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

TL;DR

This paper extends classical Laplacians to double forms on Riemannian manifolds, introduces new curvature formulas, and proves vanishing theorems that relate curvature eigenvalues to manifold topology.

Contribution

It introduces a new product on double forms, relates various Laplacians, and generalizes vanishing theorems to broader classes of double forms.

Findings

Lichnérowicz Laplacian is the average of two extended Laplacians.

New curvature formulas for Weitzenböck formulas.

Vanishing theorems link curvature eigenvalues to manifold topology.

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 $\Delta$ 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.