Let $(M,g)$ be an $n$-dimensional smooth Riemannian manifold, let $f \in C^\infty(M)$, and let $\lambda \in \mathbb{R}$. Suppose $(M,g,f)$ satisfies the gradient Ricci soliton equation

\begin{align*} \operatorname{Ric} + \operatorname{Hess}_{g} f = \lambda g \end{align*}

as an identity of smooth symmetric $(0,2)$-tensors on $M$, where $\operatorname{Ric}$ is the Ricci tensor of $g$ and $\operatorname{Hess}_{g} f$ is the Hessian of $f$ with respect to the Levi-Civita connection of $g$. Then the scalar curvature $R$ and the Laplace-Beltrami operator $\Delta f := \operatorname{tr}_g(\operatorname{Hess}_{g} f)$ satisfy the pointwise identity

\begin{align*} R + \Delta f = n\lambda \end{align*}

on $M$.