On 3-manifolds there are vector fields which vanish nowhere.
Given such a field one can ask whether all of its trajectories can be
given by geodesics. In many cases the answer is positive, yet it turns
out that if we assume that a manifold has constant negative curvature,
existence of such a vector field becomes impossible. This result was
first shown in topological setting in 1993 by A. Zeghib using theorems
about Anosov foliations. I will aim to show a simpler argument in a
smooth setting using classical tools of differential geometry and
explicit calculations of its dynamics.