Proposition 10

Let G be a Lie group and X a manifold of class $C^r$ with a law of left operation of class $C^r$ of G on X. Suppose that:

(a) the group G operates properly and freely on X;

(b) for all $x \in X$, $\rho(x)$ is an immersion (which is a consequence of (a) if K is of characteristic 0 and X is finite-dimensional).

Then the equivalence relation defined by G on X is regular (Differentiable and Analytic Manifolds, R, 5.9.5). There exists on the quotient set X/G one and only one manifold structure such that the canonical mapping $\pi : X \to X/G$ is a submersion. The underlying topology of this manifold structure is the quotient topology of that on X; it is Hausdorff. Finally, $(X, G, X/G, \pi)$ is a principal left fibre bundle.\footnote{† The principal fibre bundles defined in Differentiable and Analytic Manifolds, R, 6.2.1 are principal right fibre bundles. The definition of principal left fibre bundles can be deduced from this in an obvious way.}

Let $\theta$ be the mapping $(g, x) \mapsto (x, gx)$ of $G \times X$ into $X \times X$. This mapping is of class $C^\tau$. We show that it is an immersion. For $u \in T_g(G)$ and $v \in T_x(X)$, by (6),

But $T_g(\rho(x))$ is injective by hypothesis (b) and hence $T_{(g, x)}(\theta)$ is injective. Its image is the topological direct sum of the subspace $H_{g, x}$ consisting of the vectors $(v, T_x(\tau(g))v)$ for $v \in T_x(X)$ and the subspace

By hypothesis (b), $T_g(\rho(x))(T_g(G))$ admits a topological supplement $J_{g, x}$ in $T_{gx}(X)$. Hence the image of $T_{(g, x)}(\theta)$ admits the topological supplement $\{0\} \times J_{g, x}$. Hence we have proved that $\theta$ is an immersion of $G \times X$ in $X \times X$.

As G operates freely on X, $\theta$ is injective. Let C be the graph of the equivalence relation R defined by G on X. As G operates properly, $\theta$ is a homeomorphism of $G \times X$ onto C (General Topology, Chapter I, § 10, Proposition 2). By Differentiable and Analytic Manifolds, R, 5.8.3, it is a submanifold of $X \times X$ and $\theta$ is an isomorphism of the manifold $G \times X$ onto the manifold C. The tangent space $T_{(x, gx)}(C)$ is identified with

Let pr$_1$ and pr$_2$ be the canonical projections of $X \times X$ onto the two factors. It is immediate that $T_{(x, gx)}(\mathrm{pr}_1)$ maps $H_{g, x}$ onto $T_x(X)$ and that the kernel of $T_{(x, gx)}(\mathrm{pr}_1) | T_{(x, gx)}(C)$ is $I_{g, x}$. Thus $\mathrm{pr}_1 | C$ is a submersion of C onto X. By Differentiable and Analytic Manifolds, R, 5.9.5, R is regular. By definition, there therefore exists on the quotient set X/G one and only one manifold structure such that $\pi$ is a submersion. The underlying topology of X/G is the quotient topology of that of X (Differentiable and Analytic Manifolds, R, 5.9.4). This topology is Hausdorff (General Topology, Chapter III, § 4, Proposition 3).

For all $b \in X/G$, there exist an open neighbourhood W of b and a morphism $\sigma : W \to X$ such that $\pi \circ \sigma = \mathrm{id}_W$ (Differentiable and Analytic Manifolds, R, 5.9.1). Let $\phi$ be the bijection $(g, w) \mapsto g\sigma(w)$ of $G \times W$ onto $\pi^{-1}(W)$. It is of class $C^\tau$. Then $\pi(g\sigma(w)) = w$ and

and hence the inverse bijection of $\phi$ is of class $C^r$. Clearly $\phi(gg', w) = g\phi(g', w)$ for $w \in W,\ g \in G,\ g' \in G$. Hence $(X, G, X/G, \pi)$ is a principal left fibre bundle.