The almost cyclicity of the fundamental groups of positively curved manifolds
type: Item
instance of: scholarly article
title: The almost cyclicity of the fundamental groups of positively curved manifolds
zbMATH Open document ID: 0859.53022
published in: Inventiones Mathematicae
publication date: 1997-04-14T00:00:00Z
review text: In this paper, the author studies the fundamental groups of positively curved manifolds by using the notion of pure \(F\)-structure, which is a generalization of torus action. An invariant pure \(F\)-structure on a Riemannian manifold \(M\) consists of a flat \(T^k\)-bundle with non-trivial holonomy acting on \(M\) such that the induced vector fields are local Killing vector fields. We have a notion of orbit, and the rank of the action is then defined as the dimension of an orbit of the smallest dimension. The author proves the following result. Main Theorem. Let \(M\) be a compact positively curved \(n\)-manifold. Suppose, in addition, that \(M\) admits an invariant positive rank pure \(F\)-structure. Then the fundamental group of \(M\) has a cyclic subgroup with index less than a constant depending only on \(n\). As a consequence, he obtains the main result of the paper. Theorem 0.2. There exists a small constant, \(v(n,\delta)>0\), depending on \(n\), and \(0<\delta \leq 1\) such that, if a \(\delta\)-pinched \(n\)-manifold has volume smaller than \(v(n,\delta)\), then the fundamental group has a finite cyclic subgroup with index less than \(w_n\), a constant depending only on \(n\). In a previous paper the author has proved the existence of a finite cyclic subgroup with index less than a constant depending on \(n\) and \(\delta\); thus, this result is substantially improved in the paper under review.
reviewed by: Manuel de León
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 931983
zbMATH Keywords: \(\delta\)-pinched manifold
zbMATH Keywords: pure \(F\)-structure
zbMATH Keywords: fundamental group
author: Xiaochun Rong
MaRDI profile type: Publication
full work available at URL: s002220050088
OpenAlex ID: W1981151535
DOI: 10.1007/S002220050088
description: scientific article; zbMATH DE number 931983