Hodge-de Rham and Lichnérowicz Laplacians on double forms and some vanishing theorems

Abstract

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 $Δ$ and $\widetildeΔ$.\\ In this paper, we show that the Lichnérowicz Laplacian $Δ_L$ 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 Weitzenböck formulas corresponding to the Laplacians $Δ, \widetildeΔ$ and $Δ_L$. We prove vanishing theorems for the Hodge-de Rham Laplacian $Δ$ on $(p,0)$ double forms and for $Δ_L$ and $Δ$ 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.