On the geometric rank of homogeneous spaces of nonpositive curvature
type: Item
instance of: scholarly article
title: On the geometric rank of homogeneous spaces of nonpositive curvature
zbMATH Open document ID: 0792.53048
DOI: 10.1007/BF01232428
author: Jens Heber
published in: Inventiones Mathematicae
publication date: 1994-08-02T00:00:00Z
full work available at URL: https://eudml.org/doc/144098
review text: Let \(H\) denote a complete simply connected Riemannian manifold of sectional curvature \(K \leq 0\). \(H\) is always diffeomorphic to a Euclidean space of the same dimension. The geometric rank \(r\) of \(H\) is the largest integer \(r\) for which every geodesic of \(H\) is contained in a complete, totally geodesic imbedded copy of \(\mathbb{R}^ r\). If \(H\) is irreducible of geometric rank \(r \geq 2\), has a lower bound for the sectional curvature and admits a quotient manifold of finite volume, then W. Ballmann and K. Burns, R. Spatzier proved independently that \(H\) must be Riemannian symmetric. The author and reviewer of this article later generalized this result and in particular removed the hypothesis of a lower bound for the sectional curvature. If \(H\) is a homogeneous space, then \(H\) admits a simply transitive solvable group \(S\) of isometries, and \(H\) is isometric to \(S\) with an appropriate left invariant metric. The group \(S\) is not uniquely determined, but the codimension \(k\) of the commutator subgroup \([S,S]\) as a submanifold of \(S\) is independent of the choice of \(S\). The integer \(k\) is the algebraic rank of \(H\) and is always at least as large as the geometric rank \(r\). The main result of this article is the following analogue for homogeneous spaces of the Rank Rigidity Theorem of Ballman and Burns-Spatzier. Theorem. Let \(H\) as above be homogeneous and irreducible with geometric rank \(r \geq 2\). Then \(H\) is a Riemannian symmetric space of noncompact type. Combining this result with a result of J. Wolf one obtains the following Corollary. Every homogeneous space of sectional curvature \(K \leq 0\) admits a unique decomposition (up to order) as a Riemannian product \(T \times E \times S \times H_ 1 \times \dots \times H_ m\), where \(T\) is a flat torus, \(E\) is a Euclidean space, \(S\) is a symmetric space of noncompact type and the factors \(H_ i\) are irreducible and homogeneous with geometric rank 1. Any of the fact…
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 447096
zbMATH Keywords: homogeneous space
zbMATH Keywords: algebraic rank
zbMATH Keywords: geometric rank
zbMATH Keywords: Rank Rigidity Theorem
zbMATH Keywords: symmetric space
zbMATH Keywords: asymptotic geodesics
zbMATH Keywords: Tits pseudometric
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 447096