Proposition 16

Let G be a Lie group, X a manifold of class C^r (r ≥ 2) and (g, x) ↦ gx a law of left operation of class C^r of G on X, whence, by transporting the structure, there is a law of left operation of G on TX. Under this law, TX is a vector G-bundle of class C^{r-1}.

Let pr_1 (resp. pr_2) be the canonical projection of G × X onto G (resp. X) and let E_1 (resp. E_2) be the inverse image of TG (resp. TX) relative to pr_1 (resp. pr_2). Then the vector bundle T(G × X) is the direct sum of E_1 and E_2. Let i: E_2 → T(G × X) and q: T(G × X) → E_2 be the canonical vector bundle morphisms defined by this decomposition into a direct sum. Let φ be the mapping (g, x) ↦ (g, gx) of G × X into G × X. Then the mapping denoted by ψ in Definition 4 (where we put E = TX) is just q ∘ T(φ) ∘ i. But T(φ) is a φ-morphism of vector bundles of class C^{r-1} (Differentiable and Analytic Manifolds, R, 8.1.2).