Laplacian of a restricted ambient function
= Laplacian of a restricted ambient function
If $M\subseteq\mathbb R^{n+m}$ and $f$ is smooth in the ambient space, then
$$
\Delta_M(f|_M)=\operatorname{tr}_{TM}(\overline{\operatorname{Hess}}f)+\langle\overline\nabla f,\mathbf H\rangle.
$$