Dynamics of Riemannian $1$-foliations on $3$-manifolds

Abstract

In this paper we study several dynamical properties of a Riemannian $1$-dimensional foliation $\mathcal{L}$ on an oriented closed $3$-manifold $M$. Carrière [6] classified such pairs $(M,\mathcal{L})$. Using the classification we describe in detail recurrence points, $\omega$-limit sets and attractors. Finally, using the fact that the Poincaré map on a transversal surface for a Riemannian $1$-dimensional foliation is an isometry, we show the nonhyperbolicity of $(M,\mathcal{L})$.