Isometry (Riemannian geometry)

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.


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

:g = f^{*} g',

where f^{*} g' denotes the pullback 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 mathrm{T} M),

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

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



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