Concavity and rigidity in non-negative curvature

TL;DR

This paper investigates the geometric properties of cohomogeneity one manifolds with non-negative curvature, demonstrating the non-existence of certain invariant metrics with positive or non-negative curvature.

Contribution

It provides new examples and non-existence results for invariant metrics with non-negative or positive curvature on cohomogeneity one manifolds.

Findings

Many cohomogeneity one manifolds lack invariant analytic metrics with non-negative curvature.

No invariant metric with positive curvature exists on these manifolds.

The work introduces new applications and improves previous results.

Abstract

This is a significantly improved version with new applications. We show that there are many cohomogeneity one manifolds which do not admit an analytic invariant metric with non-negative sectional curvature, although they do have a smooth one. In particular, there are no invariant metric with positive curvature.