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

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.