- Isometry (Riemannian geometry)
In the study of
Riemannian geometryin mathematics, a local isometry from one (pseudo-) Riemannian manifoldto another is a map which pulls back the metric tensoron 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 and be two Riemannian manifolds, and let be a diffeomorphism. Then is called an isometry (or isometric isomorphism) if
where denotes the pullback of the rank (0, 2) metric tensor by . Equivalently, in terms of the push-forward , we have that for any two vector fields on (i.e. sections of the
If is a
local diffeomorphismsuch that , then is called a local isometry.
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