Corollary

Tag 0A7P, permanent. Cite it as [Bourbaki, Tag 0A7P].

Lie Groups and Lie Algebras, chapter III, § 1. Lie groups

Let $x \in X$ . If K is of characteristic 0 and X is finite-dimensional, $\rho(x)$ is a subimmersion. If further $\rho(x)$ is injective, $\rho(x)$ is an immersion.

This follows from Proposition 9 and Differentiable and Analytic Manifolds, R, 5.10.6.

Observe that, if $\eta$ denotes the mapping $(g, x) \mapsto gx$ of $G \times X$ into X, then, for $g \in G, x \in X, u \in T_g(G), v \in T_x(X)$ ,

T_{(g, x)}(\eta)(u, v) = T_{(g, x)}(\eta)(u, 0) + T_{(g, x)}(\eta)(0, x)

that is

T_{(g, x)}(\eta)(u, v) = T_g(\rho(x))u + T_x(\tau(g))v.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Proposition 14, § 1

Cites

Proposition 9