On the geometry of cohomogeneity one manifolds with positive curvature

TL;DR

This survey reviews cohomogeneity one manifolds with positive curvature, discussing known examples, their geometry, classification results, and potential candidates, including connections to Hitchin orbifolds.

Contribution

It summarizes the classification of cohomogeneity one manifolds with positive curvature and highlights unresolved cases linked to Hitchin orbifolds.

Findings

Classification by Grove-Wilking-Ziller of manifolds admitting positive curvature

Identification of two candidate series with unknown curvature status

Discussion of Hitchin orbifolds and their curvature properties

Abstract

This is a survey on cohomogeneity one manifolds with positive curvature. We discuss the known examples of this type and their geometry and the functions that describe the metric. We also describe the classification of cohomogeneity one manifolds that can admit a metric with positive curvature due to Grove-Wilking-Ziller. Two series of candidates arose in this classification for which it is not yet know if they admit positive curvature. The connection of these candidates to the Hitchin self dual Einstein orbifolds is discussed as well, including the curvature properties of the Hitchin metrics.