Weitzenböck formulas for Riemannian foliations

In this paper we study the interplay between adiabatic limits of a Riemannian foliation and the classical Weitzenböck formula. For the leafwise   part, our study leads to a vanishing result for the first order term Eˆ1 of differential spectral sequence associated with the foliation. For the transversal part we obtain a Weitzenböck type formula which is an extension of the previous formula for basic forms due to Ph. Tondeur, M. Min-Oo, and E. Ruh, and is also more general than a Weitzenböck formula for transverse fiber bundle due to Y. Kordyukov.