A remark on the Laplacian operator which acts on symmetric tensors

More than forty years ago J. H. Samson has defined the Laplacian ${\mathrm{\Delta }}_{sym}$ 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 $\mathrm{\Delta }$ 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 ${\mathrm{\Delta }}_{sym}$ is the Yano rough Laplacian and show its spectrum properties on a compact Riemannian manifold.