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

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.