Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Isometry
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
== Manifold == An isometry of a [[manifold]] is any (smooth) mapping of that manifold into itself, or into another manifold that preserves the notion of distance between points. The definition of an isometry requires the notion of a [[metric (mathematics)|metric]] on the manifold; a manifold with a (positive-definite) metric is a [[Riemannian manifold]], one with an indefinite metric is a [[pseudo-Riemannian manifold]]. Thus, isometries are studied in [[Riemannian geometry]]. A '''local isometry''' from one ([[Pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]] to another is a map which [[pullback (differential geometry)|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 theory|category]] '''Rm''' of Riemannian manifolds. === Definition === Let <math>R = (M, g) </math> and <math>R' = (M', g') </math> be two (pseudo-)Riemannian manifolds, and let <math>f : R \to R' </math> be a diffeomorphism. Then <math>f </math> is called an '''isometry''' (or '''isometric isomorphism''') if :<math>g = f^{*} g', </math> where <math>f^{*} g' </math> denotes the [[pullback (differential geometry)|pullback]] of the rank (0, 2) metric tensor <math>g' </math> by <math>f</math>. Equivalently, in terms of the [[pushforward (differential)|pushforward]] <math>f_{*},</math> we have that for any two vector fields <math>v, w </math> on <math>M </math> (i.e. sections of the [[tangent bundle]] <math>\mathrm{T} M </math>), :<math>g(v, w) = g' \left( f_{*} v, f_{*} w \right).</math> If <math>f </math> is a [[local diffeomorphism]] such that <math>g = f^{*} g',</math> then <math>f</math> is called a '''local isometry'''. ===Properties=== A collection of isometries typically form a group, the [[isometry group]]. When the group is a [[continuous group]], the [[Lie group|infinitesimal generators]] of the group are the [[Killing vector field]]s. The [[Myers–Steenrod theorem]] states that every isometry between two connected Riemannian manifolds is smooth (differentiable). A second form of this theorem states that the isometry group of a Riemannian manifold is a [[Lie group]]. [[Symmetric space]]s are important examples of [[Riemannian manifold]]s that have isometries defined at every point.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)