Corollary

Let X be a locally compact space in which a finite-dimensional real vector space V operates on the right, continuously and properly, by $(x, v) \mapsto xv$. Let $\pi$ be the canonical mapping of X onto $B = X/V$. Assume that B is paracompact.

a) There exists a continuous mapping f of X into V such that $f(xv) = f(x) + v$ for all $x \in X$ and $v \in V$.

b) If f is a mapping satisfying the conditions of a), then the mapping $x \mapsto (\pi(x), f(x))$ is a homeomorphism of X onto $B \times V$.

The assertion a) results from Prop. 2 in which g is taken to be the identity mapping of V. Let f be a mapping satisfying the conditions of a). The mapping $x \mapsto x \cdot (-f(x))$ of X into X is continuous, and is constant on each orbit, hence is of the form $\varphi \circ \pi$, where $\varphi$ is a continuous mapping of B into X; for every $b \in B$, $\pi(\varphi(b)) = b$. The mappings $x \mapsto (\pi(x), f(x))$ of X into $B \times V$ and $(b, v) \mapsto \varphi(b) \cdot v$ of $B \times V$ into X are inverse to each other, because $\varphi(\pi(x)) \cdot f(x) = x \cdot (-f(x)) \cdot (f(x)) = x$, $\pi(\varphi(b) \cdot v) = \pi(\varphi(b)) = b$, and, if $b = \pi(y)$, then

Since these mappings are continuous, they are homeomorphisms.