Manifolds of nonpositive curvature and their buildings
type: Item
instance of: scholarly article
title: Manifolds of nonpositive curvature and their buildings
zbMATH Open document ID: 0643.53037
DOI: 10.1007/BF02698934
published in: Publications Mathématiques
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104020
review text: 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 \textit{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 \textit{W. Ballmann}, \textit{M. Brin} and the reviewer [Ann. Math., II. Ser. 122, 171- 203 (1985; Zbl 0589.53047)] and \textit{W. Ballmann}, \textit{M. Brin} and \textit{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 spa…
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 51H20
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 4048573
zbMATH Keywords: nonpositive sectional curvature
zbMATH Keywords: finite volume
zbMATH Keywords: noncompact type
zbMATH Keywords: Tits buildings
zbMATH Keywords: symmetric space
author: Keith Burns
author: Ralf Spatzier
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
cites work: Nonpositively curved manifolds of higher rank
cites work: Structure of manifolds of nonpositive curvature. I
cites work: Structure of manifolds of nonpositive curvature. II
cites work: Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes
cites work: Manifolds of nonpositive curvature
cites work: On topological Tits buildings and their classification
cites work: Q3685617
cites work: Isometry groups of simply connected manifolds of nonpositive curvature. II
cites work: Visibility manifolds
cites work: Q5731354
cites work: Strong Rigidity of Locally Symmetric Spaces. (AM-78)
cites work: Discrete groups of motions of manifolds of nonpositive curvature
cites work: On the transitivity of holonomy systems
cites work: Buildings of spherical type and finite BN-pairs
OpenAlex ID: W2088927897
description: scientific article; zbMATH DE number 4048573