Consider $(M,g)$ is a Riemannian Manifold (compact).
We have a isometry $\phi:(M,\phi^*g) \to (M,g)$.
One thing I want to know is, how can I efficiently show that intrinsic quantities like $|\nabla\nabla f|^2$ and $(\Delta f)$ are invariant under the isometry.
I have showed $|\phi_* A_p|_{g}=|A_p|_{\phi^*g}$ for any vector field $A$ and similar version of equality for tensors.
Yet, I still have no idea to show my problem. Is it calculate everything out in local coordinate rly the only way to solve this problem?
Any help is appreciated, thank you.