Isometry (Riemannian geometry)

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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.

[edit] Definition

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

g = f * g',

where $f * g'$ denotes the pull-back of the rank (0, 2) metric tensor $g'$ 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 $T M$ ),