Corollary 1

Let E_0^{G_0} be the set of elements of E_0 which are invariant under G_0. For all u ∈ E_0^{G_0}, let σ_u be the mapping of X into E defined by σ_u(gx_0) = gu for all g ∈ G. (i) The G-invariant sections† of E are of class C^r. (ii) u ↦ σ_u is a bijection of E_0^{G_0} onto the set of G-invariant sections of E.

Assertion (ii) is obvious. To prove (i) it is sufficient to prove that each section σ_u is of class C^r. Let E' be the trivial G-bundle of base X and fibre E_0^{G_0}. Let f be the canonical injection of E_0^{G_0} into E_0. By Proposition 17 there exists a morphism φ of E' into E compatible with the operations of G and extending f. If u ∈ E_0^{G_0} and g ∈ G, then

and hence σ_u(x) = φ((u, x)) for all x ∈ X, which proves our assertion.

For example, let G be a finite-dimensional real Lie group, G0 a compact Lie subgroup of G and X the homogeneous space G/G_0. Let x_0 denote the canonical image of e in X. There exists a positive definite symmetric bilinear form on T_{x_0}(X) invariant under G_0 (Integration, Chapter VII, § 3, Proposition 1). Applying the above to (TX)* ⊗ (TX)*, we see that there exists on X an analytic Riemannian metric invariant under G_*.

† By a section of E we here mean a mapping σ (not necessarily of class C^r) of X into E such that p ∘ σ = Id_X, where p denotes the projection of E onto X.