Hodge-de Rham and Lichn\'erowicz Laplacians on double forms and some vanishing theorems

A $(p,q)$-double form on a Riemannian manifold $(M,g)$ can be considered simultaneously as a vector-valued differential $p$-form over $M$ or alternatively as a vector-valued $q$-form. Accordingly, the usual Hodge-de Rham Laplacian on differential forms can be extended to double forms in two ways. The differential operators obtained in this way are denoted by $\Delta$ and $\widetilde{\Delta}$.\\ In this paper, we show that the Lichn\'erowicz Laplacian $\Delta_L$ once operating on double forms, is nothing but the average of the two operators mentioned above. We introduce a new product on double forms to establish index-free formulas for the curvature terms in the Weitzenb\"ock formulas corresponding to the Laplacians $\Delta, \widetilde{\Delta}$ and $\Delta_L$. We prove vanishing theorems for the Hodge-de Rham Laplacian $\Delta$ on $(p,0)$ double forms and for $\Delta_L$ and $\Delta$ on symmetric double forms of arbitrary order. These results generalize recent results by Petersen-Wink. Our vanishing theorems reveal the impact of the role played by the rank of the eigenvectors of the curvature operator on the structure (e.g. the topology) of the manifold.