# Isometry (Riemannian geometry)

In mathematics, an isometry of a manifold is any (smooth) mapping of that manifold into itself, or into another manifold that preserves the notion of distance between points. The definition of an isometry requires the notion of a metric on the manifold; a manifold with a (positive-definite) metric is a Riemannian manifold, one with an indefinite metric is a pseudo-Riemannian manifold. Thus, isometries are studied in Riemannian geometry.

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 ${\displaystyle (M,g)}$ and ${\displaystyle (M',g')}$ be two (pseudo-)Riemannian manifolds, and let ${\displaystyle f:M\to M'}$ be a diffeomorphism. Then ${\displaystyle f}$ is called an isometry (or isometric isomorphism) if

${\displaystyle g=f^{*}g',\,}$

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

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

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