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 hyperkähler and Calabi-Yau manifolds, respectively.

arXiv commentAfter announcement Georges Habib pointed out that his article "Modified differentials and basic cohomology for Riemannian foliations" (MR3078356) with Ken Richardson contains a Bochner formula for the twisted Laplacian (Prop 6.7, Thm 6.16) which implies our Thm 4.4 even for non-harmonic foliations. The preprint will thus not be submitted for publication and will only appear in the author's thesis