Quaternion-K\"ahler manifolds with non-negative quaternionic sectional curvature

TL;DR

This paper extends Gray's characterization of Kähler manifolds with non-negative sectional curvature to quaternion-Kähler manifolds, showing they are precisely Wolf spaces using quaternionic sectional curvature and twistor space techniques.

Contribution

It introduces quaternionic sectional curvature and proves that quaternion-Kähler manifolds with non-negative curvature are exactly Wolf spaces, generalizing Gray's result.

Findings

Wolf spaces have non-negative quaternionic sectional curvature

Quaternion-Kähler manifolds with non-negative curvature are Wolf spaces

Nearly Kähler twistor spaces are crucial in the proof

Abstract

Compact Hermitian symmetric spaces are K\"ahler manifolds with constant scalar curvature and non-negative sectional curvature. A famous result by A. Gray states that, conversely, a compact simply connected K\"ahler manifold with constant scalar curvature and non-negative sectional curvature is a Hermitian symmetric space. The aim of the present article is to transpose Gray's result to the quaternion-K\"ahler setting. In order to achieve this, we introduce the quaternionic sectional curvature of quaternion-K\"ahler manifolds, we show that every Wolf space has non-negative quaternionic sectional curvature, and we prove that, conversely, every quaternion-K\"ahler manifold with non-negative quaternionic sectional curvature is a Wolf space. The proof makes crucial use of the nearly K\"ahler twistor spaces of positive quaternion-K\"ahler manifolds.