New Examples of Obstructions to Non-Negative Sectional Curvatures in Cohomogeneity One Manifolds

TL;DR

This paper extends previous work by providing new examples of cohomogeneity one manifolds that cannot admit invariant metrics with non-negative sectional curvature, and classifies key representations used in these constructions.

Contribution

It generalizes earlier examples to a broader family and classifies all class one representations for specific pairs (G;H) involving spheres.

Findings

New examples of obstructions to non-negative sectional curvature

Complete classification of class one representations for certain sphere pairs

Extension of known results to larger families of manifolds

Abstract

K. Grove, L. Verdiani, B. Wilking and W. Ziller gave the first examples of cohomogeneity one manifolds which do not carry invariant metrics with non-negative sectional curvatures. In this paper we generalize their results to a larger family. We also classified all class one representations for a pair (G;H) with G/H some sphere, which are used to construct the examples.