Minimal sets and chain recurrent sets of projective flows induced from minimal flows on $3$-manifolds

A nonsingular flow $\varphi_t$ on a $3$-manifold induces a flow on the plane bundle orthogonal to $\varphi_t$ by the derivative. This flow also induces a flow $\psi_t$ on its projectivized bundle $PX$, which is called the projective flow. In this paper, we will investigate this projective flow in order to understand the original flow $\varphi_t$, in particular, under the condition that $\varphi_t$ is minimal and $\psi_t$ has more than one minimal sets: If the projective flow $\psi_t$ has more than two minimal sets, then we will show that $\varphi_t$ is topologically equivalent to an irrational flow on the $3$-torus. In the case when $\psi_t$ has exactly two minimal sets, then we obtain several properties of the minimal sets of $\psi_t$. In particular, we construct two $C^\infty$ sections to $PX$ which separate these minimal sets (and hence $PX$ is a trivial bundle) if $\varphi_t$ is not topologically equivalent to an irrational flow on the $3$-torus. As an application of this characterization, the chain recurrent set of the projective flow is shown to be the whole $PX$.