Let $M$ be a smooth manifold, let $\bar{g}\in\operatorname{Met}(M)$ be a smooth background metric, and let $\hat{g}:[0,T]\to\operatorname{Met}(M)\subset\Gamma(S^2T^*M)$ be a smooth positive-definite symmetric $2$-tensor family. In any coordinate chart, the Ricci-DeTurck equation has the local form

\begin{align*} \partial_t\hat{g}_{ij}=\sum_{a,b}(\hat{g}^{-1})_{ab}\partial_a\partial_b\hat{g}_{ij}+Q_{ij}(\hat{g}^{-1},\partial \hat{g},\bar{g}^{-1},\partial \bar{g},\partial^2\bar{g}), \end{align*}

where $Q_{ij}$ is smooth in its arguments as long as $\hat{g}$ remains positive definite. Hence the equation is a strictly parabolic quasilinear system locally on the open cone of positive-definite symmetric $2$-tensors. On any time interval on which the eigenvalues of $\hat{g}$ relative to a fixed background metric stay between positive upper and lower bounds uniformly in space and time, this strict parabolicity is uniform.