Proposition 8

Let G_1 and G_2 be Lie groups, X_1 and X_2 manifolds of class C^r and f_i a law of left operation of class C^r of G_i on X_i (i = 1, 2). Let \phi be a morphism of G_1 into G_2 and \psi a \phi-morphism of X_1 into X_2. Then T(\psi) is a T(\phi)-morphism of T(X_1) into T(X_2).

f_2 \circ (\phi \times \psi) = \psi \circ f_1, whence

Let G be a Lie group, X a manifold of class C^r and (g, x) \mapsto gx a law of left operation of class C^r of G on X. Let I be an open subset of K containing 0 and \gamma: I \to G a mapping of class C^r such that \gamma(0) = e. Let

Let x \in X. Using (4), ax is the image under the tangent mapping to \lambda \mapsto \gamma(\lambda)x of the tangent vector 1 to I at 0. Hence the vector field x \mapsto ax on X is the vector field defined by the mapping (\lambda, x) \mapsto \gamma(\lambda)x in the sense of Differentiable and Analytic Manifolds, R, 8.4.5.