  There are very few general theorems on the kernel of the well-known
Lichnerowicz Laplacian. In the present article we consider the geometry of the
kernel of this operator restricted to covariant (not necessarily symmetric or
skew-symmetric) tensors. Our approach is based on the analytical method, due to
Bochner, of proving vanishing theorems for the null space of Laplace operator.
In particular, we pay special attention to the kernel of the Lichnerowicz
Laplacian on Riemannian symmetric spaces of compact and noncompact types. In
conclusion, we give some applications to the theories of infinitesimal Einstein
deformations and the stability of Einstein manifolds.
