Let $M$, $N$ be two Riemannian manifolds, $\Delta_{M}$ and $\Delta_{N}$ be the Laplace-Beltrami operators on $M$ and $N$ respectively and let $\phi: M \to N$ be an isometry. Suppose $u_{N}(x,t)$ is a solution to the heat equation on $N$. Then is $u_{M}(x,t) := u_{N}(\phi(y),t)$ as solution to the heat equation on $M$?
My idea:
We now that isometries commute with the Laplacian and so:
$$(\Delta_{M} u_{M})(x,t) = \Delta_{N}(u_{N}(\phi(x),t))) = (\Delta_{N}u_{N})(\phi(x),t) = (\partial_{t}u_{N})(\phi(x),t))$$
But it is not immediately clear to me why one should have $(\partial_{t}u_{N})(\phi(x),t))= \partial_{t}u_{M}(x,t)$, apart from maybe the isometry does not alter time dependence? Any references would be appreciated!
Of course I could be wrong...