Cohomogeneity one manifolds with a small family of invariant metrics

TL;DR

This paper classifies certain cohomogeneity one manifolds with limited isotropy representation complexity and shows they admit invariant metrics with non-negative sectional curvature.

Contribution

It provides a classification of compact simply connected cohomogeneity one manifolds with up to three irreducible summands in the isotropy representation, revealing their structure as bundles or symmetric spaces.

Findings

Manifolds are either bundles over homogeneous spaces or irreducible symmetric spaces.

Such manifolds admit invariant metrics with non-negative sectional curvature.

Classification up to equivariant diffeomorphism for the specified class.

Abstract

In this paper, we classify compact simply connected cohomogeneity one manifolds up to equivariant diffeomorphism whose isotropy representation by the connected component of the principal isotropy subgroup has three or less irreducible summands. The manifold is either a bundle over a homogeneous space or an irreducible symmetric space. As a corollary such manifolds admit an invariant metric with non-negative sectional curvature.