# Definition:Isometry (Riemannian Manifolds)

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*This page is about Isometry in the context of Inner Product Space. For other uses, see Isometry.*

## 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 $\struct {\tilde M, \tilde g}$)**.

## Sources

- 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.): $\S 2$: Riemannian Metrics. Definitions