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

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