Classification of cohomogeneity one manifolds in low dimensions

Corey A. Hoelscher

Vol. 246 (2010), No. 1, 129–185

DOI: 10.2140/pjm.2010.246.129

Abstract

A cohomogeneity one manifold is a manifold whose quotient by the action of a compact Lie group is one-dimensional. Such manifolds are of interest in Riemannian geometry in the context of nonnegative sectional curvature, as well as in other areas of geometry and in physics. We classify compact simply connected cohomogeneity one manifolds in dimensions 5, 6 and 7. We also show that all such manifolds admit metrics of nonnegative sectional curvature, with the possible exception of two families of manifolds.

Keywords

cohomogeneity one manifolds, nonnegative sectional curvature

Mathematical Subject Classification 2000

Primary: 53C20, 57S15

Milestones

Received: 10 March 2009

Accepted: 4 February 2010

Published: 1 May 2010