Definition:Isometry (Riemannian Manifolds)

Definition
Let $\struct {M, g}$ and $\struct {\tilde M, \tilde g}$ be Riemannian manifolds with Riemannian metrics $g$ and $\tilde g$ respectively.

Let the mapping $\phi : M \to \tilde M$ be a diffeomorphism such that:


 * $\phi^* \tilde g = g$

Then $\phi$ is called an isometry (from $\struct {M, g}$ to $\struct {\tilde M, \tilde g}$).