The marked length-spectrum of a surface of nonpositive curvature. (English) Zbl 0779.53025

Croke, C.; Fathi, A.; Feldman, J.

Topology 31, No. 4, 847-855 (1992).

If \(M\) is a manifold and \(g_ 1\), \(g_ 2\) are Riemannian metrics, then \(g_ 1\) and \(g_ 2\) are said to have the same marked length-spectrum if in each homotopy class of closed curves in \(M\) the infimum of \(g_ 1\)- lengths of curves and the infimum of \(g_ 2\)-lengths are the same. The marked length spectrum problem in general is to show that two metrics with the same marked length spectrum are isometric. One of the results in this paper states that if \(M\) is a closed surface and if \(g_ 1\) and \(g_ 2\) are Riemannian metrics on \(M\) with \(g_ 1\) of nonpositive curvature and \(g_ 2\) without conjugate points then \(g_ 1\) and \(g_ 2\) are isometric by an isometry homotopic to the identity when they have the same marked length-spectrum. The authors also show that it is possible to obtain a similar result when the Morse correspondence preserves angles.

Reviewer: A.P.Stone (Albuquerque)

MSC:

53C20

Global Riemannian geometry, including pinching

Keywords:

nonpositive curvature; conjugate points; isometry; Morse correspondence