Proposition 9

Let $x \in X$ and $g_0 \in G$.

(i) If $\rho(x)$ is an immersion (resp. a submersion, a subimmersion) at $g_0$, then, for all $g \in G$, $\rho(gx)$ is an immersion (resp. a submersion, a subimmersion).

(ii) If $\rho(x)$ is of rank k at $g_0$, then for all $g \in G$, $\rho(gx)$ is of constant rank equal to k.

This follows immediately from formulae (4) and (5) since $T_g(\delta(g)), T_x(\tau(g))$ and $T_g(\gamma(g^{-1}))$ are isomorphisms.