Slice theorem (differential geometry)
In differential geometry, the slice theorem states:[1] given a manifold M on which a Lie group G acts as diffeomorphisms, for any x in M, the map ![G/G_x \to M, \, [g] \mapsto g \cdot x](../I/m/3a7bcd76c90e7ea650815ac5e8f61a41.png)
extends to an invariant neighborhood of 
(viewed as a zero section) in 
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 
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. 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 varieties
References
- ↑ Audin 2004, Theorem I.2.1
External links
- On a proof of the existence of tubular neighborhoods
- Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004