Some topological properties of cohomogeneity one manifolds with negative curvature
type: Item
instance of: scholarly article
title: Some topological properties of cohomogeneity one manifolds with negative curvature
zbMATH Open document ID: 0846.53024
DOI: 10.1007/BF00128196
author: Andrea Spiro
author: Fabio Podestà
published in: Annals of Global Analysis and Geometry
publication date: 1996-09-22T00:00:00Z
review text: The main results of this paper are the following: 1) Let \(M\) be a complete Riemannian manifold with negative sectional curvature and let \(G\) be a closed Lie group of isometries with regular orbits of codimension 1. Then only two possibilities can occur: a) \(G \) has no singular orbit and \(M\) is diffeomorphic to \(G/K \times \mathbb{R}\), where \(K\) is the isotropy subgroup of a point \(p \in M\); b) \(G\) admits exactly one singular orbit. 2) Under the same hypotheses, if \(G\) has a singular orbit, which does not coincide with the trace of a geodesic, then \(M\) is simply connected. 3) Under the same hypotheses, if \(M\) is not simply connected and of dimension \(n \geq 3\), then either each orbit of \(G\) is regular and diffeomorphic to \(T^k \times \mathbb{R}^q\), or there exists a 1-dimensional singular orbit and each regular orbit is covered by \(S^{n - 2} \times \mathbb{R}\). Other interesting results are obtained in case when the group \(G\) is supposed to be semisimple or compact.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 854359
zbMATH Keywords: cohomogeneity one space
zbMATH Keywords: negative sectional curvature
zbMATH Keywords: Lie group of isometries
zbMATH Keywords: singular orbit
zbMATH Keywords: isotropy subgroup
reviewed by: Oldřich Kowalski
MaRDI profile type: Publication
Wikidata QID: Q125743875
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full work available at URL: https://doi.org/10.1007/bf00128196
OpenAlex ID: W2023159093
description: scientific article; zbMATH DE number 854359