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.
Definition
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 tangent bundle ),
:
If is a local diffeomorphism such that , then is called a local isometry.
References
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