Cohomogeneity one manifolds with positive Ricci curvature
type: Item
instance of: scholarly article
title: Cohomogeneity one manifolds with positive Ricci curvature
zbMATH Open document ID: 1038.53034
author: Karsten Grove
author: Wolfgang Ziller
published in: Inventiones Mathematicae
publication date: 2003-08-17T00:00:00Z
review text: A connected manifold \(M\) is said to have cohomogeneity one if \(M\) supports a smooth action by a compact Lie group \(G\) such that the orbit space \(M/G\) is one-dimensional. In this paper, the authors prove the following theorem A: A compact cohomogeneity one manifold admits an invariant metric with positive Ricci curvature if and only if its fundamental group is finite. A special case of Theorem A (the case of Kervaire spheres) was treated earlier by J. Cheeger (1973) and by A. Back and W. Y. Hsiang (1987). The authors obtain also the following result, which in fact was the starting point for Theorem A. Theorem B: Any cohomogeneity one manifold admits a complete invariant metric with nonnegative Ricci curvature. Theorem B establishes a special case of a conjecture which was made by the authors in their paper [Ann. Math. (2) 152, 331--367 (2000; Zbl 0991.53016)].
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 57S15
zbMATH DE Number: 1965434
zbMATH Keywords: Ricci curvature
zbMATH Keywords: sectional curvature
zbMATH Keywords: cohomegeneity one manifold
zbMATH Keywords: Lie group
zbMATH Keywords: homogeneous bundle
zbMATH Keywords: invariant metric.
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
full work available at URL: s002220200225
OpenAlex ID: W2046176786
DOI: 10.1007/S002220200225
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description: scientific article; zbMATH DE number 1965434