Let $M$ be a smooth manifold, let $I \subset \mathbb{R}$ be an open interval, and let $(g(t))_{t \in I}$ be a smooth one-parameter family of Riemannian metrics on $M$. Fix a coordinate chart $(U,\varphi)$ on $M$ with coordinates $(x_1,\dots,x_n)$. Write $g_{ij}(t)$ for the local components of $g(t)$ on $U$, write $g^{ij}(t)$ for the local components of the inverse metric, and define the symmetric tensor $v(t)$ on $U$ by

\begin{align*} v_{ij}(t) := \partial_t g_{ij}(t). \end{align*}

Then, for every $t \in I$ and every point of $U$,

\begin{align*} \partial_t g^{ij}(t) = -g^{ik}(t)g^{j\ell}(t)v_{k\ell}(t). \end{align*}

In particular, if $g(t)$ evolves by Ricci flow,

\begin{align*} \partial_t g_{ij}(t) = -2\operatorname{Ric}_{ij}(t), \end{align*}

then

\begin{align*} \partial_t g^{ij}(t) = 2\operatorname{Ric}^{ij}(t), \end{align*}

where $\operatorname{Ric}^{ij}(t) := g^{ik}(t)g^{j\ell}(t)\operatorname{Ric}_{k\ell}(t)$.