Quaternion-Kähler manifolds with non-negative quaternionic sectional curvature
Andrei Moroianu, Uwe Semmelmann, Gregor Weingart
TL;DR
The paper establishes a quaternionic analogue of Gray’s result by introducing quaternionic sectional curvature and analyzing the twistor space of a positive quaternion-Kähler manifold with a nearly Kähler structure. By relating the base curvature to the canonical curvature on the nearly Kähler twistor space and applying a Weitzenböck-type framework, it shows that non-negative quaternionic sectional curvature forces the manifold to be a Wolf space, i.e., symmetric. Conversely, Wolf spaces are shown to have non-negative quaternionic sectional curvature, with sharp bounds expressed in terms of the Wirtinger angle and scalar curvature. Together, these results provide a curvature-based characterization of Wolf spaces and reinforce the view that positivity conditions strongly constrain quaternion-Kähler geometry.
Abstract
Compact Hermitian symmetric spaces are Kähler manifolds with constant scalar curvature and non-negative sectional curvature. A famous result by A. Gray states that, conversely, a compact simply connected Kähler 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-Kähler setting. In order to achieve this, we introduce the quaternionic sectional curvature of quaternion-Kähler manifolds, we show that every Wolf space has non-negative quaternionic sectional curvature, and we prove that, conversely, every quaternion-Kähler manifold with non-negative quaternionic sectional curvature is a Wolf space. The proof makes crucial use of the nearly Kähler twistor spaces of positive quaternion-Kähler manifolds.