Surfaces with the same marked length spectrum

AbstractVarious authors have shown that isotopy classes of nonpositively curved Riemannian metrics on surfaces are characterized by their marked length spectrum. We show by an example that, although this property makes sense in a non-Riemannian setting, it does not hold for every metric space structure on a surface. More precisely, given two arbitrary negatively curved Riemannian metrics on a compact surface, we construct a metric which has the same geodesics as the first one, has the same marked length spectrum as the second one, and is in general not isometric to either one