# Transitively twisted flows of 3-manifolds

### H. Nakayama

Hiroshima University, Higashi-Hiroshima, Japan

## Abstract

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