Flag manifolds with strongly positive curvature

TL;DR

This paper classifies all simply-connected homogeneous spaces, specifically Wallach flag manifolds, that admit homogeneous metrics with strongly positive curvature, completing the broader classification effort.

Contribution

It provides a complete description of the moduli spaces of such metrics on Wallach flag manifolds, extending previous work to finalize the classification.

Findings

Complete description of moduli spaces for $W^6$, $W^{12}$, $W^{24}$

Classification of homogeneous spaces with strongly positive curvature

Closure of the classification of such spaces

Abstract

We obtain a complete description of the moduli spaces of homogeneous metrics with strongly positive curvature on the Wallach flag manifolds $W^6$, $W^{12}$ and $W^{24}$, which are respectively the manifolds of complete flags in $\mathbb C^3$, $\mathbb H^3$ and $\mathbb{Ca}^3$. Together with our earlier work, this concludes the classification of simply-connected homogeneous spaces that admit a homogeneous metric with strongly positive curvature.