On quaternionic bisectional curvature

TL;DR

This paper investigates quaternionic bisectional curvature on quaternion-Kähler manifolds, showing that only quaternionic projective space has non-negative curvature and providing new insights into curvature properties and classifications.

Contribution

It demonstrates that non-negative quaternionic bisectional curvature is exclusive to quaternionic projective space and offers a new proof of Gray's classification of Kähler manifolds with non-negative sectional curvature.

Findings

Non-negative quaternionic bisectional curvature only occurs in quaternionic projective space.

Symmetric quaternion-Kähler manifolds other than quaternionic projective space have negative quaternionic bisectional curvature.

Non-negative sectional curvature does not imply non-negative quaternionic bisectional curvature.

Abstract

In this article we study the concept of quaternionic bisectional curvature introduced by B. Chow and D. Yang for quaternion-K\"ahler manifolds. We show that non-negative quaternionic bisectional curvature is only realized for the quaternionic projective space. We also show that all symmetric quaternion-K\"ahler manifolds different from the quaternionic projective space admit quaternionic lines of negative quaternionic bisectional curvature. In particular this implies that non-negative sectional curvature does not imply non-negative quaternionic bisectional curvature. Moreover we give a new and rather short proof of a classification result by A. Gray on compact K\"ahler manifolds of non-negative sectional curvature.