Hessian on Riemannian product manifold

Question

Let $(M_1,g_1) \times (M_2,g_2)$ be a Riemannian product-manifold, and let $f:(M_1 \times M_2) \rightarrow \mathbb{R}^+$ a positive scalar function on the product manifold.

If $H^f$ is the Hessian of the fuction $f$ on the product manifold, can it be written as $H^f=H_1^f +H_2^f$?

(where with $H_1^f$ and $H_2^f$, I mean the Hessian of $f$ on $(M_1, g_1)$ and $(M_2, g_2)$ respecrively).