stylistic:style_transfer, structural:expansion
Isometries

An isometry is a function between Riemannian manifolds which preserves all of the structure of Riemannian manifolds. If two Riemannian manifolds have an isometry between them, they are called isometric, and they are considered to be the same manifold for the purpose of Riemannian geometry.

Specifically, if (M, g) {\displaystyle (M,g)} and (N, h) {\displaystyle (N,h)} are two Riemannian manifolds, a diffeomorphism f: M → N {\displaystyle f:M\to N} is called an isometry if g = f ∗ h {\displaystyle g=f^{\ast }h}, that is, if

for all p ∈ M {\displaystyle p\in M} and u, v ∈ T p M. {\displaystyle u,v\in T_{p}M.} For example, translations and rotations are both isometries from Euclidean space (to be defined soon) to itself.

One says that a smooth map f: M → N, {\displaystyle f:M\to N,} not assumed to be a diffeomorphism, is a local isometry if every p ∈ M {\displaystyle p\in M} has an open neighborhood U {\displaystyle U} such that f: U → f (U) {\displaystyle f:U\to f(U)} is an isometry (and thus a diffeomorphism).