Cohomogeneity one manifolds with quasipositive curvature

TL;DR

This paper classifies cohomogeneity one manifolds with quasipositive curvature, extending known positive curvature results and utilizing key geometric tools to identify such manifolds with few exceptions.

Contribution

It generalizes classification techniques from positive to quasipositive curvature for cohomogeneity one manifolds, covering almost all cases.

Findings

Most cohomogeneity one manifolds admit quasipositive curvature.

Two 7-dimensional families are exceptions.

Key tools include Wilking's Chain Theorem and the Rank Lemma.

Abstract

In this paper we give a classification of cohomogeneity one manifolds admitting an invariant metric with quasipositive sectional curvature except for two $7$-dimensional families. The main result carries over almost verbatim from the classification results in positive curvature carried out by Verdiani and Grove, Wilking and Ziller. Three main tools used in the positively curved case that we generalized to quasipositively curved cohomogeneity one manifolds are Wilking's Chain Theorem, the classification of positively curved fixed point homogeneous manifolds by Grove and Searle and the Rank Lemma.