Surfaces with the same marked length spectrum (Q685045)

\textit{C. Croke}, \textit{A. Fathi}, and \textit{J. Feldman} [Topology 31, 847- 855 (1992; Zbl 0779.53025)]have proved: Theorem 0: Let \(M\) be a closed surface and let \(g_ 0\) and \(g_ 1\) be two Riemannian metrics on \(M\) where \(g_ 1\) has non positive curvature and where \(g_ 0\) has no conjugate points. If \(g_ 0\) and \(g_ 1\) have the same marked length spectrum, then they are isometric by an isometry isotopic to the identity. The author shows that the hypothesis that \(g_ 0\) is a Riemannian metric is crucial to the proof of Theorem 0 by showing: Theorem 1: Let \(g_ 1\) and \(g_ 2\) be two Riemannian metrics of negative curvature on the closed surface \(M\). There exists a metric \(m_ 0\) on \(M\) which is compatible with the topology of \(M\) which has the same marked length spectrum as \(g_ 1\) and such that the geodesics of \(g_ 0\) are identical to the geodesics of \(g_ 2\). In particular \(m_ 0\) has minimizing geodesics and defines a \(G\)-space structure on \(M\). The author notes his construction was discovered independently by Fathi and that Otal has also observed a similar phenomenon with nonsymmetric Finsler metrics.