Sasaki manifolds with positive transverse orthogonal bisectional curvature

TL;DR

This paper proves that compact Sasaki manifolds with positive transverse orthogonal bisectional curvature have finite fundamental groups and universal covers isomorphic to weighted Sasaki spheres, extending recent results using Sasaki-Ricci flow.

Contribution

It establishes the finiteness of the fundamental group and characterizes the universal cover of such Sasaki manifolds, extending prior work by He and Sun.

Findings

Fundamental group of the manifold is finite.

Universal cover is isomorphic to a weighted Sasaki sphere.

Results extend to nonnegative curvature under additional conditions.

Abstract

In this short note we show the following result: Let $(M^{2n+1},g)$ ($n \geq 2$) be a compact Sasaki manifold with positive transverse orthogonal bisectional curvature. Then $\pi_1(M)$ is finite, and the universal cover of $(M^{2n+1},g)$ is isomorphic to a weighted Sasaki sphere. We also get some results in the case of nonnegative transverse orthogonal bisectional curvature under some additional conditions. This extends recent work of He and Sun. The proof uses Sasaki-Ricci flow.