A non-singular C1 vector field X of a closed 3-manifold M generating a flow induces a flow of the bundle N X orthogonal to X. This flow further induces a flow of the projectivized bundle of N X. In this paper, we assume that the projectivized bundle is a trivial bundle, and study the lift of to the infinite cyclic covering . We prove that the flow is not minimal, and construct an example of such that has a dense orbit. If is almost periodic and minimal, then is shown to be classified into three cases: (1) All the orbits of are bounded. (2) All the orbits of are proper. (3) is transitive.
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H. Nakayama, Transitively twisted flows of 3-manifolds. Comment. Math. Helv. 76 (2001), no. 4, pp. 577–588DOI 10.1007/S00014-001-8321-Z