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In this paper we study several dynamical properties of the riemannian $1$-dimensional foliation $\mathcal{L}$ on an oriented closed 3-manifold $M$. Carri\`ere \cite{Ca} classified such pairs $(M,\mathcal{L})$. Using the classification we prove the nonhyperbolicity of $(M,\mathcal{L})$. Also we describe in detail recurrence points, $\omega$-limit sets and attractors.

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