Isometry Invariance of Riemannian Distance Function

Theorem
Let $\struct {M, g}$ and $\struct {\tilde M, \tilde g}$ be connected Riemannian manifolds with or without boundaries.

Let $\phi : M \to \tilde M$ be an isometry.

Let $x, y \in M$ be points.

Let $\map {d_g} {x, y}$ be the Riemannian distance between $x$ and $y$.

Then:


 * $\map {d_{\tilde g} } {\map \phi x, \map \phi y} = \map {d_g} {x, y}$