A note on quasi-positive curvature conditions

TL;DR

This paper classifies certain nested Lie group triples that guarantee the existence of metrics with quasi-positive curvature on the associated homogeneous spaces, leading to new examples of such spaces.

Contribution

It provides a classification of triples of compact Lie groups satisfying the positive triple condition, resulting in new examples of quasi-positively curved manifolds.

Findings

Classified all triples H ⊂ K ⊂ G satisfying the positive triple condition.

Identified new examples of quasi-positively curved spaces, including bundles over spheres and projective spaces.

Expanded the known catalog of manifolds admitting quasi-positive curvature.

Abstract

We classify the triples $H \subset K \subset G$ of nested compact Lie groups which satisfy the "positive triple" condition that was shown by the second author to ensure that $G/H$ admits a metric with quasi-positive curvature. A few new examples of spaces that admit quasi-positively curved metrics emerge from this classification; namely, a $\CP^2$-bundle over $S^6$, a $B^7$-bundle over $\HP^2$, a $\CP^{2n-1}$-bundle over $\HP^{n}$ for each $n\geq 2$, and a family of finite quotients of $T^1S^6$.