Transverse Weitzenb\"ock formulas and de Rham cohomology of totally geodesic foliations

TL;DR

This paper establishes transverse Weitzenb"ock formulas for totally geodesic foliations, leading to new nullity theorems for de Rham cohomology under positive transverse curvature conditions.

Contribution

It introduces transverse Weitzenb"ock identities for horizontal Laplacians and derives nullity theorems based on transverse curvature positivity.

Findings

Nullity theorems for de Rham cohomology under positive transverse curvature

Transverse Weitzenb"ock identities for horizontal Laplacians

Curvature quantities in the adiabatic limit of metric variations

Abstract

We prove transverse Weitzenb\"ock identities for the horizontal Laplacians of a totally geodesic foliation. As a consequence, we obtain nullity theorems for the de Rham cohomology assuming only the positivity of curvature quantities transverse to the leaves. Those curvature quantities appear in the adiabatic limit of the canonical variation of the metric.