Transitively twisted flows of 3-manifolds

Abstract

A non-singular C1 vector field X of a closed 3-manifold M generating a flow ${\phi }_t$ induces a flow of the bundle N X orthogonal to X. This flow further induces a flow $P{\phi }_t$ of the projectivized bundle of N X. In this paper, we assume that the projectivized bundle is a trivial bundle, and study the lift $\angle {\phi }_t$ of $P{\phi }_t$ to the infinite cyclic covering $M\times \mathbb{R}$. We prove that the flow $\angle {\phi }_t$ is not minimal, and construct an example of ${\phi }_t$ such that $\angle {\phi }_t$ has a dense orbit. If ${\phi }_t$ is almost periodic and minimal, then $\angle {\phi }_t$ is shown to be classified into three cases: (1) All the orbits of $\angle {\phi }_t$ are bounded. (2) All the orbits of $\angle {\phi }_t$ are proper. (3) $\angle {\phi }_t$ is transitive.