Rank and symmetry of Riemannian manifolds
type: Item
instance of: scholarly article
title: Rank and symmetry of Riemannian manifolds
zbMATH Open document ID: 0822.53032
DOI: 10.1007/BF02564499
author: J.-H. Eschenburg
author: Carlos Olmos
published in: Commentarii Mathematici Helvetici
publication date: 1995-10-05T00:00:00Z
full work available at URL: https://eudml.org/doc/140350
review text: Let \(M\) be a complete irreducible Riemannian manifold. A \(k\)-flat in \(M\) is a complete connected flat totally geodesic immersed submanifold in \(M\) of dimension \(k\). The rank of \(M\) is the maximal dimension \(k\) such that every geodesic in \(M\) lies in a \(k\)-flat. Examples of manifolds of rank \(k\) are locally symmetric spaces of rank \(k\). There is a conjecture that these are the only examples for \(k \geq 2\), which is open for the case that \(M\) has sectional curvature \(K \leq 0\). Concerning this conjecture, the authors give a partial positive answer: If \(M\) is a complete irreducible Riemannian manifold of rank \(k\) satisfying the condition that there is a \(k\)-flat \(F\) in \(M\) and a Lie subgroup \(G\) of the full isometry group \(I(M)\) of \(M\) with bounded Killing fields such that \(G(TF) = TM\), then \(M\) is locally symmetric of rank \(k\). Moreover, if any \(k\)-flat is one-to-one immersed and intrinsically symmetric, then \(M\) is globally symmetric. This result is an improvement of the recent result by \textit{E. Heintze, R. Palais, C.-L. Terng} and \textit{G. Thorbergsson} [J. Reine Angew. Math. 454, 163-179 (1994; Zbl 0804.53074)]. Further, for the case \(k = 1\), the authors prove that a complete Riemannian manifold \(M\), such that \(I(M)\) acts transitively on the set of geodesics, is globally symmetric.
reviewed by: Kouei Sekigawa
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 718593
zbMATH Keywords: rank of Riemannian manifolds
zbMATH Keywords: \(k\)-flat
zbMATH Keywords: locally symmetric spaces
zbMATH Keywords: isometry group
zbMATH Keywords: geodesics
MaRDI profile type: Publication
Wikidata QID: Q115392056
OpenAlex ID: W2057670709
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description: scientific article; zbMATH DE number 718593