# When does Laplace operator commute with isometries? [closed]

let $(M,g)$ be a smooth compact Riemannian manifold and $$\phi:M\rightarrow M$$ an isometry. Let $u\in C^{\infty}(M)$, are there general conditions on $u$ and $\phi$ such that the following relation holds? $$\Delta_{g}\left( u\circ \phi \right)=(\Delta_{g}u)\circ\phi$$

Always: the definition of $\Delta_g$ is in terms of $g$, so any diffeomorphism preserving $g$ preserves $\Delta_g$, and this is what preserves'' means.