A New Strong Laplacian on Differential Forms

We construct a strong Laplacian $D^*D$ by using the third operator in the basis $\{d,d^*,D\}$ of the space of natural first-order operators acting on the differential forms of a Riemannian manifold $(M,g)$. We study the properties of the Laplacian $D^*D$ and obtain Weitzenbock's formula relating the three strong Laplacians $dd^*$, $d^*d$, and $D^*D$ to the curvature of the manifold $(M,g)$.