On intermediate Ricci curvature and fundamental groups
type: Item
instance of: scholarly article
title: On intermediate Ricci curvature and fundamental groups
zbMATH Open document ID: 0883.53042
published in: Illinois Journal of Mathematics
publication date: 1997-08-03T00:00:00Z
review text: Synge's theorem states that a closed Riemannian \(n\)-manifold with positive sectional curvature is orientable if either \(n\) is even and has fundamental group of order 1 or 2. This result is extended here for any manifold with positive Ricci curvature provided the first systole (i.e., the length of the shortest closed noncontractible curve) is sufficiently large. Synge's theorem is implied by Weinstein's result which states that any isometry \(f\) of a compact, oriented, Riemannian \(n\)-manifold with positive sectional curvature has a fixed point, provided \(n\) is even (resp. odd) and \(f\) preserves (resp. reverses) orientation. An analogous result is given here. If \(f\) is an isometry of a complete, oriented, Riemannian \(n\)-manifold with \(\text{Ric}_k\geq k\), whose minimal displacement is sufficiently large, then \(f\) preserves (resp. reverses) orientation provided \(n\) is odd (resp. even).
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1043161
zbMATH Keywords: Synge's theorem
zbMATH Keywords: fundamental group
zbMATH Keywords: positive Ricci curvature
zbMATH Keywords: first systole
zbMATH Keywords: isometry
author: Frederick Wilhelm
reviewed by: Cornelia-Livia Bejan
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1043161