THE MARKED LENGTH-SPECTRUM OF A

SURFACE OF NONPOSITIVE CURVATURE^{1}

by C. Croke, A. Fathi & J. Feldman

University of Pennsylvania University of Florida & CNRS University of California

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 g_{1}, g_{2} 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 g_{1}-lengths of curves and the infimum of g_{2}-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 g_{1} and g_{2} 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 g_{1}, g_{2} be Riemannian metrics on M, with g_{1} of nonpositive curvature and g_{2} without conjugate points. If g_{1} and g_{2} 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 g_{1}, g_{2} be Riemannian metrics without conjugate points on M. If g_{1} and g_{2} have the same marked lengthspectrum and the Morse correspondence preserves angles then they are isometric by an isometry homotopic to the identity.

Typeset by AMS-TEX

Finally, we obtain a third result of a more dynamical nature. This is a generalization of a question raised in OF, 6.3 page 70, see also [Cr] and CF where this question is solved.

Theorem C. Let (M_{1}, g_{1}) and (M_{2}, g_{2}) be Riemannian closed surfaces of genus ≥ 2 without conjugate points. If one of the two metrics has nonpositive curvature, then any time preserving semi-conjugacy from the geodesic flow of (M_{1}, g_{1}) to the geodesic flow of (M_{2}, g_{2}) comes from a Riemannian submersion composed with a shift by some fixed time.

We introduce here some definitions and notation. If g is a Riemannian metric on the surface M we will denote by κ_{g}(m) the curvature of g at a point m ∈ M. The lift of g to the universal cover M̃ of M will be denoted by g̃. A g̃-strip in M̃ is a closed subset of M̃ homeomorphic to R × [0, 1] whose boundary consists of two g̃-geodesics which remain at bounded distance from each other. Any two disjoint g̃-geodesics which remain at bounded distance from one another and are closed as subsets of M̃ bound a g̃-strip. A g̃-strip is flat if the curvature of g̃ is zero on the strip. If two g̃-geodesics G and G′ intersect at a unique point, we will denote by ∠g̃(G, G′) ∈]0, π[ the angle at the point of intersection.

1. Background.

We fix a reference Riemannian metric g_{0} of (strictly) negative curvature on M. The following theorem is due to Morse [Mr].

Theorem 1.1. (Morse). Let g be a Riemannian metric on M. Let g̃ and g̃_{0} be the lifts of g and of the Riemannian metric g_{0} of (strictly) negative curvature to the universal cover M̃ of M. Then there exists a constant K > 0, which depends only on g and g_{0}, such that any g̃_{0}-geodesic contains in its K-neighborhood a minimizing g̃-geodesic, and any minimizing g̃-geodesic contains a unique g̃_{0}-geodesic in its K-neighborhood. The map S̃ : M̃ → G̃_{0} from the space M̃ of g̃-minimizing geodesics to the space G̃_{0} of g̃_{0}-geodesics which sends a minimizing g̃-geodesic to the g̃_{0}-geodesic in its K-neighborhood is continuous and proper. (When g has no conjugate points then M̃ is of course the space G̃ of all g̃-geodesics.) Moreover, the map S̃ is π_{1}(M) equivariant.

We will call the map S̃ a Morse map.

Corollary 1.2. For every Riemannian metric g without conjugate points on M, there is a constant C such that any g̃ strip in M̃ has width ≤ C.

Proof. The Morse map sends all g̃-geodesics entirely contained in a given strip to the same g̃_{0}-geodesic.

It will be convenient to introduce the following concept. If g_{1} and g_{2} are metrics on M, we say that the g̃_{1}-geodesic G_{1} and the g̃_{2}-geodesic G_{2}, both in M̃,are Morse correspondent and we will write G_{1} ∼ G_{2}, if they remain at bounded distance, i.e. if we have:

sup_{m∈G_{1}} d(m, G_{2}) < ∞ and sup_{m∈G_{2}} d(m, G_{1}) < ∞,

where d is a metric on M̃ coming from a Riemannian metric on M. Since M is compact, the condition does not depend on the choice of d. It is not assumed that the Riemannian metrics g_{1} and g_{2} are distinct. As before g_{0} will be a fixed reference Riemannian metric of (strictly) negative curvature on M. Suppose g_{1} and g_{2} are Riemannian metrics without conjugate points on M. Call S̃_{1} : G̃_{1} → G̃_{0} (resp. S̃_{2} : G̃_{2} → G̃_{0}) the Morse map from the set of g̃_{1}-geodesics (resp. g̃_{2}-geodesics) onto the set of g̃_{0}-geodesics. If G_{1} is a g̃_{1}-geodesic and G_{2} is a g̃_{2}-geodesic, then G_{1} ∼ G_{2} if and only if S̃_{1}(G_{1}) = S̃_{2}(G_{2}). The following two statements are easy consequences of the work of Leon Green [Gr, corollary 3.2 page 536 and theorem 4.1 page 559]:

Proposition 1.3. Let g be a Riemmanian metric without conjugate points on M. Let G and G′ be two g̃-geodesics (in M̃). If G ∼ G′ then either G = G′ or G and G′ do not intersect; moreover, in the latter case they bound a g̃-strip, and through each point y of this g̃-strip there passes a unique g̃-geodesic G_{y} such that G ∼ G_{y}. If g is of nonpositive curvature then all g̃-strips are flat.

Corollary 1.4. Let g_{1} and g_{2} be Riemannian metrics without conjugate points on M. Let G_{1} and G′_{1} (resp. G_{2} and G′_{2}) be two g̃_{1}-geodesics (resp. g̃_{2}-geodesics) in M̃. If G_{1} ∼ G_{2} and G′_{1} ∼ G′_{2} then G_{1} and G′_{1} intersect transversally if and only if G_{2} and G′_{2} intersect transversally.

We will now show how to adapt Otal’s arguments [To] to prove the following lemma:

Lemma 1.5. Let g_{1} and g_{2} be two Riemannian metrics without conjugate points on M. Suppose that g_{2} is of nonpositive curvature and that g_{1} and g_{2} have the same length spectrum. Then:

For every pair (G_{1}, G′_{1}) of g̃_{1}-geodesics, and every pair (G_{2}, G′_{2}) of g̃_{2}-geodesics,

if G_{1} ∼ G_{2} and G′_{1} ∼ G′_{2}, then ∠g̃_{1}(G_{1}, G′_{1}) = ∠g̃_{2}(G_{2}, G′_{2}).

Consequently the metric g_{1} also has nonpositive curvature.

Sketch of proof. We will use the setting of [Fa] to show how to adapt the arguments of Otal. Let g_{0} be a metric of (strictly) negative curvature on M. Let S̃_{1} : G̃_{1} → G̃ (resp. S̃_{2} : G̃_{1} → G̃) be the Morse map from the space of g̃_{1}-geodesics (resp. g̃_{2}-geodesics) onto the space of g̃_{0}-geodesics, as described above. As in [Fa], using S̃_{1} and S̃_{2}, we obtain, from the Liouville measures, geodesic currents λ̄_{g_{1}} and λ̄_{g_{2}}. By[To, théorème 2], we obtain λ̄_{g_{1}} = λ̄_{g_{2}}. We define for each pair (G, G′) of transversally intersecting g̃_{0}-geodesics the angle θ′(G, G′) as the g̃_{2} angle of any pair (G_{2}, G′_{2}) of g̃_{2}-geodesics such that S̃_{2}(G_{2}) = G and S̃_{2}(G′_{2}) = G′. The fact that this angle is independent of the choices follows from the flatness of the g̃_{2}-strips—see lemma 1.3. It is not very difficult now to adapt Otal’s arguments [To] as in [Fa], to prove the angle condition of the lemma.

This angle condition, taken with the fact that g_{2} has nonpositive curvature, implies that the sum of the angles of any triangle whose sides are g̃_{1}-geodesics is ≤ π. It follows from the Gauss-Bonnet theorem that g_{1} also has nonpositive curvature.

2. Proof of theorem A.

Because, the sphere is simply connected the genus of M has to be ≥ 1. A theorem of Hopf says that a metric without conjugate points on a torus or a Klein bottle is flat. Theorem A follows for the torus a nd the Klein bottle—see [Cr, pages 167–168]. So we assume for the rest of the section that the genus of M is ≥ 2. Since we want to use the work done in this section to prove other theorems, we will use general arguments as often as possible.

Definition 2.1. Suppose g_{1} and g_{2} are Riemannian metrics without conjugate points on M. We define a partial relation R on M̃ in the following way mRm′, if every g̃_{2}-geodesic through m′ is at bounded distance from some g̃_{1}-geodesic through m.

Lemma 2.2. Suppose g_{1} and g_{2} are Riemannian metrics without conjugate points on M. If mRm′, then every g̃_{1}-geodesic through m is at bounded distance from some g̃_{2}-geodesic through m′.

Proof. Let g_{0} be a metric of (strictly) negative curvature on M. As before, let S̃_{1} : G̃_{1} → G̃ (resp. S̃_{2} : G̃_{2} → G̃) be the Morse map from the space of g̃_{1}-geodesics (resp. g̃_{2}-geodesics) onto the space of g̃_{0}-geodesics. We have mRm′ if and only if S̃_{2}{G ∈ G̃_{2} | m′ ∈ G} ⊂ S̃_{1}{G ∈ G̃_{1} | m ∈ G}. But by proposition 1.3, the map S̃_{1} (resp. S̃_{2}) is injective on {G ∈ G̃_{1} | m ∈ G} (resp. {G ∈ G̃_{2} | m′ ∈ G}). Hence we have a natural 1-1 continuous map from the circle of g̃_{2}-geodesics through m′ to the circle of g̃_{1}-geodesics through m. Such a map must be a homeomorphism, so we are done.

Lemma 2.3. Suppose g_{1} and g_{2} are Riemannian metrics without conjugate points on M, and every g̃_{1}-strip is flat. If m_{1}Rm′ and m_{2}Rm′, then m_{1} = m_{2}. If every g̃_{2}-strip is also flat, then R is the graph of a bijection φ between the domain D̃_{1} of R and its range D̃_{2}. Both D̃_{1} and D̃_{2} are invariant under the action of π_{1}(M), and moreover, the map φ is π_{1}(M) equivariant.

Proof. From Lemma 2.2 and the definition of R, it follows that if G is a g̃_{1}-geodesic through m_{1} it is at bounded distance from some g̃_{1}-geodesic G′ through m_{2}. If m_{1} ≠ m_{2} and G is not the geodesic through m_{1} and m_{2}, then G and G′ bound a g̃_{1}-strip which by hypothesis must be flat. This implies that the curvature of g̃_{1} along every g̃_{1}-geodesic through m_{1} is 0 (by continuity this is also true for the geodesic between m_{1} and m_{2}). The completeness of g_{1} implies then that g_{1} is flat which is impossible since the genus of M is ≥ 2. The equivariance is obvious.

Let us now suppose that the Riemannian metrics g_{1} and g_{2} without conjugate points on M have the same length spectrum and one of them has non positive curvature. By 1.5, both of them have nonpositive curvature, and by 1.3 all g̃_{1} and g̃_{2} strips are flat and by 2.3 the relation R is the graph of a bijection.

Let Ũ_{i}, i = 1, 2, be {m ∈ M̃ | κg̃_{i}(m) ≠ 0}.

Lemma 2.4. Under the hypothesis of Theorem A, let G_{1} and G′_{1} be g̃_{1}-geodesics which intersect transversally at a point m which is in Ũ_{1}. Suppose that G_{2} and G′_{2} are g̃_{2}-geodesics with G_{1} ∼ G_{2} and G′_{1} ∼ G′_{2}. If m′ is the point of intersection of G_{2} and G′_{2} then mRm′. In particular, the set Ũ_{1} is contained in the domain D̃_{1} of the map φ given by 2.3.

Moreover, every g̃_{1}-geodesic which is at bounded distance from a g̃_{1}-geodesic through m must also pass through m, and every g̃_{2}-geodesic which is at bounded distance from a g̃_{2}-geodesic through m′ must also pass through m′.

Proof. If H is a g̃_{1}-geodesic which is at bounded distance from some g̃_{2}-geodesic which passes through m′, the angle condition of Lemma 1.5 shows that G_{1}, G′_{1} and H bound a triangle T whose sum of angles is π. But g_{1} is of nonpositive curvature and one of the vertices of the triangle, namely m, satisfies κg̃_{1}(m) ≠ 0, so from the Gauss-Bonnet theorem it follows that T is degenerate and that H goes through m. This proves that mRm′.

Since g̃_{1}-strips are flat, the point m cannot be contained in a g̃_{1}-strip. So every g̃_{1}-geodesic which is at bounded distance from some g̃_{1}-geodesic through m must also pass through m.

It remains to prove that every g̃_{2}-geodesic which is at bounded distance from a g̃_{2}-geodesic through m′ must also pass through m′. Let H′_{2} and H′′_{2} be two g̃_{2}-geodesics that remain at bounded distance and suppose that m′ ∈ H′_{2}, m′ ∉ H′′_{2} and m′′ ∈ H′′_{2}. We know from the first part that both H′_{2} and K, the g̃_{2}-geodesic through m′ and m′′, are at bounded distance from g̃_{1}-geodesics that pass through m. It follows that the pair H′′_{2}, K of transverse g̃_{2}-geodesics are at bounded distance from g̃_{1}-geodesics through m. By the first part of the lemma mRm′′. From Lemma 2.3, we obtain m′ = m′′. This is a contradiction.

Lemma 2.5. Under the hypothesis of Theorem A, if m, m′ ∈ Ũ_{1}, then dg̃_{1}(m, m′) = dg̃_{2}(φ(m), φ(m′)), where φ is given by Lemma 2.3. In particular, the map φ induces an isometry between Ũ_{1} and Ũ_{2}.

Proof. Fix a Riemannian metric g_{0} on M of (strictly) negative curvature and let S̃_{1} : G̃_{1} → G̃ and S̃_{2} : G̃_{2} → G̃ the Morse maps described above. It is not difficult to see, using lemma 2.4, that there exists a set A ⊂ G̃ such that S̃_{1}^{−1}(A) (resp. S̃_{2}^{−1}(A)) is the subset of G̃_{1} (resp. G̃_{2}) consisting of g̃_{1}-geodesics (resp. g̃_{2}-geodesics) that intersect the g̃_{1}-geodesic (resp. g̃_{2}-geodesic) segment between m and m′ (resp. φ(m) and φ(m′)). Using the fact that the Liouville currents obtained from g_{1} and g_{2} are the same, an application of the Crofton formula finishes the proof of the first part.

It follows from [MS] that φ is differentiable on Ũ_{1} and hence it is also a Riemannian isometry on U_{1}. This implies φ(Ũ_{1}) ⊂ Ũ_{2}. Exchanging the role of g_{1} and g_{2} gives φ(Ũ_{1}) = Ũ_{2}.

Lemma 2.6. Let g_{1} and g_{2} be Riemannian metrics without conjugate points on M, for which g̃_{1} and g̃_{2} strips are flat. Suppose that the map φ induces a bijection between Ũ_{1} = {m ∈ M̃ | κ_{g_{1}}(m) ≠ 0} and Ũ_{2} = {m ∈ M̃ | κ_{g_{2}}(m) ≠ 0}, and that for every m, m′ ∈ Ũ_{1}, we have dg̃_{1}(m, m′) = dg̃_{2}(φ(m), φ(m′)). Then φ extends to a Riemannian isometry of (M̃, g̃_{1}) onto (M̃, g̃_{2}) which is equivariant under the action of π_{1}. Hence the Riemannian metrics g_{1} and g_{2} are isometric by an isometry homotopic to the identity of M.

Proof. As above φ induces a Riemannian isometry. If p ∈ Ũ_{1}, call T_{p}φ the derivative of φ at p. One can check that the map φ̄ = exp_{φp}^{g_{2}} T_{p}φ(exp_{p}^{g_{1}})^{−1} extends φ to M̃. Moreover, the extension φ̄ preserves curvature, since along any geodesic through p the map will be an isometry at points of Ũ_{1}, namely φ, and will take points of zero curvature to points of zero curvature. From the well-known relation between Jacobi fields and the derivative of the exponential map—see [Kl, lemma 5.4.3 page 102]—it follows that φ̄ : (M̃, g̃_{1}) → (M̃, g̃_{2}) is an isometry. The fact that φ̄ is equivariant under π_{1}(M) follows from the fact that φ is invariant under the same action.

The proof of theorem A follows from the above lemmas.

3. Proof of theorem B.

In this section we assume that g_{1} and g_{2} are Riemannian metrics without conjugate points on M. and that the angle hypothesis of Theorem B is satisfied, i.e.:

For every pair (G_{1}, G′_{1}) of g̃_{1}-geodesics, and every pair (G_{2}, G′_{2}) of g̃_{2}-geodesics,

G_{1} ∼ G_{2} & G′_{1} ∼ G′_{2} ⟹ ∠g̃_{1}(G_{1}, G′_{1}) = ∠g̃_{2}(G_{2}, G′_{2}).

As before let Ũ_{=}{m ∈ M̃ | κg̃_{i}(m) ≠ 0}, i = 1, 2.

We first prove three more lemmas.

Lemma 3.1. All strips for g̃_{1} and g̃_{2} are flat. Consequently, no point of Ũ_{1} (resp. Ũ_{2}) is contained in a g̃_{1} (resp. g̃_{2}) strip.

Proof. We will show the result for g̃_{1}. A consequence of the hypothesis of theorem 3.1, is that if the g̃_{1}-geodesics G, G′ remain at bounded distance then any other g̃_{1}-geodesic cuts them at the same angle. By the result of Leon Green, Proposition 1.3, any strip bounded by two g̃_{1}-geodesics can be foliated by infinite g̃_{1}-geodesics. It is easy to deduce that any point inside the strip is contained in arbitrarily small geodesic triangles whose sum of interior angles is π. It follows from the Gauss-Bonnet theorem that the strip is flat.

Lemma 3.2. Let Ũ_{1} = {G_{1} ∈ G̃_{1} | G_{1} ∩ Ũ_{1} ≠ ∅} and Ũ′_{2} = {G_{2} ∈ G̃_{2} | ∃G_{1} ∈ Ũ_{1}, G_{1} ∼ G_{2}}. The formula G_{2} ∼ S̃G_{2} defines a continuous surjective map S̃ : Ũ′_{2} → Ũ_{1}. Of course Ũ_{1} is open; moreover, the set Ũ′_{2} is also open.

Proof. By Lemma 3.1, if G, G′ are g̃_{1}-geodesics with G ∈ Ũ_{1} and G ∼ G′ then G = G′. As above, let S̃_{1} : G̃_{1} → G̃ and S̃_{2} : G̃_{2} → G̃ the Morse maps obtained in 1.1. From the observation just made, S̃_{1} induces a bijection from Ũ_{1} → S̃_{1}(Ũ_{1}) and S̃_{1}^{−1}S̃_{1}(Ũ_{1}) = Ũ_{1}. It is not difficult, using the fact that S̃_{1} is continuous and proper, to conclude that S̃_{1}(Ũ_{1}) is open and that S̃_{1} restricts to a homeomorphism from Ũ_{1} onto S̃_{1}(Ũ_{1}). The lemma follows since Ũ′_{2} = S̃_{2}^{−1}(S̃_{1}(Ũ_{1})) and S̃ = S̃_{1}^{−1}S̃_{2}.

Lemma 3.3. Suppose that the g̃_{2}-geodesics G_{2}, G′_{2} are in Ũ′_{2} and that S̃G_{2} and S̃G′_{2} intersect transversally at a point m which is in Ũ_{1}. If m′ is the point of intersection of G_{2} and G′_{2} then mRm′. Moreover, every g̃_{1}-geodesic which is at bounded distance from some g̃_{1}-geodesic through m must also pass through m, and every g̃_{2}-geodesic which is at bounded distance from some g̃_{2}-geodesic through m′ must also pass through m′.

Proof. Let G_{2} and G′_{2} be g̃_{2}-geodesics passing through m′ such that G_{2} ∼ G_{1} and G′_{2} ∼ G′_{1} where G_{1} and G′_{1} are g̃_{1}-geodesics passing trough m. For θ ∈ [0, π], let G_{2}^{θ} (resp. G′_{2}^{θ}) be the g̃_{2}-geodesic through m′ making an angle θ with G_{2} (resp. G′_{2}). Let T be the set of θ ∈ [0, π] such that there exists g̃_{1}-geodesics G_{1}^{θ} and G′_{1}^{θ} through m with G_{2}^{θ} ∼ G_{1}^{θ} and G′_{2}^{θ} ∼ G′_{1}^{θ}.

Since two geodesics that stay at a bounded distance must stay within a constant distance depending only on g_{0}, g_{1} and g_{2} we see that T must be closed.

Since T is non-empty we need only show that T is open. Let ψ ∈ T. Since G_{2}^{ψ} and G′_{2}^{ψ} are in Ũ′_{2} and Ũ′_{2} is open, if θ is close enough to ψ, the g̃_{2}-geodesics G_{2}^{θ} and G′_{2}^{θ} are also in Ũ′_{2} and hence there exists unique geodesics G_{1}^{θ} and G′_{1}^{θ} with G_{2}^{θ} ∼ G_{1}^{θ} and G′_{2}^{θ} ∼ G′_{1}^{θ}.

Let ϵ > 0 be so small that B(m, ϵ), the ϵ-ball for g̃_{1} about m, is convex and has non-zero curvature for g̃_{1} at every point. By transversality and continuity of the map S̃: Ũ′_{2} → Ũ_{1} we see that for all θ sufficiently close to ψ the intersection points of G_{1}^{ψ} and G′_{1}^{θ}, G_{1}^{ψ} and G′_{1}^{ψ}, G_{1}^{θ} and G′_{1}^{θ}, and G_{1}^{θ} and G′_{1}^{ψ} all lie in B(m, ϵ).

We consider two cases. First assume G_{1}^{θ} intersects G_{1}^{ψ} inside B(m, ϵ) (or similarly that G′_{1}^{θ} intersects G′_{1}^{ψ} inside B(m, ϵ)). Then the geodesic triangles G_{1}^{θ}, G_{1}^{ψ}, G′_{1}^{ψ} and G_{1}^{θ}, G_{1}^{ψ}, G′_{1}^{θ} both lie inside B(m, ϵ). By preservation of angles both have interior angles that sum to π, but since the curvature is never zero in these triangles, the Gauss-Bonnet theorem forces them to be degenerate triangles which forces all these geodesics to pass through the common point m, so mRm′.

If on the other hand, both the intersections of G_{1}^{θ} with G_{1}^{ψ} and G′_{1}^{θ} with G′_{1}^{ψ} occur outside B(m, ϵ) we see that G_{1}^{θ}, G_{1}^{ψ}, G′_{1}^{θ}, G′_{1}^{ψ} form a quadrilateral inside B(m, ϵ). Again the Gauss-Bonnet theorem forces this quadrilateral to be degenerate and all geodesics pass through m, and again mRm′.

Again by flatness of strips, every g̃_{1}-geodesic at a bounded distance from some g̃_{1}-geodesic through m passes through m. Let G′_{2} and G′′_{2} be g̃_{2}-geodesics with G′_{2} ∼ G′′_{2} and m′ ∈ G′_{2}, m′ ∉ G′′_{2}. Let G_{1} be the g̃_{1}-geodesic through m with G_{1} ∼ G′_{2} hence G_{1} ∼ G′′_{2}. Pick m′′ ∈ G′′_{2} and let H_{2} be the g̃_{2}-geodesic through m′ and m′′. We know that there is a g̃_{1}-geodesic through m such that H_{1} ∼ H_{2} since mRm′. On the other hand, since m′′ = H ∩ G′′_{2} the first part of the lemma yields mRm′′. Now Lemma 2.3 yields m′ = m′′.

Corollary 3.4. Under the assumptions of this section Ũ_{1} is contained in the domain D̃_{1} of the relation R and Ũ_{2} is contained in is range D̃_{2}.

Proof. This is immediate from Lemma 3.3.

Proof of theorem B. We now proceed as in the proof of theorem A, using lemma 2.6.

4. Proof of theorem C.

Part of the arguments are already in [CF] and [Cr]. It is easy to see, by taking an orientable cover, that one can reduce the proof to the case where M_{1} is orientable. We will assume that this is the case in the sequel.

Lemma 4.1. Suppose S(M) is the unit tangent bundle of the closed surface M of genus ≥ 2. If a subgroup of π_{1}(M) has non trivial center then it is isomorphic to Z. Call p_{M} : S(M) → M the canonical projection. The center of π_{1}(S(M)) is contained the kernel of the induced map p_{M}_{∗} : π_{1}(S(M)) → π_{1}(M). Moreover, if M is orientable the kernel of p_{M}_{∗} is precisely the center of π_{1}(S(M))

Proof. This is well-known and can be proven using elementary hyperbolic geometry.

Lemma 4.2. Suppose M_{1} and M_{2} are closed surfaces of genus ≥ 2 endowed respectively with Riemannian metrics g_{1} and g_{2}. Suppose that h : S(M_{1}) → S(M_{2}) is a time preserving semi-conjugacy between the geodesic flow g_{1}^{t} of g_{1} and the geodesic g_{2}^{t} of g_{2}. If g_{2} has no conjugate points, then h maps the center of π_{1}(S(M_{1})) in the center of π_{1}(S(M_{2})), hence it induces a map h_{#} : π_{1}(M) → π_{1}(M_{2}). The map h_{#} is injective.

Proof. Let us look at the composition θ : S(M_{1}) → S(M_{2}) → M_{2}, where the first arrow is h and the second one is p_{M_{2}}. We want to show that θ_{∗} : π_{1}S((M_{1})) → π_{1}(M_{2}) sends the center of π_{1}S((M_{1})) to 0. Suppose this is not the case, then by 4.1 the image G of θ_{∗} is a cyclic subgroup of π_{1}(M_{2}) which is isomorphic to Z. Let us call P : C → M_{2} the covering of M_{2} such that P_{∗}(π_{1}(C)) = G. It is easy to see that h can be written as a composition S(P)h̃ where h̃ : S(M_{1}) → S(C) and S(P) : S(C) → S(M_{2}) is the tangent map obtained from P. If we lift the metric g_{2} to a metric g̃_{2} on C via P, we obtain that h̃ is a time preserving semi-conjugacy between flows. Using a little bit of the theory described in §1 and the fact that C is a cylinder or an open Möbius band without conjugate points, it is not difficult to realize that h̃ sends each g_{1}-geodesic to a g̃_{2}-geodesic that remain in the strip associated to a non-trivial closed g̃_{2}-geodesic G of minimum length in C. If H is a g̃_{2}-geodesic transversal to G, using the fact that g_{2} has no conjugate points, all g̃_{2}-geodesics that remain in the strip of G are also transversal to H. By the connectedness of S(M), we conclude that h̃ sends each oriented g_{1}-geodesic to a g̃_{2}-geodesic that always raps around C in the same sense. This is impossible, because a closed oriented g_{1}-geodesic and its opposite are in opposite homotopy classes of closed curves in S(M).

To show that h_{#} is injective, let us start with γ in π_{1}(M_{1}), we can find a closed g_{1}-geodesic G_{1} in the free homotopy class of γ. Since h is a semi-conjugacy the image h(G_{1}) = is a closed g_{2}-geodesic. Since g_{2} has no conjugate points it cannot be homotopic to 0.

Suppose M_{1} and M_{2} are closed surfaces of genus ≥ 2 endowed respectively with Riemannian metrics g_{1} and g_{2}. We assume that g_{2} has no conjugate points. Since by 4.2, the map h_{#} is injective and M_{1} and M_{2} are closed surfaces, the subgroup h_{#}(π_{1}(M_{1})) has finite index in π_{1}(M_{2}) (if not then by covering theory π_{1}(M_{1}) would be the fundamental group of a connected non-compact surface, but such a group is free and the fundamental group of a closed surface is never free). Hence it is easy to reduce to the case where h_{#} is an isomorphism. Since all automorphisms of the fundamental group of a surface can be realized by diffeomorphisms. we can find a diffeomorphism f : M_{2} → M_{1} such that the induced map f_{∗} on π_{1} is h_{#}^{−1}. If we use the diffeomorphism f to transport the metric g_{2} to a metric ĝ_{2} on M_{1}, it is not difficult to see that ĝ_{2} has no conjugate points and that g_{1} and ĝ_{2} have the same marked length-spectrum. If both g_{1} and g_{2} are without conjugate points and one of them of nonpositive curvature, then we can apply theorem A to g_{1} and ĝ_{2}, so if we compose f with an isometry homotopic to the identity, we see that the proof of theorem C is reduced to:

Lemma 4.3. Let g be a Riemannian metric with nonpositive curvature on a closed surface M. If h : S(M) → S(M) is a self semiconjugacy of the geodesic flow of g such that h_{#} : π_{1}(M) → π_{1}(M) is the identity then h = g^{t_{0}} for some fixed time t_{0}.

Proof. It is not difficult to see from the hypothesis on h_{#} that we can lift h to a map h̃ : S(M̃) → S(M̃) homotopic to the identity by a bounded homotopy, where M̃ is the universal cover of M. It follows that for any geodesic G of the lift g̃ of g to the universal cover M̃ the geodesic h̃(G) is bounded distance from G. By Proposition 1.3, the geodesics G and h̃(G) either coincide or bound a flat strip. Suppose that G is the lift to M̃ of a geodesic dense in S(M); then the second case cannot happen because G has to go through points of negative curvature. In fact, the geodesic G and h̃(G) have to coincide as oriented geodesics since h̃ preserves time and is homotopic to the identity by a bounded homotopy. The lemma follows easily using the denseness of the image of G in M and the fact that h preserves time.

Remark 4.4. Suppose M_{1} and M_{2} are closed surfaces of genus ≥ 2 endowed respectively with Riemannian metrics g_{1} and g_{2}. We assume that g_{2} has nonpositive curvature. If there exists a time preserving conjugacy (not necessarily C^{1}) between the geodesic flows of g_{1} and g_{2}, then the arguments in [Cr, lemma 3.2] show that g_{1} has no conjugate points and we can apply theorem C to obtain [Cr, theorem B] without the assumption that the conjugacy is C^{1}

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