- Isometry (Riemannian geometry)
In the study of

Riemannian geometry inmathematics , a**local isometry**from one (pseudo-)Riemannian manifold to another is a map which pulls back themetric tensor on the second manifold to the metric tensor on the first. When such a map is also adiffeomorphism , such a map is called an**isometry**(or**isometric isomorphism**), and provides a notion ofisomorphism ("sameness") in the category**Rm**of Riemannian manifolds.**Definition**Let $(M,\; g)$ and $(M\text{'},\; g\text{'})$ be two Riemannian manifolds, and let $f\; :\; M\; o\; M\text{'}$ be a diffeomorphism. Then $f$ is called an

**isometry**(or**isometric isomorphism**) if:$g\; =\; f^\{*\}\; g\text{'},$

where $f^\{*\}\; g\text{'}$ denotes the pullback of the rank (0, 2) metric tensor $g\text{'}$ by $f$. Equivalently, in terms of the push-forward $f\_\{*\}$, we have that for any two vector fields $v,\; w$ on $M$ (i.e. sections of the

tangent bundle $mathrm\{T\}\; M$),:$g(v,\; w)\; =\; g\text{'}\; left(\; f\_\{*\}\; v,\; f\_\{*\}\; w\; ight).$

If $f$ is a

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

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2010.*

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