  A conjecture of Burns and Knieper asks whether a 2-plane with a metric
without conjugate points, and with a geodesic foliation whose lines are at
bounded Hausdorff distance, is necessarily flat. We prove this conjecture in
two cases: under the hypothesis that the plane admits total curvature, and
under the hypothesis of visibility at some point. Along the way, we show that
all geodesic line foliations on a Riemannian 2-plane must be homeomorphic to
the standard one.
