Slice theorem (differential geometry)

Short description: On extending a Lie group action on a manifold to an equivariant diffeomorphism

In differential geometry, the slice theorem states: 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

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.