Revisiting homogeneous spaces with positive curvature

TL;DR

This paper revisits the classification of positively curved homogeneous spaces, providing a comprehensive proof that confirms no additional examples exist beyond those previously known, especially addressing gaps in earlier classifications.

Contribution

It offers a modern, complete, and self-contained proof of the classification, closing gaps in Berard Bergery's work on odd-dimensional spaces.

Findings

Confirmed no new positively curved homogeneous spaces exist

Provided a complete proof for both odd and even dimensions

Addressed and filled gaps in previous classifications

Abstract

As was recently observed by M. Xu and J. Wolf, there is a gap in Berard Bergery's classification of odd dimensional positively curved homogeneous spaces. Since this classification has been used in other papers as well, we give a modern, complete and self contained proof (in odd as well as even dimensions), confirming that there are indeed no new examples.