Slice theorem

Remarks

The idea of the proof is to average a Riemannian metric on $M$ over $G$, which is possible because $G$ is compact. The new metric is $G$-invariant, so the exponential map on the normal bundle of the orbit $G x$ is $G$-equivariant. The orbit is a compact submanifold diffeomorphic to $G / H$, and its normal bundle is $G \times_H V$. The tubular neighbourhood theorem gives $\Phi$ from this exponential map on vectors of small length. The image $S = \Phi(\set{[e, v] : v \in V})$ is called a slice at $x$: it is $H$-invariant, and $g S \cap S \neq \emptyset$ implies $g \in H$. Example: let $G = SO(2)$ act on $\R^2$ by rotations and let $x = (1, 0)$. Then $H$ is trivial, $V$ is the radial line $\R$, and $\Phi([g, t]) = g (e^t, 0)$ is an equivariant diffeomorphism from $SO(2) \times \R$ onto $U = \R^2 \setminus \set{0}$. Compactness can be replaced by properness of the action, a result of Palais. Without such a hypothesis the theorem fails. Let $\R$ act on the torus $\R^2 / \Z^2$ by translations along a line of irrational slope. Every orbit is dense, so no orbit has a neighbourhood of the form $G \times_H V$ in which it is closed.

Going deeper

For a student

The theorem reduces local questions about smooth actions of compact groups to linear algebra. It shows that every orbit has a neighbourhood that retracts onto it equivariantly, and that the stabilisers of nearby points are conjugate to subgroups of $H$. From it one derives the principal orbit type theorem. It also shows that the quotient of a manifold by a free action of a compact Lie group is again a manifold. When $G$ is finite and acts faithfully, it shows that $M / G$ is an orbifold.