Proposition 14

Let G be a Lie group, X an analytic manifold and (g, x) ↦ gx a law of analytic left operation of G on X. Let x ∈ X. Suppose that the corresponding orbital mapping ρ(x) is a subimmersion (which is always the case if K is of characteristic 0 and X is finite-dimensional (Corollary to Proposition 9)). Let G_x be the stabilizer of x in G. (i) G_x is a Lie subgroup and T_e(G_x) = Ker T_e(ρ(x)). (ii) The canonical mapping i_x of the homogeneous space G/G_x into X is an immersion with image Gx. (iii) If further the orbit Gx is locally closed and the topology on G admits a countable base, then Gx is a submanifold of X, i_x is an isomorphism of the manifold G/G_x onto the manifold Gx and T_x(Gx) = Im T_e(ρ(x)).

The inverse image of x under ρ(x) is G_x. As ρ(x) is a subimmersion, G_x is a submanifold and, for all g ∈ G, the tangent space J to gG_x = ρ_{(x)}^{-1}(gx) at g is Ker T_g(ρ(x)) (Differentiable and Analytic Manifolds, R, 5.10.5), whence (i). Let π: G → G/G_x be the canonical mapping. Then i_x ∘ π = ρ(x). As G/G_x is a quotient manifold of G, this equality proves that i_x is analytic. Further, the kernels of T_g(ρ(x)) and T_g(π) are both equal to J. Hence T_{π(g)}(i_x) is injective. The image of T_{π(g)}(i_x) is equal to the image of T_g(ρ(x)) and hence admits a topological supplement. This proves (ii).

Suppose that Gx is locally closed. Every point of Gx then has a neighbourhood in Gx which is homeomorphic to a closed subspace of a complete metric space and hence is a Baire space. Hence Gx is a Baire space (General Topology, Chapter IX, § 5, Proposition 4). If G has a countable base, i_x is therefore a homeomorphism of G/G_x onto Gx (General Topology, Chapter IX, § 5). Then by (ii) and Differentiable and Analytic Manifolds, R, 5.8.3, i_x is an isomorphism of the manifolds G/G_x onto the manifold Gx and