It is true that the restiction of the Laplace operator on $\mathbb R^n$ to functions on the sphere is the Laplacian for the round metric on the sphere. Is this true for any Riemannian metric $g$ on $\mathbb R^n$?

I mean, is it true that the restriction of $\Delta_g$ to functions on the sphere is the Laplacian on $S^{n-1}$ of the metric induced by $g$?

Thanks in advance.