Example

Let $K = R$ and $X = R^2$. Let the discrete group $\Gamma = Z^2$ operate on X by translations. Let $m \in R$ and let $(X, Y_m)$ be the foliation defined by the submersion $p : (x_1, x_2) \mapsto x_2 - mx_1$ of X onto R. Put $X' = X/\Gamma$ and $Y'_m = Y_m/\Gamma$; then $Y'_m$ is a foliation of the torus $X'$ (cf. 9.2.4, example d)). If m is rational, there exists a submersion $p'$ of $X'$ onto $R/Z$ such that $Y'_m = X'_{p'}$. If m is irrational, every maximal connected leaf of $(X', Y'_m)$ is dense in $X'$, and there does not exist a submersion $p'$ of $X'$ into a manifold S such that $Y'_m = X'_{p'}$.

9.2.10. Let Y be a foliation of X, and let $\pi : X \to S$ be a morphism of manifolds such that the composite $Y \to X \xrightarrow{\pi} S$ is étale. If S’ is an open subset of S, a section $\sigma : S' \to X$ of $\pi$ over S’ is said to be horizontal (with respect to the foliation Y) if it is a morphism from S’ into Y; it amounts to saying that $\sigma(S')$ is a leaf of (X, Y), or again that $T(\sigma)$ maps $T(S')$ into $T(X, Y)$. For every $s_0 \in S$ and every $x_0 \in \pi^{-1}(s_0)$, there exists a horizontal section defined in a neighbourhood of $s_0$ and taking the value $x_0$ at $s_0$; two such sections coincide on a neighbourhood of $s_0$.

More generally, let $s_0 \in S$, let T be a manifold, let f be a morphism from T into the submanifold $\pi^{-1}(s_0)$ of X, and let $t_0 \in T$. There then exists an open neighbourhood T’ (resp. S’) of $t_0$ in T (resp. of $s_0$ in S), and a morphism

satisfying the following condition: for every $t \in T'$, the mapping $s \mapsto F(s, t)$ is a horizontal section of $\pi$ over S’ taking the value $f(t)$ at the point $s_0$. Two such mappings F coincide on a neighbourhood of $(s_0, t_0)$.