Ser. 122, 171- 203 (1985; Zbl 0589.53047)] and W. Ballmann, M. Brin and R. Spatzier [ibid., 205-235 (1985; Zbl 0598.53046)]. Following Ballmann, Brin and the reviewer [loc. cit.], a vector $$v\in S\tilde M$$ is regular if $$r(v)=rank(\tilde M)$$. From this work one knows that the set $${\mathcal R}$$ of regular unit vectors is dense and open in $$S\tilde M$$ and if $$v\in {\mathcal R}$$ then the set F(v), the union of all geodesics of $$\tilde M$$ parallel to $$\gamma_ v$$, is a complete, totally geodesic flat submanifold of $$\tilde M$$ of dimension $$k=\text{rank}(\tilde M)$$. Every regular vector $$v\in S\tilde M$$ determines a Weylsimplex $${\mathcal C}(v)\subseteq S_ vF(v)$$ with $$v\in {\mathcal C}(v)$$. More generally one may define $${\mathcal C}(v)$$ for any vector $$v\in S\tilde M$$ that is asymptotic to a regular vector of $$S\tilde M$$. If $$\tilde M$$ is a symmetric space of noncompact type and rank $$k\geq 2$$, then these Weyl simplices are all isometric.
The authors in this paper and Ballmann in loc. cit. prove similar weakened versions of this fact. Ballmann continues by showing that if v is the center of gravity of an appropriate Weyl simplex $${\mathcal C}(w)$$, then the orbit of v under the holonomy group at the base point $$\pi(v)$$ is a proper closed subset of the unit vectors at $$\pi(v)$$. The theorem stated above then follows from a result of M. Berger. The authors follow a different path and define Weyl simplices in the boundary sphere at infinity $$\tilde M(\infty)$$ by projecting the Weyl simplices $${\mathcal C}(v)$$ onto $$\tilde M(\infty)$$. The Weyl simplices in $$\tilde M(\infty)$$ form a topological Tits building $$\Delta(\tilde M)$$ covering $$\tilde M(\infty)$$ and the main result of the authors’ paper “On topological Tits buildings and their classification” [see the review above (Zbl 0643.53036)] shows that $$\Delta(\tilde M)$$ is the Tits building associated to a connected, noncompact simple Lie group G. The group G is the connected group of all automorphisms of $$\Delta(\tilde M)$$ that are also homeomorphisms of $$\Delta(\tilde M)$$. Using arguments similar to those used by Gromov in his extension of the Mostow rigidity theorem the authors then show that $$\tilde M$$ is isometric to the symmetric space G/K, where K is a maximal compact subgroup of G.
Reviewer: P.Eberlein

MSC:
 53C35 Differential geometry of symmetric spaces 51H20 Topological geometries on manifolds 22E46 Semisimple Lie groups and their representations
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References:
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