Let $(M,g)$ be a smooth Riemannian manifold and let $h\in \Gamma(S^2T^*M)$ be a symmetric $2$-tensor. The principal symbol of the linearization of $g\mapsto -2\operatorname{Ric}(g)$ is
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\begin{align*}
\sigma_\xi(D(-2\operatorname{Ric})_g)(h)_{ij}
&= |\xi|_g^2h_{ij}+\xi_i\xi_j\operatorname{tr}_g h
-\sum_{k,\ell}\xi_i(g^{-1})_{k\ell}\xi_\ell h_{kj}
-\sum_{k,\ell}\xi_j(g^{-1})_{k\ell}\xi_\ell h_{ki}.
\end{align*}
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For every $p\in M$ and every non-zero $\xi\in T_p^*M$, the [linear map](/page/Linear%20Map) $\sigma_\xi(D(-2\operatorname{Ric})_g)$ has a non-trivial kernel.