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.

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.

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.

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.