Bounded distance geodesic foliations in Riemannian planes

TL;DR

This paper proves a conjecture by Burns and Knieper that a 2-plane with a metric without conjugate points and a bounded geodesic foliation is flat, under certain conditions, and characterizes all such foliations.

Contribution

It confirms the conjecture in specific cases and shows all geodesic line foliations on a Riemannian 2-plane are topologically standard.

Findings

The conjecture holds if the plane admits total curvature.

The conjecture holds under the visibility condition at some point.

All geodesic line foliations are homeomorphic to the standard foliation.

Abstract

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.