Let $\overline\nabla f$ and $\overline{\operatorname{Hess}}f$ be the ambient gradient and Hessian. The <Laplacian of a restricted ambient function> is
$$
\Delta_M(f|_M)
=\operatorname{tr}_{TM}(\overline{\operatorname{Hess}}f)
+\langle\overline\nabla f,\mathbf H\rangle.
$$
To prove it, choose a local orthonormal tangent frame $e_1,\ldots,e_n$ with $\nabla^M_{e_i}e_j=0$ at the point under consideration. There,
$$
\begin{aligned}
\Delta_M(f|_M)
&=\sum_i e_i(e_i f)\\
&=\sum_i\left(\overline{\operatorname{Hess}}f(e_i,e_i)
+\langle\overline\nabla f,\overline\nabla_{e_i}e_i\rangle\right)\\
&=\sum_i\overline{\operatorname{Hess}}f(e_i,e_i)
+\left\langle\overline\nabla f,\sum_iA(e_i,e_i)\right\rangle.
\end{aligned}
$$
Both sides are intrinsic scalars, so the pointwise calculation proves the formula everywhere.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 115 / 4 / b / Solution

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


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