# Isometry (Riemannian geometry)

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Isometry (Riemannian geometry)

In the study of Riemannian geometry in mathematics, a local isometry from one (pseudo-)Riemannian manifold to another is a map which pulls back the metric tensor on the second manifold to the metric tensor on the first. When such a map is also a diffeomorphism, such a map is called an isometry (or isometric isomorphism), and provides a notion of isomorphism ("sameness") in the category Rm of Riemannian manifolds.

Definition

Let $\left(M, g\right)$ and $\left(M\text{'}, g\text{'}\right)$ be two Riemannian manifolds, and let $f : M o M\text{'}$ be a diffeomorphism. Then $f$ is called an isometry (or isometric isomorphism) if

:$g = f^\left\{*\right\} g\text{'},$

where $f^\left\{*\right\} g\text{'}$ denotes the pullback of the rank (0, 2) metric tensor $g\text{'}$ by $f$. Equivalently, in terms of the push-forward $f_\left\{*\right\}$, we have that for any two vector fields $v, w$ on $M$ (i.e. sections of the tangent bundle $mathrm\left\{T\right\} M$),

:$g\left(v, w\right) = g\text{'} left\left( f_\left\{*\right\} v, f_\left\{*\right\} w ight\right).$

If $f$ is a local diffeomorphism such that $g = f^\left\{*\right\} g\text{'},$, then $f$ is called a local isometry.

References

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