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2013-09-20T09:10:09Z

Removing "ObjectiveStressRates.png", it has been deleted from Commons by Fastily because: No license since 11 September 2013: you may re-upload, but please include a license tag.

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{{Underlinked|date=July 2013}}

In [[differential geometry]], the '''slice theorem''' states:<ref>{{harvnb|Audin|2004|loc=Theorem I.2.1}}</ref> given a manifold ''M'' on which a Lie group ''G'' [[Lie group action|acts]] as [[diffeomorphism]]s, for any ''x'' in ''M'', the map <math>G/G_x \to M, \, [g] \mapsto g \cdot x</math> extends to an invariant neighborhood of <math>G/G_x</math> (viewed as a zero section) in <math>G \times_{G_x} T_x M / T_x(G \cdot x)</math> so that it defines an equivariant diffeomorphism from the neighborhood to its image, which contains the orbit of ''x''.

The important application of the theorem is a proof of the fact that the quotient <math>M/G</math> admits a manifold structure when ''G'' is compact and the action is free.

In [[algebraic geometry]], there is an analog of the slice theorem; it is called the [[Luna's slice theorem]].

== Idea of proof when ''G'' is compact ==
Since ''G'' is compact, there exists an invariant metric; i.e., ''G'' acts as [[Isometries|isometries]]. One then adopts the usual proof of the existence of a tubular neighborhood using this metric.

==See also==
*[[Luna's slice theorem]], an analogous result for [[reductive algebraic group]] actions on [[algebraic variety|algebraic varieties]]

== References ==
{{reflist}}

==External links==
* [http://mathoverflow.net/questions/54799/on-a-proof-of-the-existence-of-tubular-neighborhoods On a proof of the existence of tubular neighborhoods]
*[[Michele Audin]], Torus actions on symplectic manifolds, Birkhauser, 2004

[[Category:Differential geometry]]
[[Category:Mathematical theorems]]

{{geometry-stub}}

Objective stress rates - Revision history

en>CommonsDelinker: Removing "ObjectiveStressRates.png", it has been deleted from Commons by Fastily because: No license since 11 September 2013: you may re-upload, but please include a license tag.