Remark

With the above hypotheses, further let $H$ be a manifold of class $C^r$ and $(x, h) \mapsto m(x, h)$ a mapping of class $C^r$ of $X \times H$ into $X$ such that $m(gx, h) = gm(x, h)$ for $x \in X,\ g \in G,\ h \in H$. Let $n$ be the mapping of $(X/G) \times H$ into $X/G$ derived from $m$ by taking quotients. We show that $n$ is of class $C^r$. Consider the diagram

It is commutative, $\pi \circ m$ is of class $C^r$ and $\pi \times 1$ is a surjective submersion; it then suffices to apply Differentiable and Analytic Manifolds, R, 5.9.5.

Let $G$ be a Lie group, $X$ a manifold of class $C^r$ and $(g, x) \mapsto xg$ a law of right operation of class $C^r$ of $G$ on $X$. Let $\tau(g)x = \rho(x)g = xg$ for $g \in G,\ x \in X$. Then this time

and hence

On the other hand, if $\eta$ denotes the mapping $(g, x) \mapsto xg$ of $G \times X$ into $X$, formula (6) remains valid. Proposition 9, its Corollary and Proposition 10 remain equally true (with "principal left fibre bundle" replaced by "principal right fibre bundle" in the last).