Lower curvature bounds and cohomogeneity one manifolds.
type: Item
instance of: scholarly article
title: Lower curvature bounds and cohomogeneity one manifolds.
zbMATH Open document ID: 1035.53043
DOI: 10.1016/S0926-2245(02)00108-0
author: Lorenz Schwachhöfer
published in: Differential Geometry and its Applications
publication date: 2003-08-06T00:00:00Z
review text: \textit{K. Grove} and \textit{W. Ziller} [Ann. Math. (2) 152, 331--367 (2000; Zbl 0991.53016)] discovered a class of new examples of closed manifolds with nonnegative sectional curvature. All these manifolds admit a cohomogeneity one action, i.e. a smooth action of a compact Lie group whose principal orbit has codimension one. They conjectured that any cohomogeneity one manifold admits an invariant metric with nonnegative sectional curvature. Moreover, \textit{W. Tuschmann} and \textit{L. J. Schwachhöfer} [Curvature and cohomogeneity one'', preprint 62, Max Planck Inst., 2001, available at \texttt{http://www.mis.mpg.de/preprints/2001/prepr6201-abstr.html}] showed that any cohomogeneity one manifold admits invariant metrics of almost nonnegative sectional curvature. In the paper under review the author considers examples which give positive answer to the above conjecture. In particular he discusses metrics on homogeneous vector bundels, generalizing some ideas of J. Cheeger, and discusses when such bundles admit invariant metrics with normal homogeneous collar. Moreover, examples of manifolds with nonnegative or almost nonegative sectional curvature are discussed. Some survey of known obstructions for the existance of metrics of almost nonnegative curvature is also given.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 57R60
Mathematics Subject Classification ID: 58B20
zbMATH DE Number: 1959901
zbMATH Keywords: cohomogeneity one
zbMATH Keywords: nonnegative sectional curvature
zbMATH Keywords: lower curvature bounds
zbMATH Keywords: Kervaire spheres
Wikidata QID: Q115338136
reviewed by: Andrzej Szczepański
MaRDI profile type: Publication
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full work available at URL: s0926-2245(02)00108-0
OpenAlex ID: W2086645874
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description: scientific article; zbMATH DE number 1959901