The generalized Frankel conjecture in Sasaki geometry

TL;DR

This paper proves structure results for transverse reducible Sasaki manifolds, showing positive Ricci curvature implies transverse irreducibility, and classifies compact Sasaki manifolds with non-negative transverse bisectional curvature, extending Mok's theorem to Sasaki geometry.

Contribution

It establishes the transverse irreducibility of Sasaki manifolds with positive Ricci curvature and classifies those with non-negative transverse bisectional curvature, generalizing Mok's theorem.

Findings

Sasaki manifolds with positive Ricci curvature are transversely irreducible.

No join construction exists for irregular Sasaki-Einstein manifolds.

Classification of compact Sasaki manifolds with non-negative transverse bisectional curvature.

Abstract

We prove some structure results for \emph{transverse reducible} Sasaki manifolds. In particular, we show Sasaki manifolds with positive Ricci curvature is transversely irreducible, and so there is no join (product) construction for irregular Sasaki-Einstein manifolds, as opposed to the quasi-regular case done by Wang-Ziller and Boyer-Galicki. As an application, we classify compact Sasaki manifolds with non-negative transverse bisectional curvature, which can be viewed as the generalized Frankel conjecture (N. Mok's theorem) in Sasaki geometry.