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 $G/G_{x}\to M$ extends to a neighborhood of $G/G_{x}$ 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.

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