Slice theorem (differential geometry)

In differential geometry, the slice theorem states: given a manifold M on which a Lie group G acts as diffeomorphisms, for any x in M, the map $G/G_{x}\to M,\,[g]\mapsto g\cdot x$ 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 diffeomorphism from the neighborhood to its image, which contains the orbit of x.

The theorem is also called the equivariant tubular neighborhood theorem.

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. For more details, see http://mathoverflow.net/questions/54799/on-a-proof-of-the-existence-of-tubular-neighborhoods

References

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