Let $\Phi: M \times \mathbf{R} \rightarrow M$ be a smooth dynamical system having no periodic orbits, i.e. such that the canonical map $\pi:M \rightarrow \mathbf{R}\backslash M$ is a principal $\mathbf{R}$-bundle. Is $\pi$ always locally trivial? If not, are there any nice (not too contrived/complicated) counterexamples? Which conditions ensure $\pi$ to be locally trivial?
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The answer is negative. Take, for instance, the irrational foliation of the flat 2-torus by geodesics with the obvious ${\mathbb R}$ action via translations along leaves. Note that every fiber bundle is locally trivial (by definition), so this should not have been one of the assumptions, only nonexistence of periodic orbits. A necessary and sufficient condition for the ${\mathbb R}$ action to come from a fibration is that the action is proper, this follows, for instance, from the slice theorem. |
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