A differential geometric characterization of symmetric spaces of higher rank
type: Item
instance of: scholarly article
title: A differential geometric characterization of symmetric spaces of higher rank
zbMATH Open document ID: 0722.53041
DOI: 10.1007/BF02699876
author: Patrick Eberlein
author: Jens Heber
published in: Publications Mathématiques
publication date: 1990-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104066
review text: A simply connected complete Riemannian manifold M of nonpositive curvature (Hadamard manifold) is of higher rank if every geodesic lies in a totally geodesic flat subspace (``flat'') of dimension \(\geq 2\). The only known irreducible examples of such manifolds are symmetric spaces of higher rank. Ballmann and Burns/Spatzier have shown that these are in fact the only possible examples provided that (i) the curvature is bounded from below and (ii) there exists a discrete subgroup of the isometry group I(M) such that its orbit space has finite volume. Using a different proof, the present paper generalizes this result in two ways: The authors omit the assumption (i) and replace (ii) by the more general duality condition for I(M) saying that for any geodesic \(\gamma\), there exists a sequence \((\phi_ n)\) in I(M) such that \(\phi_ n(p)\to \gamma (\infty)\) and \(\phi_ n^{-1}(p)\to \gamma (-\infty)\) for any \(p\in M.\) One of the main tools of the proof is taken from a previous paper of the first author, based on an idea of Ballmann: If the duality condition holds, the orbits of the holonomy group, projected to M(\(\infty)\), are contained in the closures of isometry orbits on M(\(\infty)\). Thus, by Berger's holonomy theorem, it is enough to find a closed nonempty proper subset \(A\subset M(\infty)\) which is invariant under I(M). It is shown that \(A=M_{\alpha}(\infty)\) for some \(\alpha\in (0,\pi)\) does the job, where \[ M_{\alpha}(\infty)=\{x\in M(\infty);\quad y\in M(\infty);\quad p\in M;\quad\sphericalangle_ p(x,y)=\alpha \} \] and \(\sphericalangle_ p(x,y)\) is the angle between the geodesics from p to x and to y. To show that \(M_{\alpha}(\infty)\neq \phi\), the authors use the Tits distance at M(\(\infty)\) which was introduced by Gromov. The main observation is (Lemma 4.2): If \(v\in T^ 1M\) is a regular recurrent vector tangent to a flat F, then the Tits distance ball \(B_{\alpha}(x)\) for \(x=\gamma \vee (\infty)\) and for small \(\alpha >0\) is containe…
reviewed by: J.-H. Eschenburg
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 4190620
zbMATH Keywords: rank rigidity theorem
zbMATH Keywords: Hadamard manifold
zbMATH Keywords: isometry group
zbMATH Keywords: duality condition
zbMATH Keywords: Tits distance
Wikidata QID: Q115391866
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4190620