A remark on the Laplacian operator which acts on symmetric tensors

Abstract

More than forty years ago J. H. Samson has defined the Laplacian $\Delta_{sym}$ acting on the space of symmetric covariant $p$-tensors on an $n$-dimensional Riemannian manifold $(M, g)$. This operator is an analogue of the well known Hodge-de Rham Laplacian $\Delta$ which acts on the space of exterior differential $p$-forms ($1 \le p \le n$) on $(M, g)$. In the present paper we will prove that for $n > p = 1$ the operator $\Delta_{sym}$ is the Yano rough Laplacian and show its spectrum properties on a compact Riemannian manifold.