# Seven dimensional cohomogeneity one manifolds with nonnegative curvature

## TL;DR

This paper investigates seven-dimensional cohomogeneity one manifolds, demonstrating that most do not support nonnegative curvature metrics unless they admit positive curvature, thereby narrowing the classification of such manifolds.

## Contribution

It identifies specific families of 7-dimensional cohomogeneity one manifolds that cannot have nonnegative curvature unless they admit positive curvature, refining the classification.

## Key findings

- Most seven-dimensional cohomogeneity one manifolds lack nonnegative curvature metrics.
- A reduction in candidate manifolds for nonnegative curvature classification in dimension 7.
- Identification of a unique family of candidates for nonnegative curvature in dimension 7.

## Abstract

We show that a certain family of cohomogeneity one manifolds does not admit an invariant metric of nonnegative sectional curvature, unless it admits one with positive curvature. As a consequence, the classification of nonnegatively curved cohomogeneity one manifolds in dimension 7 is reduced to only one further family of candidates