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 and be two (pseudo-)Riemannian manifolds, and let be a diffeomorphism. Then is called an isometry (or isometric isomorphism) if
If is a local diffeomorphism such that , then is called a local isometry.
- Lee, Jeffrey M. (2000). Differential Geometry, Analysis and Physics.