Positively Curved Cohomogeneity One Manifolds and 3-Sasakian Geometry

TL;DR

This paper classifies all simply connected odd-dimensional cohomogeneity one manifolds that can support invariant metrics with positive sectional curvature, identifying known examples and potential new candidates.

Contribution

It provides an exhaustive classification of such manifolds, highlighting the limited known examples and potential new ones in odd dimensions.

Findings

Identifies known positively curved manifolds: spheres, Eschenburg spaces, Bazaikin spaces, Berger space.

Shows only a few isolated and infinite families potentially admit positive curvature.

Provides a complete description of cohomogeneity one manifolds supporting positive curvature.

Abstract

We give an exhaustive description of all simply connected odd dimensional cohomogeneity one manifolds that can possibly support an invariant metric with positive sectional curvature. Among the known examples of odd dimensional manifolds with positive curvature, apart from spheres, there are two infinite families among the 7-dimensional Eschenburg spaces and 13-dimensional Bazaikin spaces and one isolated 7-dimensional Berger space with this property. In addition to these examples, it turns out that only one isolated 7-manifold, and two infinite 7-dimensional families, potentially admit invariant cohomogeneity one metrics of positive curvature.