Transitively twisted flows of 3-manifolds

  • H. Nakayama

    Hiroshima University, Higashi-Hiroshima, Japan


A non-singular C1 vector field X of a closed 3-manifold M generating a flow φt\varphi_t induces a flow of the bundle N X orthogonal to X. This flow further induces a flow PφtP \varphi_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 φt\angle\varphi_t of PφtP \varphi_t to the infinite cyclic covering M×RM \times \mathbb{R}. We prove that the flow φt\angle\varphi_t is not minimal, and construct an example of φt\varphi_t such that φt\angle\varphi_t has a dense orbit. If φt\varphi_t is almost periodic and minimal, then φt\angle\varphi_t is shown to be classified into three cases: (1) All the orbits of φt\angle\varphi_t are bounded. (2) All the orbits of φt\angle\varphi_t are proper. (3) φt\angle\varphi_t is transitive.

Cite this article

H. Nakayama, Transitively twisted flows of 3-manifolds. Comment. Math. Helv. 76 (2001), no. 4, pp. 577–588

DOI 10.1007/S00014-001-8321-Z