On continuous geodesic foliations of hyperbolic manifolds
We prove the following result: Theorem: A compact hyperbolic 3-manifold does not possess a \(\mathbb{C}^ 0\)- geodesic foliation. (Here hyperbolic means of constant curvature \(=- 1\).) The theorem will be proved for foliations of dimension 1, because the codimension 1 case is well known [see for instance: \textit{E. Ghys}, Comment. Math. Helv. 58, 543-572 (1983; Zbl 0534.57015); or the author, Ann. Sci. Éc. Norm. Supér., IV. Sér. 24, No. 2, 171-188 (1991; Zbl 0738.53019)]. The proof is done in 3 steps. We suppose that such a foliation exists. We can assume it oriented by a unit tangent vector field. The first step of the proof is dynamic. We remark that the generated flow is expansive, and then of pseudo-Anosov type. From this, we deduce, in the second step, that the flow is Lipschitz. In the final step, we treat Lipschitz foliations.
scientific article; zbMATH DE number 549645