Isometry (Riemannian geometry)

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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 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

References

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

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