Curvature on Eschenburg spaces

TL;DR

This paper analyzes the curvature properties of Eschenburg spaces under two different metrics, providing a complete characterization for one and exploring curvature behavior in specific cases for the other.

Contribution

It offers a full curvature classification for Eschenburg spaces with Eschenburg's metric and examines curvature phenomena in Wilking's metric for cohomogeneity two examples.

Findings

Complete curvature characterization for Eschenburg metric.

Identification of zero-curvature regions in Wilking's metric.

Confirmation of almost positive curvature in known examples.

Abstract

We investigate the curvature of Eschenburg spaces with respect to two different metrics, one constructed by Eschenburg and the other by Wilking. With respect to the Eschenburg metric, we obtain a simple complete characterization of the curvature of every Eschenburg space in terms of the triples of integers defining the space. With respect to Wilking's metric, we study all the examples whose natural isometry group acts with cohomogeneity two. Here, we find that apart from the previously known examples with almost positive curvature, all the remaining examples have open sets of points with zero-curvature planes.