The marked length-spectrum of a surface of nonpositive curvature

Abstract

We generalize work of J.P. Otal and C. Croke on the marked length spectrum of surfaces to the case where one of the metrics is of nonpositive curvature and the other one has no conjugate points. If M is a manifold and g1, g2 are two Riemannian metrics, we say that they have the same marked length spectrum if in each homotopy class of closed curves in M the infimum of g1-lengths of curves and the infimum of g2-lengths of curves are the same. The marked length spectrum problem in general is to show that two metrics with the same marked length spectrum are isometric. Of course, this cannot hold for arbitrary metrics (for example if M is simply connected). This problem was stated as a conjecture in [BK] in the case where M is a closed surface and g1 and g2 are of negative curvature. This conjecture was solved by J.P. Otal [To] and independently by C. Croke [Cr]–see also [Fa]. Previous work on the problem was done by Guillemin and Kazhdan [GK]. In this work, using Otal’s approach, we improve some of these results by proving the following theorem: Theorem A. Let M be a closed surface and let g1, g2 be Riemannian metrics on M , with g1 of nonpositive curvature and g2 without conjugate points. If g1 and g2 have the same marked length-spectrum then they are isometric by an isometry homotopic to the identity. We will also prove the following fact, which reduces the length spectrum and curvature condition to the assumption that the Morse correspondence preserves angles—see §1 for the definition of the Morse correspondence. Theorem B. Let M be a closed surface of genus ≥ 2, and let g1, g2 be Riemannian metrics without conjugate points on M . If g1 and g2 have the same marked lengthspectrum and the Morse correspondence preserves angles then they are isometric by an isometry homotopic to the identity.