Theorem 2

Let G be a Lie group operating properly on X, such that the law of operation $(g, x) \rightarrow gx$ is of class $C^r$. Assume that X is paracompact.

a) The map which associates to any point of X its orbit type has the following semi-continuity property: let $x\in X$ and let $t\in \mathscr{T}$ be its orbit type; there exists a stable open neighbourhood U of $x$ such that, for any $u\in U$, the type of $u$ is $\geq t$.

b) For all $t\in \mathscr{T},X_{(t)}$ is a submanifold of X, the equivalence relation on $X_{(t)}$ induced by the operation of G is regular (Differentiable and Analytic Manifolds, Results, 5.9.5), and the morphism $X_{(t)}\rightarrow X_{(t)}/G$ is a bundle.

c) Assume that $X/G$ is connected. Then the set of orbit types of elements of X has a largest element $\tau$; moreover, $X_{(\tau)}$ is a dense open subset of X and $X_{(\tau)}/G$ is connected.

Let $x$ be a point of X and $t\in \mathscr{T}$ its type. To prove a) and b), we can replace X by a stable open set containing $x$, and hence (Prop. 6) can assume that X is of the form $G\times^HW$, where W is the space of a finite dimensional analytic linear representation of a compact subgroup H of G, the point $x$ being the image $p(e,0)$ of $(e,0)\in G\times W$ under the canonical projection $p: G\times W\rightarrow G\times^HW$. If $u=p(g, y)\in G\times^HW$ and $a\in G$, then $au=u$ if and only if there exists $h\in H$ with $(ag, y) = (gh^{-1}, hy)$, that is, if $a\in gH_yg^{-1}$. Thus, $G_u=gH_yg^{-1}$; in particular, $G_x= H$, so $G_u$ is conjugate to a subgroup of $G_x$, which proves that the type of $u$ is $\geq t$, hence a).

Moreover, $u$ is of type $t$ if and only if $G_u$ is conjugate to H in G, or equivalently that $H_y$ is conjugate to H in G; by Lemma $6b)$, this means that $H_y= H$, and hence that $y$ is fixed by H. If $W'$ is the vector subspace of W consisting of the elements fixed by H, it follows that $X_{(t)}$ can be identified with $G\times^HW'$, and hence also with $G/H\times W'$, hence b).

To prove c), observe that the assumption that $X/G$ is connected implies that X is pure of finite dimension: indeed, for all $k\geq 0$, denote by $X_k$ the set of points $x\in X$ such that dim$_xX =k$; then $X_k$ is open and closed in X, and stable under G, so X is equal to one of the $X_k$.

We now prove c) by induction on the dimension of X, the assertion being clear for dim X = 0. Let $\tau$ be a maximal element among the orbit types of the points of X (such an element exists by Lemma $6c$)$)$. We shall prove the following:

$c')$ For every subset A of $X_{(t)}$, open and closed in $X_{(\tau)}$ and stable under G, the closure $\overline{A}$ of A in X is open.

This assertion implies c). Indeed, note first that $X_{(\tau)}$ is open in X, by a); assertion $c')$ implies that $\overline{X}_{(\tau)}$ is open and closed in X, hence equal to X since it is stable under G and $X/G$ is connected. Let A be a non-empty open and closed subset of $X_{(\tau)}$ stable under G; by $c'),\overline{A}$ is open and closed in X and stable under G, hence equal to X; this implies that A is dense in $X_{(\tau)}$, hence equal to $X_{(\tau)}$. Consequently, every non-empty open and closed subset of $X_{(\tau)}/G$ is equal to $X_{(\tau)}/G$, which proves that $X_{(\tau)}/G$ is connected. Finally, since $X_{(\tau)}$ is dense in X, it follows from a) that every point of X is of type $\leq \tau$; in other words, $\tau$ is the largest element among the orbit types of the points of X.

We now prove $c')$. We can assume that A is non-empty; let $x\in A$. It suffices to prove that A is a neighbourhood of $x$. For this we can, as above, assume that $X = G\times^HW$ with H compact, $x$ being the canonical image of $(e,0)$. Assume first that H operates trivially on W: then X can be identified with $(G/H)\times W$, and $X_{(\tau)}/G = X/G$ is homeomorphic to W, hence connected; thus, $A/G = X/G$, so A = X. Assume from now on that H does not operate trivially on W. Choose a scalar product on W invariant under the compact group H; let S be the unit sphere in W (the set of vectors of norm 1). Note that $S/H$ is connected: indeed, if dim(W) $\geq 2, S$ is connected, and if dim(W) = 1, S is a space of two points on which H operates non-trivially. Put $Y = G\times^HS$; this is a closed submanifold of X, stable under G, of codimension 1, and $Y/G$, which is homeomorphic to $S/H$, is connected. Thus, by the induction hypothesis, there exists a maximal orbit type $\theta$ for Y, the set $Y_{(\theta)}$ is open and dense in Y, and $Y_{(\theta)}/G$ is connected.

Consider the operation of $\mathbf{R}^*_+$ on X induced by passage to the quotient by the law of operation $(\lambda ,(g, w)) \rightarrow (g, \lambda w)$ of $\mathbf{R}^*_+$ on $G\times W$. Two points of X conjugate under this operation are of the same orbit type; consequently, $X_{(\theta)}$ contains $\mathbf{R}^*_+Y_{(\theta)}$, which is a dense open subset of X. But $X_{(\tau)}$ is open by a), and hence meets $X_{(\theta)}$, so $\theta =\tau$.

On the other hand, the homeomorphism $(\lambda , w) \rightarrow \lambda w$ from $\mathbf{R}^*_+\times S$ to W -- $\{0\}($General Topology, Chap. VI, §2, no. 3, Prop. 3) induces a homeomorphism from $\mathbf{R}^*_+\times (S/H)$ to $(\mathbf{R}^*_+S)/H$, hence also from $\mathbf{R}^*_+\times (Y/G)$ to $(\mathbf{R}^*_+Y)/G$, and from $\mathbf{R}^*_+\times (Y_{(\theta)}/G)$ to $(\mathbf{R}^*_+Y_{(\theta)})/G$. Thus, $(\mathbf{R}^*_+Y_{(\theta)})/G$ is connected, and $X_{(\tau)}/G$, which contains a connected dense subset, is itself connected (General Topology, Chap. I, §11, no. 1, Prop. 1). Consequently, A is equal to $X_{(\tau)}$, hence is dense in X, and A is a neighbourhood of $x$. This completes the proof of the theorem.

With the notations in Th. $2c)$, the points of $X_{(\tau)}$ are said to be principal and their orbits are called principal orbits. If $x$ is a point of X, and if $G\times^{G_x}W$ is a linear tube in X around the orbit of $x$, the point $x$ is principal if and only if $G_x$ operates trivially on W, that is, if the tube is of the form $(G/G_x)\times W$.