Slice theorem (differential geometry)
In differential geometry, the slice theorem states:[1] given a manifold [math]\displaystyle{ M }[/math] on which a Lie group [math]\displaystyle{ G }[/math] acts as diffeomorphisms, for any [math]\displaystyle{ x }[/math] in [math]\displaystyle{ M }[/math], the map [math]\displaystyle{ G/G_x \to M, \, [g] \mapsto g \cdot x }[/math] extends to an invariant neighborhood of [math]\displaystyle{ G/G_x }[/math] (viewed as a zero section) in [math]\displaystyle{ 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 [math]\displaystyle{ x }[/math].
The important application of the theorem is a proof of the fact that the quotient [math]\displaystyle{ M/G }[/math] admits a manifold structure when [math]\displaystyle{ G }[/math] is compact and the action is free.
In algebraic geometry, there is an analog of the slice theorem; it is called Luna's slice theorem.
Idea of proof when G is compact
Since [math]\displaystyle{ G }[/math] is compact, there exists an invariant metric; i.e., [math]\displaystyle{ G }[/math] acts as isometries. One then adapts 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 varieties
References
- ↑ Audin 2004, Theorem I.2.1
External links
- On a proof of the existence of tubular neighborhoods
- Audin, Michèle (2004) (in de). Torus Actions on Symplectic Manifolds. Birkhauser. doi:10.1007/978-3-0348-7960-6. ISBN 978-3-0348-7960-6. OCLC 863697782.
Original source: https://en.wikipedia.org/wiki/Slice theorem (differential geometry).
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