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.

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

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