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.
Read more about this topic: Isometry (Riemannian Geometry)
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