A Bochner Technique For Foliations With Non-Negative Transverse Ricci Curvature

TL;DR

This paper extends the Bochner technique to foliations with non-negative transverse Ricci curvature, leading to new vanishing theorems for basic cohomology and applications to specific geometric structures like Sasaki manifolds.

Contribution

It introduces a generalized Bochner technique for foliations with non-negative transverse Ricci curvature, resulting in novel vanishing theorems and applications to Sasaki and Boothby-Wang manifolds.

Findings

New vanishing theorem for basic cohomology

Applications to degenerate 3-$(eta, heta)$-Sasaki manifolds

Applications to Sasaki-$ ext{eta}$-Einstein manifolds

Abstract

We generalize the Bochner technique to foliations with non-negative transverse Ricci curvature. In particular, we obtain a new vanishing theorem for basic cohomology. Subsequently, we provide two natural applications, namely to degenerate 3-$(\alpha,\delta) $-Sasaki and certain Sasaki-$ \eta $-Einstein manifolds, which arise for example as Boothby-Wang bundles over hyperk\"ahler and Calabi-Yau manifolds, respectively.