---- > [!definition] Definition. ([[Riemannian isometry]]) > A [[smooth maps between manifolds|smooth map]] $F:(M,g) \to (N,h)$ between [[Riemannian manifold|Riemannian manifolds]] is called a **(Riemannian) isometry** if $F$ is a [[diffeomorphism]] [[pullback of a Riemannian metric|for which]] $F^{*}h=g$. > The isometries of $(M,g)$ form a [[subgroup|sub]][[group]] of $\operatorname{Diff}(M)$ which we denote $\operatorname{Isom}(M,g)$. > We say the [[diffeomorphism]] $F$ is a **local isometry at $p \in M$** if there exist [[neighborhood|neighborhoods]] $U \ni p$, $V \ni F(p)$ such that $F :U \to V$ is an isometry. We call $F$ a **local isometry** if it is a local isometry at $p$ for all $p \in M$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```