Proposition 17

Let G be a Lie group, X a left Lie homogeneous space of G, x0 a point of X, G_0 the stabilizer of x_0 in G, E and E' left vector G-bundles of class C^r and base space X, E_0 (resp. E'_0) the fibre of E (resp. E') at x_0 and f an element of $\mathscr{L}(E_0, E'_0)$ such that f(gu) = gf(u) for all u ∈ E_0 and g ∈ G_0. Then there exists one and only one morphism of E into E' compatible with the operations of G and extending f.*

The uniqueness of this morphism is obvious. We prove its existence. Let g, g' elements of G and u ∈ E_0 be such that gu = g'u. Then g'^{-1}g ∈ G_0 and g'^{-1}gu = u and hence g'^{-1}gf(u) = f(u), that is gf(u) = g'f(u). Hence a mapping φ is defined of E into E' by writing φ(gu) = gf(u). Clearly this mapping extends f and it is compatible with the operations of G. We show that φ is a vector bundle morphism of class C^r. Let x_1 ∈ X. There exists an open neighbourhood V of x_1 in X and a submanifold W of G such that the mapping g ↦ gx_0 is an isomorphism θ of class C^r of W onto V. By shrinking V and W it can be assumed that: (1) E | V (resp. E' | V) is identified with a trivial vector bundle of fibre M (resp. M'); (2) if ψ_g (resp. ψ'_g) denotes the mapping u ↦ gu of E_0 (resp. E'_0) into E_{gx_0} (resp. E'_{gx_0}), then the mappings g ↦ ψ_g and g ↦ ψ_g^{-1} (resp. g ↦ ψ'_g and g ↦ ψ'_g^{-1}) of W into $\mathscr{L}(E_0, M)$ and $\mathscr{L}(M, E_0)$ (resp. $\mathscr{L}(E'_0, M')$ and $\mathscr{L}(M', E'_0)$) are of class C^r.

For x ∈ V, let φ_x : M → N be the restriction of φ to E_x = M. Then φ_x is obtained by composing the following mappings: (1) the mapping (ψ_θ^{-1,x})^{-1} of M into E_0; (2) the mapping f of E_0 into E_0; (3) the mapping ψ'_θ^{-1,x} of E_0 into M'. Hence we see that the mapping x ↦ φ_x of V into $\mathscr{L}(M, M')$ is of class C^r.