A metric on a sphere that is geodesically equivalent to itself a metric of constant curvature is a metric of constant curvature (Q1406431)

Consider a surface \(P^2\) with two smooth metrics \(G\) and \(\widehat G\). The metrics \(G\) and \(\widehat G\) are called geodesically equivalent if geodesic lines of the metric \(G\) coincide with geodesic lines of the metric \(\widehat G\) as sets of points. The main result of this paper is the proof of the following: Theorem. Any metric on a sphere that is geodesically equivalent to a metric of constant curvature is itself a metric of constant curvature.