Torus actions, maximality, and non-negative curvature
type: Item
instance of: scholarly article
title: Torus actions, maximality, and non-negative curvature
zbMATH Open document ID: 1493.53049
author: Christine M. Escher
author: Catherine Searle
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2021-12-01T00:00:00Z
review text: The question of which closed smooth manifolds admit a Riemannian metric with positive or non-negative sectional curvature is a classical long-standing problem in Riemannian geometry. In the 1990's, Grove advocated for a specialization of this problem in the presence of a large symmetry group. A prototypical result in this direction is [\textit{K. Grove} and the second author, J. Pure Appl. Algebra 91, No. 1--3, 137--142 (1994; Zbl 0793.53040)], which bounds the rank of a torus acting on a closed simply connected manifold with positive sectional curvature and establishes an equivariant diffeomorphism classification when the bound is saturated. Many other topological classification results of varying strength have been established over the years and a generalisation of the result of Grove-Searle to the case of non-negative curvature was proposed in [\textit{F. Galaz-Garcia} and the second author, Proc. Am. Math. Soc. 139, No. 7, 2559--2564 (2011; Zbl 1220.53044)] as the Maximal Symmetry Rank Conjecture. The paper under review considers closed simply connected non-negatively curved Riemannian manifolds admitting an isometric \emph{isotropy-maximal} torus action. The main result of the paper (Theorem A) gives bounds for the rank of the torus in terms of the dimension and a classification up to equivariant diffeomorphism (any such manifold is equivariantly diffeomorphic to the quotient of a product of spheres of dimension larger or equal than 3 by a free linear action of a torus). Consequences of this result for the Maximal Symmetry Rank Conjecture in low dimensions are also discussed. The type of torus actions considered in the paper imply that there exists a sub-torus \(T'\) of the torus \(T\) acting on the non-negatively curved manifold \(M^n\), such that the quotient \(X^{2m}=M/T'\) is an even-dimensional orbifold with non-negative curvature (in the Alexandrov sense) and an isometric action of an \(m\)-torus \(T^m=T/T'\). Under a smoothness assumption, a classificati…
reviewed by: Lorenzo Foscolo
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 7436420
zbMATH Keywords: non-negative sectional curvature
zbMATH Keywords: torus actions
zbMATH Keywords: maximal symmetry rank conjecture
zbMATH Keywords: torus orbifolds
MaRDI profile type: Publication
full work available at URL: crelle-2021-0035
OpenAlex ID: W3198140231
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DOI: 10.1515/CRELLE-2021-0035
description: scientific article; zbMATH DE number 7436420