Positively curved manifolds with almost maximal symmetry rank
type: Item
instance of: scholarly article
title: Positively curved manifolds with almost maximal symmetry rank
zbMATH Open document ID: 1032.53025
DOI: 10.1023/A:1021242512463
published in: Geometriae Dedicata
publication date: 2003-03-11T00:00:00Z
review text: The author is interested in the classification, up to homeomorphism, of those positively curved closed manifolds \((M^n,g)\) whose isometry groups are large, in the sense that the rank of these isometry groups is high. This rank is called the \textit{symmetry rank} of the manifold. In [J. Pure Appl.\ Algebra 91, 137-142 (1994; Zbl 0793.53040)], \textit{K.~Grove} and \textit{C. Searle} showed that the symmetry rank can at most be equal to \(\lfloor (n+1)/2\rfloor\), where \(n\)~is the dimension of the manifold. Moreover, equality occurs only for spaces diffeomorphic to a sphere, a real or complex projective space, or a lens space. Here, the author studies the topology of positively curved manifolds with \textit{almost} maximal symmetry rank \(\lfloor (n-1)/2\rfloor\). He shows that a simply connected positively curved manifold \((M^5,g)\) with symmetry rank~\(2\) is homeomorphic to a \(5\)-sphere and provides information on the Euler characteristic and on the rational homotopy groups in a number of cases. He also investigates in how far his results extend to positively curved manifolds with only suitably large symmetry rank 0.
reviewed by: Eric Boeckx
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 1879691
zbMATH Keywords: positive curvature
zbMATH Keywords: isometric torus action
zbMATH Keywords: symmetry rank
author: Xiaochun Rong
MaRDI profile type: Publication
full work available at URL: a:1021242512463
OpenAlex ID: W6475584
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description: scientific article; zbMATH DE number 1879691