Slice theorem (differential geometry)

In differential geometry, the slice theorem states: given a manifold M on which a Lie group G acts as diffomorphisms, for any x in M, the map $G/G_{x}\to M,\,[g]\mapsto g\cdot x$ extends to an invariant neighborhood of $G/G_{x}$ (viewed as a zero section) in $G\times _{G_{x}}T_{x}M/T_{x}(G\cdot x)$ so that it defines an equivariant diffomorphism from the neighborhood to its image, which contains the the orbit of x.

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

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.

References

• Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004