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 $(M, g)$ and $(M', g')$ be two (pseudo-)Riemannian manifolds, and let $f : M \to M'$ be a diffeomorphism. Then $f$ is called an isometry (or isometric isomorphism) if

$g = f^{*} g', \,$

where $f^{*} g'$ denotes the pullback of the rank (0, 2) metric tensor $g'$ by $f$. Equivalently, in terms of the push-forward $f_{*}$, we have that for any two vector fields $v, w$ on $M$ (i.e. sections of the tangent bundle $\mathrm{T} M$),

$g(v, w) = g' \left( f_{*} v, f_{*} w \right). \,$

If $f$ is a local diffeomorphism such that $g = f^{*} g'$, then $f$ is called a local isometry.

See also

• Myers–Steenrod theorem

References

• Lee, Jeffrey M. (2000). Differential Geometry, Analysis and Physics.