On tori without conjugate points

Christopher B. Croke^{∗}  Bruce Kleiner^{†}

August, 1993

Abstract In this paper we consider Riemannian metrics without conjugate points on an n-torus. Recent work of J. Heber established that the gradient vector fields of Busemann functions on the universal cover of such a manifold induce a natural foliation (akin to the weak stable foliation for a Riemannian manifold with negative sectional curvature) on the unit tangent bundle. The main result in the paper is that the metric is flat if this foliation is Lipschitz. We also prove that this foliation is Lipschitz if and only if the metric has bounded asymptotes. This confirms a conjecture of E. Hopf in this case.

1 Introduction

By definition, a complete Riemannian manifold M has no conjugate points if no geodesic in M contains a pair of conjugate points. It follows from the proof of the Cartan-Hadamard theorem [ChEb] that M has no conjugate points if and only if every pair of points in the universal cover M̃ is joined by a unique geodesic segment. The most basic examples of manifolds without conjugate points are manifolds with nonpositive sectional curvature, and in many ways manifolds without conjugate points closely resemble nonpositively curved manifolds; in particular, a number of theorems about nonpositively curved manifolds can be proved using only synthetic arguments and the condition above [CS, Kl].

∗Supported in part by NSF grant #DMS90-01707 and #DMS85-05550 while at MSRI.

†Supported by an NSF Postdoctoral Fellowship.

There is increasing evidence that metrics without conjugate points – like nonpositively curved metrics (see [GrWo, LY, GrWu, BGS, Ba, BuSp]) – can be very rigid:

• In [H], E. Hopf showed that every Riemannian 2-torus without conjugate points is flat.

• Appealing to Hopf’s theorem, Green and Gulliver proved in [GrGu] that a metric on R^{2} which has no conjugate points and coincides with the standard metric outside a compact set, is flat. Using a different approach, the first author proved an n-dimensional generalization of this in [Cr1, Cr2].

• In [CrKl], the authors show that if M is any complete, 1-connected Riemannian manifold without conjugate points, and g is a metric on M × R which has no conjugate points and agrees with the product metric outside a compact set, then g is isometric to the product metric. See [CrKl] for other rigidity theorems.

In this paper we study metrics without conjugate points on n-tori; our main results support the rigidity conjecture of E. Hopf: Conjecture Every Riemannian metric without conjugate points on T^{n} is flat.

Statement of results

Let M be a Riemannian manifold without conjugate points diffeomorphic to an n-torus T^{n}. It was shown in [He] (cf. section 3) that the gradient of any Busemann function on M̃ descends to a geodesic vector field on M (i.e. a vector field whose integral curves are geodesics), and that the images (in the unit tangent bundle SM) of all such vector fields form the leaves of a continuous foliation F of SM.

Before stating our main result, we recall that a foliation is Lipschitz if it admits a system of bilipschitz foliation charts.

Theorem 1.1 If the foliation F is Lipschitz then M is flat.

We prove that the Lipschitz regularity of F is equivalent to a well-known condition on Jacobi fields of M:

Proposition 1.2 The foliation F is Lipschitz if and only if M has bounded asymptotes.

We recall that a manifold without conjugate points M has bounded asymptotes if all stable Jacobi tensors are uniformly bounded (see [E] and section 5). Combining theorem 1.1 and proposition 1.2 we have:

Corollary 1.3 Any Riemannian metric on the n-torus which has no conjugate points and bounded asymptotes is flat.

To describe the structure of the proof of theorem 1.1 we introduce the Cohomological Image of a metric g on T^{n}: CI(g) = {[ω] ∈ H^{1}(T^{n}; R)|[ω] can be represented by a weakly closed^{1} Lipschitz one-form ω and |ω(x)| ≤ 1 for all x ∈ T^{n}}. CI(g) is a convex subset of H^{1}(T^{n}; R), and we let vol(CI(g)) be the Lebesgue measure of CI(g) (we normalize Lebesgue measure on H^{1}(T^{n}; R) so that the induced measure on the torus H^{1}(T^{n}; R)/H^{1}(T^{n}; Z) has total measure 1).

The proof of theorem 1.1 indicates a method for proving the full Hopf conjecture. Our proof breaks up into two parts, the first of which is valid for any torus without conjugate points:

Proposition 1.4 If g is a Riemannian metric without conjugate points on an n-torus then

V ol(CI(g)) ≤ β(n)V ol(g)

where β(n) is the volume of the unit ball in E^{n}; equality holds if and only if g is flat.

Thus we are reduced to showing that V ol(CI(g)) = β(n)V ol(g). In section 5 we first show that V ol(CI(g)) = β(n)V ol(g) if F is smooth. Using a standard smoothing argument, we then obtain the same equality assuming F is Lipschitz:

Proposition 1.5 If g is a metric without conjugate points on an n-torus such that the foliation F is Lipschitz then

V ol(CI(g)) = β(n)V ol(g).

1A 1-form is weakly closed if it integrates to zero over any small loop.

As far as we know, a more elaborate approximation method might work for general F. Our foliations F are not completely arbitrary as they can be locally C^{0} approximated by smooth foliations whose leaves are the images in SM of gradients of locally defined smooth functions on T^{n}. If such approximations can be made global then our arguments prove the Hopf conjecture.

In section 6 we discuss some open questions and possible approaches to improving theorem 1.1 to a proof of the Hopf conjecture.

Remarks

1. It has been known for some time (see [Bus, §32]) that for each nontrivial free homotopy class α ∈ [S^{1}, M] there is a smooth unit vector field V_{α} on M whose integral curves are precisely the set of closed geodesics in the free homotopy class α. We call the unit vectors V_{α}(x) for x ∈ M “rational directions”. More recently, it was shown by Eberlein (unpublished) and Heber [He] (see also proposition 3.2) that rational directions are dense in SM. By studying limits of sequences V_{α_{k}}, α_{k} ∈ [S^{1}, M], one finds that each element of SM lies in a Lipschitz geodesic vector field on M (a vector field whose integral curves are geodesics); furthermore, the collection of all such limits is a family of Lipschitz geodesic vector fields whose images (in SM) form the leaves of a continuous foliation F of SM. This was first proved by Heber (see [He]), and we present an independent proof in section 3. Theorem 1.1 evolved from an earlier result which answered the question: Must g be flat if F is C^{∞}? The smoothness of F implies that the geodesic flow of M is smoothly conjugate to the geodesic flow of a flat Finsler torus, i.e. R^{n}/Z^{n} where R^{n} is endowed with a translation invariant Finsler metric. The authors had shown shortly before that a Riemannian manifold with geodesic flow conjugate to the geodesic flow of a flat Riemannian torus, is flat; theorem 1.1 developed naturally from these ingredients.

2. Most of the results in this paper extend in a straightforward way to Finsler metrics on tori. In particular, the vector fields V_{α}, the foliation F, and proposition 1.5 have Finsler analogs. The Finsler version of proposition 1.5 says that V ol(CI(g)) = V ol(D^{∗}T^{n}) where D^{∗}T^{n} ⊂ T^{∗}T^{n} is the dual unit disk bundle, and V ol(D^{∗}T^{n}) is the (appropriately normalized) symplectic volume of D^{∗}T^{n}. However, the Riemannian relation V ol(D^{∗}T^{n}) = β(n)V ol(g) breaks down in the Finsler case, as does the Hopf conjecture itself: there are many nonflat Finsler metrics on T^{n} (i.e. metrics which are not locally isometric to a translation invariant Finsler metric on R^{n}) which don’t have conjugate points. This was first observed by Busemann in [Bus, §33]. We also mention that there are nonflat, complete, torsion-free connections on T^{n} which have no conjugate points (i.e. the exponential maps have full rank everywhere).

3. The only property of the n-torus which is crucial for the theorems stated above is the fact that π_{1}(T^{n}) ≃ Z^{n}. Consequently, all the statements still hold (with appropriate small modifications) if the n-torus is replaced by a Riemannian manifold M^{n} with no conjugate points and solvable fundamental group: by [CS, Kl] such manifolds have finite covers with fundamental group isomorphic to Z^{n}.

Historical comments

To the best of our knowledge, the question of whether tori without conjugate points must be flat first appeared in the literature in [HeMo]. This was proved in two dimensions by E. Hopf [H]. His method was extended to n-dimensions by L. Green [G] who showed that any closed n-dimensional manifold without conjugate points has total scalar curvature ≤ 0, with equality only if the metric is flat. The Hopf-Green approach was extended in [Cr3] to the statement that for any r > 0, the average volume of geodesics balls of radius r in M̃ is greater than in E^{n}. The Hopf conjecture is known to hold if the manifold admits a cohomogeneity two isometric group action (see [Van]), if the metric is conformally flat (see [CF] and [K]), or if the metric has no focal points [Av]. There are numerous interesting references on manifolds without conjugate points; we give a partial list of those which have some historical/mathematical connection with the ideas in the present paper: [Bus, EsOS, OS, FrMa, BBB, Klbg].

2 Properties of Busemann functions

In this section we collect the well-known facts about Busemann functions on universal covers of compact Riemannian manifolds M without conjugate points that we will use in the other sections.

For ṽ ∈ SM̃ we let Bṽ : M̃ → R be the Busemann function associated to ṽ, i.e. Bṽ(x̃) = lim_{t→∞} d(x̃, γ̃ṽ(t))−t, where γ̃ṽ is the geodesic with γ̃′ ; ṽ(0) = ṽ. The triangle inequality implies that d(x̃, γ̃ṽ(t)) − t is nonincreasing in t so in particular we see that Bṽ is well defined and

(i) Bṽ(x̃) ≤ d(x̃, γ̃ṽ(t)) − t for all t > 0 and all x̃ ∈ M̃.

It is not hard to see that Busemann functions are C^{1} and that their gradients are the limits of the gradients of the appropriate distance functions (see [E]). Thus |∇Bṽ(x̃)| = 1 for all x̃ ∈ M̃ and in particular, Bṽ is Lipschitz with Lipschitz constant 1. The integral curves of the gradient are thus easily seen to be geodesics and further

(ii) Let w̃ = −∇Bṽ(x̃). Then for every t > 0, γ̃w̃ is the unique geodesic through x̃ such that Bṽ(γ̃w̃(t)) = Bṽ(γ̃w̃(0)) − t.

When M̃ is the universal cover of a compact Riemannian manifold without conjugate points then it is not hard to see that Busemann functions are in fact C^{1,1} since the Hessians of the approximating functions can be controlled by the second fundamental forms of unit spheres.

If α is an isometry of M̃, we define the displacement function of α by d_{α}(x̃) = d(x̃, α(x̃)). If β is an isometry that commutes with α then d_{α}(β(x)) = d_{α}(x). An axis of α is a geodesic γ̃ such that for all t, α(γ(t)) = γ(t + c) for some constant c. The first variation formula tells us that the set of points on axes of α coincide with the critical points of d_{α} (see lemma 2.1 of [CS]). Further it is not hard to show (see lemma 2.1 of [CS]) that these must be minima of d_{α}.

We now wish to outline the proof of the existence of the smooth vector fields V_{α} mentioned in the introduction so we turn to the case where M is a torus and α is a deck transformation. Since α commutes with all other deck transformations, and since there are compact fundamental domains, we see that d_{α} has a maximum. Since all critical points of d_{α} are minima we see that d_{α} is constant. Thus there is an axis of α through each x̃ ∈ M̃. Taking the unit tangent vector to the axis at each point gives us our C^{∞} vector field Ṽ_{α} (i.e. Ṽ_{α}(x̃) = γ̃′(0) where γ̃ is the geodesic from x̃ to α(x̃)). Again since α commutes with the full deck group Ṽ_{α} is invariant under the full deck group and hence projects to a C^{∞} unit vector field V_{α} on M.

For ṽ = Ṽ_{α}(x̃) it is easy to check that Bṽ(α(ỹ))−Bṽ(ỹ) = −d_{α} (recall that d_{α} is constant). Thus, by equation (ii), Ṽ_{α}(ỹ) = −∇Bṽ(ỹ) for all x̃, ỹ ∈ M̃. The Hessian of Bṽ is bounded by the maximum of the second fundamental forms of unit balls (which exists by compactness) hence there is a uniform Lipschitz constant C for the smooth vector fields Ṽ_{α} and V_{α}. We note that each closed geodesic γ is an orbit of V_{[γ]}.

3 The foliation F

There has been quite a bit of recent interest and progress in the study of the properties of the geodesic flow on tori without conjugate points. Several people have contributed independent proofs of some of the results in this section (either published or unpublished.) We wish to mention in particular V. Bangert, P. Eberlein, and J. Heber. For the sake of completeness we give all proofs, even those (i.e.proposition 3.2) for which we claim no credit.

A geodesic γ is called recurrent if there exists a sequence t_{i} increasing to ∞ such that lim_{i→∞}γ′(t_{i}) = γ′(0). A unit vector v will be called recurrent if the associated geodesic γ_{v} is recurrent.

Lemma 3.1 Let γ be recurrent geodesic on M a torus without conjugate points and let γ̃ a lift to M̃. Then there exists a sequence γ_{i} of closed geodesics of periods l_{i} with γ_{i}(0) = γ(0) such that the lifts γ̃_{i} with γ̃_{i}(0) = γ̃(0) satisfy lim_{i→∞}d(γ̃_{i}(l_{i}), γ̃_{v}(l_{i})) = 0.

Proof: Since γ is recurrent there is a sequence t_{i} increasing to ∞ such that γ′(t_{i}) → γ′(0). This means that there is a sequence α_{i} ∈ π_{1}(M) such that α_{i∗}(γ̃′(t_{i})) → γ̃′(0) where the α_{i} act as deck transformations on M̃. Hence d(α_{i}^{−1}(γ̃(0)), γ̃(t_{i})) → 0. Let γ_{i} be the closed geodesic of length l_{i} through γ(0) representing α_{i}^{−1}. Now γ̃_{i}(l_{i}) = α_{i}^{−1}(γ̃(0)) so we see that d(γ̃_{i}(l_{i}), γ̃(t_{i})) → 0. Since all geodesic are minimizing in M̃ we see that |l_{i} − t_{i}| → 0 and the lemma follows. □

One conclusion of this is:

Proposition 3.2 (Eberlein, Heber) The rational directions are dense in SM.

Proof: Since the recurrent vectors are dense (see [Man, Theorem 2.3]) it is sufficient to show that each recurrent vector v is in the closure of the rational directions. To see this we need only show that lim_{i→∞}γ̃′_{i}(0) = γ̃′_{v}(0) for the sequence of closed geodesics guaranteed by lemma 3.1. By taking a subsequence we may assume that lim_{i→∞}γ̃′_{i}(0) = w̃ for some unit vector w̃ at γ̃_{v}(0). We need to show that w̃ = ṽ ≡ γ̃′_{v}(0); i.e. that w̃ = −∇Bṽ. If we fix t > 0 it is sufficient to show (by equation (ii)) that Bṽ(γ̃w̃(t)) = −t. To see this fix ϵ > 0 and (by lemma 3.1) choose i so large that l_{i} > t, d(γ̃_{i}(t), γ̃w̃(t)) < ϵ/2 and d(γ̃_{i}(l_{i}), γ̃_{v}(l_{i})) < ϵ/2. Then we see (using equation (i))

Bṽ(γ̃w̃(t)) ≤ d(γ̃_{v}(l_{i}), γ̃w̃(t)) − l_{i} ≤ d(γ̃_{v}(l_{i}), γ̃_{i}(t)) − l_{i} + ϵ/2 ≤
≤ d(γ̃_{i}(l_{i}), γ̃_{i}(t)) − l_{i} + ϵ = −t + ϵ.

Letting ϵ go to 0 and using the fact that Bṽ has Lipschitz constant 1 yields the result. □

We are now in a position to prove the existence of the foliation F.

Proposition 3.3 Let M be a Riemannian manifold without conjugate points diffeomorphic to the torus. Then there is a unique continuous foliation F of SM for which the V_{α} for α ∈ [S^{1}, M] are leaves. The leaves of F are invariant under the geodesic flow, and are the images of Lipschitz (with uniform Lipschitz constant) vector fields on M for which the orthogonal distributions are integrable.

For an arbitary unit vector v ∈ SM at x ∈ M, by the proposition 3.2, there is a sequence α_{i} of homotopy classes such that v = lim_{i→∞} V_{α_{i}}(x). By the Arzelà - Ascoli theorem, after passing to a subsequence, V_{α_{i}} converges to a Lipschitz vector field L_{v} which has Lipschitz constant C. The orbits of L_{v} will also be geodesics (although they won’t be closed geodesics in general), and the one-form dual to L_{v} is closed (i.e. the orthogonal distributions are integrable), being the limit of closed one-forms.

To see that this yields a geodesic flow invariant, continuous foliation with uniformly Lipschitz leaves, F, the only thing left to show is that if V_{α_{i}^{1}} → L_{v}^{1} and V_{α_{i}^{2}} → L_{v}^{2} are two convergent sequences to vector fields L_{v}^{1} and L_{v}^{2} such that L_{v}^{j}(x) = v then L_{v}^{1} = L_{v}^{2}. This we show in the following lemma.

Lemma 3.4 If M is a torus without conjugate points, and v ∈ SM is a unit vector, then there is at most one leaf L_{v} which is a limit of the V_{α}.

Proof: Assume there are two such L_{v}^{1} and L_{v}^{2}. Let ω_{v}^{j} be the dual closed Lipschitz one forms.

We first prove the lemma for recurrent vectors v. Let γ_{i} be a sequence of closed geodesics as in lemma 3.1 converging to γ_{v} and let ϵ_{i} = d(γ̃_{v}(l_{i}), γ̃_{i}(l_{i})).

Then for j = 1 or 2, ω_{v}^{j}(γ_{i}) ≥ l_{i} − ϵ_{i} where lim_{i→∞} ϵ_{i} = 0. Now for fixed y ∈ M let τ_{i} be the closed geodesic (of length l_{i}) through y homotopic to γ_{i}. Since ω_{v}^{j} is closed we see that ω_{v}^{j}(τ_{i}) = ω_{v}^{j}(γ_{i}) ≥ l_{i} − ϵ_{i}. In order to show that L_{v}^{1}(y) = L_{v}^{2}(y) it is sufficient to show that lim_{i→∞}ω_{v}^{j}(τ′_{i}(0)) = 1 since ω_{v}^{j}(y) are the covectors dual to the unit vectors L_{v}^{j}(y). We may thus assume that there is a 2 ≥ δ > 0, a unit vector w at y and a sequence τ_{i} (a subsequence of the original sequence) such that lim_{i→∞}τ′_{i}(0) = w and ω_{v}^{1}(w) = 1 − δ (or possibly ω^{2} instead of ω^{1}). Since there is a neighborhood of w in SM on which ω_{v}^{1} is less than 1 − δ/2 there is a t_{0} > 0 such that for all sufficiently large i and for 0 < t < t_{0} we have ω_{v}^{1}(τ′_{i}(t)) < 1 − δ/2 and hence ( since |ω_{v}^{1}(τ′_{i}(t))| ≤ 1 for all t) l_{i} − t_{0}δ/2 ≥ ω_{v}^{1}(τ_{i}) ≥ l_{i} − ϵ_{i}. This contradicts the condition that ϵ_{i} goes to zero and the lemma follows for recurrent v.

For an arbitrary v ∈ SM, w ∈ L_{v}^{j} for all w in the closure of the orbit of v under the geodesic flow, since L_{v}^{j} is invariant under geodesic flow and v ∈ L_{v}^{j}. But in a compact manifold the closure of the orbit of a unit vector under the geodesic flow always contains a recurrent vector w (see [PdM, pp. 138-9]). Since w ∈ L_{v}^{j} the argument for recurrent vectors gives L_{v}^{1} = L_{v}^{2} and the lemma follows. □

For v ∈ SM we will let L_{v} be the vector field on M corresponding to the leaf of F passing through v and let L̃_{v} be its lift to the universal cover M̃.

The construction of F in fact shows that there is a homeomorphism Ψ̂ : M ×S^{n−1} ⟶ SM covering the identity on M such that Ψ̂(M ×{p}) ⊂ SM is the image of a Lipschitz vector field with integrable orthogonal distribution.

Lemma 3.5 For each v ∈ SM, L̃_{v}(x̃) = −∇Bṽ(x̃) for every x̃ ∈ M̃ and every lift ṽ of v.

Proof: The properties of L̃_{v} guarantee that there is a unique C^{1,1} function f_{v} on M̃ such that L̃_{v} = −∇f and f_{v}(γ̃ṽ(t)) = −t . Assume L̃_{v}(x̃) ≠ −∇Bṽ(x̃). Let w̃ = −∇Bṽ(x̃) and w the projection of w̃ to M. By the compactness of M there is a minimum angle θ_{0} > 0 between L_{v} and L_{w} and hence between L̃_{v} and L̃_{w}. Thus f_{v}(γ̃w̃(t)) − f_{v}(γ̃w̃(0)) ≥ −t · cos(θ_{0}). Since f_{v} has Lipschitz constant 1 we see that

d(γ̃ṽ(l), γ̃w̃(t)) − l ≥ f_{v}(γ̃w̃(t)) − f_{v}(γ̃ṽ(l)) − l
= f_{v}(γ̃w̃(t)) − f_{v}(γ̃w̃(0)) + f_{v}(γ̃w̃(0))

≥ f_{v}(γ̃w̃(0)) − t · cos(θ_{0}).

Hence letting l go to ∞ we see that Bṽ(γ̃w̃(t)) ≥ f_{v}(γ̃w̃(0)) − t · cos(θ_{0}). But this is a contradiction since the definition of w̃ implies (equation (ii)) that Bṽ(γ̃w̃(t)) = Bṽ(γ̃w̃(0)) − t. □

4 The asymptotic metric

When g has no conjugate points then each leaf L of F gives rise to a closed Lipschitz one form ω_{L} with |ω_{L}(x)| = 1. It is clear that for any Riemannian metric g on T^{n}, the cohomological image (cf. the introduction) of g satisfies CI(g) ⊆ {[ω] ∈ H^{1}(T^{n}; R)| [ω](γ) ≤ l(γ) for all closed curves γ of T^{n}} (the inclusion is actually an equality (see [Grm]), but we will not use this fact). We use this to show:

Lemma 4.1 If (T^{n}, g) is a Riemannian torus without conjugate points then CI(g) is a strictly convex set whose boundary consists of {[ω_{L}]|L ∈ F}.

Proof: From the convexity of the unit coballs it is clear that CI(g) is convex and contains an open neighborhood of 0 in H^{1}(T^{n}) ≅ R^{n}. It is clear from the inclusion mentioned in the preceding paragraph that for rational leaves L, c[ω_{L}] ∈ CI if and only if c ∈ [−1, 1]. This holds for the irrational leaves as well by continuity and hence [ω_{L}] lies on the boundary of CI. Fixing a base point x ∈ T^{n} we can parametrize the leaf space of F by the unit vectors v at x. The induced map v → [ω_{L_{v}}] is a one to one continuous map from an (n-1)-sphere to an (n-1)-sphere (the boundary of the convex set CI) and hence is onto. The strict convexity of CI(g) follows from the fact that different leaves correspond to closed one-forms which are different at every point in T^{n}. □

As a background metric we will let g_{0} be the standard (flat) product metric on T^{n} = {(ϕ_{1}, ϕ_{2}, ..., ϕ_{n}) ∈ S^{1}×S^{1}×...×S^{1}} (where S^{1} = [0, 1]/0 ∼ 1). The universal cover can be naturally identified with H_{1}(T^{n}; R) ≅ R^{n} and hence supports a Banach metric | · |_{b} whose unit ball B_{b}(1) is the dual to CI(g) ⊂ H^{1}(T^{n}; R).

Lemma 4.2 |·|_{b} is the (unique) Banach norm such that the induced distance between points on the integer lattice H_{1}(T^{n}; Z) is the length of the closed geodesics in the corresponding homotopy class.

Proof: The only thing that needs to be shown is that for each lattice point z ∈ H_{1}(T^{n}; Z), |z|_{b} is the length l of the shortest closed geodesic γ_{z} representing z. By definition |z|_{b} = max{[ω](z)|[ω] ∈ CI}. But since [ω] can be represented by a form ω with |ω(m)| ≤ 1 for all m ∈ M we see that ω(γ_{z}) ≤ l. On the other hand γ_{z} is an orbit of the “rational” vector field V_{z}. If we let ω_{z} be the dual closed one form we see that [ω_{z}] ∈ CI and ω_{z}(γ_{z}) = l. □

We let g̃_{0} and g̃ be the lifts of g_{0} and g to the universal cover. g̃_{0} is the standard metric on R^{n}. It will simplify the arguments if we choose our identification with the standard T^{n} using Moser’s theorem [Mos] so that dvol_{1} = V ol(g)dvol_{0} where dvol_{1} is the volume form associated to g̃ and dvol_{0} is the Euclidean volume form. (All the arguments below would go through without such a choice by studying fundamental domains.)

If B is a centrally symmetric convex body in Euclidean space R^{n} and B^{∗} is the dual convex body (i.e. the polar of B), then Mahler’s inequality (whose sharp version was proved by Santaló [San]) says that V ol(B)V ol(B^{∗}) ≤ β^{2}(n) with equality holding if and only if B is an ellipsoid. Hence by our choice above we can conclude that

V ol_{0}(B_{b}(1))V ol(CI(g)) ≤ β^{2}(n) (1)

The distance function d_{1} associated to g̃ restricted to the integer points Z^{n} ⊂ R^{n} is the length of the closed geodesic in the corresponding homotopy class and hence by lemma 4.2 is the same as the Banach distance | · |_{b}.

For each integer a, let h_{a} : R^{n} → R^{n} be the dilation h_{a}(x) = ax and g̃_{a} = \frac{1}{a^{2}} h_{a}^{∗}(g̃). We note that the associated volume forms dvol_{a} are all the same (i.e.V ol(g)dvol_{0}) and that the associated distance functions d_{a} coincide with the Banach norm | · |_{b} when restricted to 1/a times the integer lattice. Thus it is easy to see that the pointed metric spaces (R^{n}, g̃_{a}, 0) converge in the Gromov-Hausdorff topology to the Banach space (R^{n}, | · |_{b}, 0). Hence the unit ball B_{a}(1) in g̃_{a} centered at 0 converges to B_{b}(1) and hence V ol_{a}(B_{a}(1)) converges to V ol(g)V ol_{0}(B_{b}(1)) where V ol_{a} is the volume with respect to g̃_{a} and V ol_{0} is the Euclidean volume. On the other hand V ol_{a}(B_{a}(1)) = V ol_{1}(B_{1}(a))/a^{n}. Theorem 1 of [Cr3] says

β(n) ≤ lim inf_{a→∞} \frac{V ol_{1}(B_{1}(a))}{a^{n}} = V ol(g)V ol_{0}(B_{b}(1)) (2)

with equality holding if and only if g is flat. Combining equations 1 and 2 yields proposition 1.4.

5 Lipschitz regularity of F and its consequences

In this section we prove propositions 1.5 and 1.2.

We prove the next lemma in preparation for the proof of proposition 1.5.

Lemma 5.1 Suppose (T^{n}, g) is a Riemannian torus without conjugate points, and suppose the foliation F is Lipschitz. Then there is a Lipschitz homeomorphism Ψ̂ : T^{n} × S^{n−1} ⟶ ST^{n} such that π ∘ Ψ̂ = π_{1} and Ψ̂(T^{n} × {p}) is a leaf of the foliation F for every p ∈ S^{n−1} .

Proof: We put the product metric on T^{n} × S^{n−1} and the Sasaki metric on ST^{n}. Pick p_{0} ∈ T^{n}, and identify S_{p_{0}}T^{n} with S^{n−1} by a diffeomorphism. Define Ψ̂ : T^{n} × S^{n−1} ⟶ ST^{n} by letting Ψ̂(p, v) be the unique element of S_{p}T^{n} lying in the same leaf as v ∈ S_{p_{0}}T^{n}. Ψ̂ is a homeomorphism (cf. the paragraph preceding lemma 3.5), and we will show that Ψ̂ is Lipschitz provided F is a Lipschitz foliation. Since the leaves of F are the images of uniformly Lipschitz sections (cf. prop. 3.3), it is enough to show that there is a constant C_{1} such that for every p ∈ T^{n} the holonomy homeomorphism S_{p_{0}}T^{n} ⟶ S_{p}T^{n} of the foliation F and its inverse both have Lipschitz constant ≤ C_{1}. We will prove this via the following two claims:

Claim A: There is a constant C_{2} such that for every pair of leaves L, L′ of F, and every pair of points p, q ∈ L, we have

C_{2}^{−1}d(p, L′) ≤ d(q, L′) ≤ C_{2}d(p, L′). (3)

Claim B: There is a constant C_{3} such that for any pair of leaves L, L′ ∈ F and any p ∈ T^{n}

d(S_{p}T^{n} ∩ L, S_{p}T^{n} ∩ L′) ≤ C_{3}d(S_{p}T^{n} ∩ L, L′). (4)

Together these two inequalities imply that the maps S_{p}T^{n} ⟶ L are uniformly bi-Lipschitz maps, where L is the leaf space of the foliation F with the quotient metric. This easily implies that the holonomy homeomorphisms are uniformly bi-Lipschitz as desired.

Proof of Claim A: Since F is a Lipschitz foliation there is a finite collection of open sets U_{i} ⊂ ST^{n} and bi-Lipschitz foliation charts ϕ_{i} : U_{i} ⟶ D^{n} × D^{n−1} carrying F|_{U_{⟩}} to the standard foliation of D^{n} × D^{n−1}. Let

C_{4} = sup_{i}(Lip(ϕ_{i}), Lip(ϕ_{i}^{−1})). Pick a slightly smaller open covering {Ū_{i}}_{i∈I} refining {U_{i}}, such that dist(ST^{n}−U_{i}, Ū_{i}) ≥ ϵ for some ϵ > 0 and every i ∈ I. Hence if p ∈ Ū_{i}, Σ ⊆ ST^{n}, and d(p, Σ) < ϵ, then d(p, Σ ∩ U_{i}) = d(p, Σ).

Let L be a leaf of F, and suppose p, q ∈ L ∩ Ū_{i}. If L′ is another leaf of F such that d(p, L′), d(q, L′) < ϵ, then d(p, L′∩U_{i}) = d(p, L′) and d(q, L′∩U_{i}) = d(q, L′) and so

d(p, L′) ≤ C_{4}d(ϕ_{i}(p), ϕ_{i}(L′ ∩ U_{i}))
= C_{4}d(ϕ_{i}(q), ϕ_{i}(L′ ∩ U_{i}))
≤ C_{4}^{2}d(q, L′ ∩ U_{i})
= C_{4}^{2}d(q, L′).

If p, q ∈ L and d(p, L′), d(q, L′) < \frac{ϵ}{(C_{4}^{2})^{|I|}}, then

C̄_{2}^{−1}d(p, L′) ≤ d(q, L′) ≤ C̄_{2}d(p, L′)

for C̄_{2} = (C_{4}^{2})^{|I|} since we may find a sequence p = p_{1}, p_{2}, ..., p_{k} = q with p_{i} ∈ L, k ≤ |I| + 1, and such that both p_{i} and p_{i+1} lie in some element of the cover {Ū_{i}} for every 1 ≤ i ≤ k − 1. Therefore inequality 3 holds for a possibly larger constant C_{2} ≥ C̄_{2}.

Proof of Claim B: Let v = S_{p}T^{n} ∩ L, let w = S_{p}T^{n} ∩ L′, and let z ∈ L′ be the point in L′ closest to v. Put a smooth product metric on ST^{n} (which may be used to measure distances since any two smooth Riemannian metrics on ST^{n} define Lipschitz equivalent distance functions) and let π_{1} : ST^{n} ⟶ T^{n} and π_{2} : ST^{n} ⟶ S^{n−1} be the projection maps associated with this product structure. Pick C_{5} ∈ R so that for every leaf of F, the corresponding section of π_{1} : ST^{n} ⟶ T^{n} has Lipschitz constant ≤ C_{5} with respect to the chosen product structure. Let

a = d(π_{1}(v), π_{1}(z))
b = d(π_{2}(v), π_{2}(z))
c = d(v, z)
h = d(v, w).

Then c = \sqrt{a^{2} + b^{2}} and b ≥ max(h−C_{5}a, 0) which implies that \frac{h}{c} is uniformly bounded by a function of C_{5}.

□

Given a smooth n-parameter family of closed one-forms on a compact manifold M^{n}, there is a relation between the volume that the family sweeps out in T^{∗}M^{n} and the volume swept out by the corresponding family of cohomology classes. More precisely, let M^{n} be a compact manifold, and define vol_{∧}^{0} ∈ ∧^{n}(H^{1}(M^{n}; R))^{∗} by vol_{∧}^{0}(α_{1}, ..., α_{n}) = ⟨α_{1} ∧ ... ∧ α_{n}, [M^{n}]⟩ ∈ R. Let vol_{∧} ∈ Ω^{n}(H^{1}(M^{n}; R)) denote the corresponding translation invariant n-form on H^{1}(M^{n}; R). Now suppose Ψ : M^{n} × D^{n} ⟶ T^{∗}M^{n} determines a family of closed one-forms on M^{n}, i.e.

1. π ∘ Ψ = π_{1},

2. For every v ∈ D^{n}, Ψ(M^{n} × {v}) ⊆ T^{∗}M^{n} is the image of a closed one-form on M^{n}.

Proposition 5.2 Define Φ : D^{n} ⟶ H^{1}(M^{n}; R) by Φ(v) = [(id_{M^{n}}×v)^{∗}Ψ^{∗}θ], where θ ∈ Ω^{1}(T^{∗}M^{n}) is the canonical one-form. Φ(v) is the cohomology class of the one-form with image Ψ(M^{n} × {v}). If Ψ is smooth then

∫_{M^{n}×D^{n}} Ψ^{∗}vol_{T^{∗}T^{n}} = ∫_{D^{n}} Φ^{∗}vol_{∧} (5)

where vol_{T^{∗}M^{n}} is the volume form on T^{∗}M^{n}.

Proof: Let θ̄ = Ψ^{∗}θ ∈ Ω^{1}(M^{n} × D^{n}) and ω̄ = dθ̄ = Ψ^{∗}dθ, so we have Φ(v) = [(id × \{v\})^{∗}θ̄] ∈ H^{1}(M^{n}; R). Since vol_{T^{∗}M^{n}} = (−1)^{\frac{n(n+1)}{2}} \frac{1}{n!}(dθ)^{n}, by Fubini’s theorem we have

∫_{M^{n}×D^{n}} Ψ^{∗}vol_{T^{∗}M^{n}} = (−1)^{\frac{n(n+1)}{2}} ∫_{M^{n}×D^{n}} \frac{1}{n!}ω̄^{n}

= ∫_{D^{n}} [∫_{M^{n}×\{p\}} i_{X_{1}}...i_{X_{n}}(\frac{1}{n!}ω̄^{n})] dp

where X_{i} = \frac{∂}{∂x_{i}} is the lift of the ith coordinate vector field on D^{n} to M^{n}×D^{n}. Now

i_{X_{1}}i_{X_{2}}...i_{X_{n}}(\frac{1}{n!}ω̄^{n})|_{M^{n}×\{p\}}

= [(i_{X_{1}} ω̄) ∧ ... ∧ (i_{X_{n}} ω̄)]|_{M^{n}×{p}} by the derivation property of i_{X_{i}} and since ω̄|_{M^{n}×{p}} = 0 by property 2 of Ψ. But i_{X_{i}} ω̄ = \frac{∂}{∂x_{i}} θ̄ − di_{X_{i}} θ̄ so

i_{X_{1}}i_{X_{2}}...i_{X_{n}}(\frac{1}{n!}ω̄^{n})|_{M^{n}×\{p\}}

= ∏_{i}(\frac{∂}{∂x_{i}} θ̄ − di_{X_{i}} θ̄).

Being the derivative of a family of closed one-forms, \frac{∂}{∂x_{i}} θ̄|_{M^{n}×\{p\}} is closed, while the forms di_{X_{i}} θ̄ are exact; consequently

∫_{M^{n}×\{p\}} ∏_{i}(\frac{∂}{∂x_{i}} θ̄ − di_{X_{i}} θ̄)

= ∫_{M^{n}×\{p\}} ∏_{i} \frac{∂}{∂x_{i}} θ̄

= ⟨∏_{i}[\frac{∂}{∂x_{i}} θ̄], [M^{n}]⟩

= ⟨∏_{i}[\frac{∂}{∂x_{i}} θ̄], vol_{∧}⟩.

On the other hand

∫_{D^{n}} Φ^{∗}vol_{∧} = ∫_{D^{n}}⟨Φ_{p∗}X_{1} ∧ ... ∧ Φ_{p∗}X_{n}, vol_{∧}⟩dp

= ∫_{D^{n}}⟨∏_{i} \frac{∂}{∂x_{i}} [θ̄], vol_{∧}⟩

so equation 5 holds. □

Proof of proposition 1.5: Using standard properties of Lipschitz functions and a smoothing argument, we will reduce proposition 1.5 to proposition 5.2. The Lipschitz homeomorphism Ψ̂^{∗} : T^{n} × S^{n−1} ⟶ S^{∗}T^{n} defined by composing Ψ̂ : T^{n} × S^{n−1} ⟶ ST^{n} (see lemma 5.1) with the duality diffeomorphism ST^{n} ⟶ S^{∗}T^{n} extends uniquely to a locally Lipschitz homeomorphism Ψ : T^{n} × R^{n} ⟶ T^{∗}T^{n} satisfying

1. Ψ(x, λv) = λΨ(x, v) for λ ≥ 0, 2. Ψ covers id_{T^{n}}, i.e. π ∘ Ψ = π_{1},

3. Ψ(T^{n} × {v}) is the image of a weakly closed (in the sense that the integral around small loops is zero) Lipschitz 1-form for each v ∈ R^{n}.

Define Φ : R^{n} ⟶ H^{1}(T^{n}; R) by letting Φ(v) be the cohomology class of the one-form with image Ψ(T^{n} × {v}); Φ is a locally Lipschitz homeomorphism when H^{1}(T^{n}; R) is given any Euclidean metric. Let vol_{T^{∗}T^{n}} be the natural volume form on T^{∗}T^{n} arising from the Riemannian structure. Now

β(n)V ol(g) = V ol(D^{∗}T^{n}) = ∫_{D^{∗}T^{n}} vol_{T^{∗}T^{n}}
= ∫_{T^{n}×D^{n}} Ψ^{∗}vol_{T^{∗}T^{n}}

where D^{∗}T^{n} = {α ∈ T^{∗}T^{n}||α| ≤ 1}, and

V ol(CI(g)) = ∫_{CI(g)} vol_{∧} = ∫_{D^{n}} Φ^{∗}vol_{∧}

where vol_{∧} is the translation invariant volume form on H^{1}(T^{n}; R) defined before proposition 5.2 (for which the associated measure coincides with the normalized Lebesgue measure defined in the introduction). It now suffices to show that

∫_{T^{n}×D^{n}} Ψ^{∗}vol_{T^{∗}T^{n}} = ∫_{D^{n}} Φ^{∗}vol_{∧}.

Let T^{n} × R^{n} act on itself by translation and on T^{∗}T^{n} by the formula (t, v) · α = L_{−t}^{∗}α for (t, v) ∈ T^{n} × R^{n}, α ∈ T^{∗}T^{n}, where L_{−t} is left translation by −t. Finally, let T^{n} × R^{n} act on Map(T^{n} × R^{n}, T^{∗}T^{n}) by

(g · Ξ)(ḡ) = g · (Ξ(g^{−1} · ḡ))

where g, ḡ ∈ T^{n} × R^{n} and Ξ ∈ Map(T^{n} × R^{n}, T^{∗}T^{n}). If we smooth our map Ψ using a compactly supported bump function ρ ∈ C^{∞}(T^{n} × R^{n}):

Ψ_{ρ}(ḡ) = ∫_{T^{n}×R^{n}} ρ(g)(g · Ψ)(ḡ)dg

then Ψ_{ρ} will still have properties 1 and 2 of proposition 5.2; in particular, we may define Φ_{ρ} : R^{n} ⟶ H^{1}(T^{n}; R) by Φ_{ρ}(v) = the cohomology class of the 1-form on T^{n} with image Ψ_{ρ}(T^{n} × {v}), and this coincides with the cohomology class of (id × v)^{∗}Ψ_{ρ}^{∗}θ ∈ Ω^{1}(T^{n}). Applying proposition 5.2 we get

∫_{D^{n}} Φ_{ρ}^{∗}vol_{∧} = ∫_{T^{n}×D^{n}} Ψ_{ρ}^{∗}(vol_{T^{∗}T^{n}}).

Letting the support of ρ shrink to the identity while preserving ∫_{T^{n}×R^{n}} ρ = 1, we get

∫_{D^{n}} Φ_{ρ}^{∗}vol_{∧} → ∫_{D^{n}} Φ^{∗}vol_{∧}

and

∫_{T^{n}×D^{n}} Ψ_{ρ}^{∗}vol_{T^{∗}T^{n}} → ∫_{T^{n}×D^{n}} Ψ^{∗}vol_{T^{∗}T^{n}}

since Ψ and Φ are Lipschitz; the proposition follows. □

Proof of proposition 1.2: We will show that F is Lipschitz if (T^{n}, g) has bounded asymptotes; the converse follows immediately from theorem 1.1 (there are also direct arguments for the converse). To do this, we will prove in three steps that the holonomy maps S_{p}T^{n} ⟶ S_{q}T^{n} of the foliation F are uniformly Lipschitz. It then follows as in lemma 5.1 that there is a bilipschitz homeomorphism Ψ̂ : T^{n} × S^{n−1} ⟶ ST^{n} with the properties described in lemma 5.1, and this implies that F is Lipschitz.

Step 1. We will use the notation and terminology from section 1 in [E]. Let γ : R ⟶ T̃^{n} be a geodesic covering a closed geodesic γ̄ in T^{n}. Let J(t) be the Lagrange tensor along γ which corresponds to the variations of γ through lifts of closed geodesics freely homotopic to γ̄, with the normalization J(0) = id. Since J corresponds to periodic Jacobi fields along γ̄, there is a constant C_{J} such that |J(t)| ≤ C_{J} for every t. If D_{s} is the Jacobi tensor along γ satisfying D_{s}(s) = 0 and D_{s}(0) = id, then by definition (see [E, p.240]) the stable Jacobi tensor is D(t) = lim_{s→∞} D_{s}(t). Using the equation

D_{s}(t) = J(t) ∫_{s}^{t} J^{−1}J^{∗−1}du (J(0) ∫_{s}^{0} J^{−1}J^{∗−1}du)^{−1} (see [E, equation2]) (6)

and the bound on J, it follows that D(t) = J(t). Hence the stable and unstable Jacobi tensors coincide. We now invoke the bounded asymptotes condition: the Jacobi tensors J_{s}(t) = J(t)J^{−1}(s), being the stable Jacobi tensors for the different points of γ, are uniformly bounded by some constant C (independent of γ); consequently, the stable Jacobi tensor D = J satisfies the upper and lower bounds

|D(t)|, |D^{−1}(t)| ≤ C (7)

for every t. Since the velocities of lifts of periodic geodesics are dense in the unit sphere bundle of T̃^{n} (see proposition 3.2), every geodesic carries a Jacobi tensor which is bounded above and below; by the reasoning above this is (up to a constant matrix factor) the stable Jacobi tensor, so inequality 7 holds for stable Jacobi tensors along arbitrary geodesics in T̃^{n}.

Step 2. Let γ : R ⟶ T̃^{n} be a geodesic, and let A(t) be the Lagrange tensor along γ with A(0) = 0 and A′(0) = id. Then using [E, equation 2] we have

A(t) = D(t) ∫_{0}^{t} D^{−1}D^{∗−1}du

and so inequality 7 leads to

C^{−3}|v||t| ≤ |A(t)v| ≤ C^{3}|v||t|

for every v. Therefore if j is a Jacobi field along γ satisfying j(0) = 0 and |j′(0)| = 1, then j has linear growth:

C^{−3}|t| ≤ |j(t)| ≤ C^{3}|t|. (8)

Step 3. Pick p̄, q̄ ∈ T^{n}, v̄_{1}, v̄_{2} ∈ Sp̄T^{n}. Let w̄_{i} ∈ Sq̄T^{n} be the unique element in Sq̄T^{n} which belongs to the same leaf of F as v̄_{i}, i = 1, 2. Choose p, q ∈ T̃^{n} so that π(p) = p̄, π(q) = q̄, and define v_{i}, w_{i} (with footpoints p and q respectively) by π_{∗}(v_{i}) = v̄_{i}, and π_{∗}(w_{i}) = w̄_{i}. For i = 1, 2, let γ_{i} : R ⟶ T̃^{n} be the geodesic with γ′_{i}(0) = v_{i}, let η_{i} : R ⟶ T̃^{n} be the geodesic with η′_{i}(0) = w_{i}, and let η_{i}^{s} : [0, l_{i}^{s}] ⟶ T̃^{n} be the geodesic segment from q to γ_{i}(s). By lemma 3.5, we have lim_{s→∞}(η_{i}^{s})′(0) = η′_{i}(0) = w_{i}. Also, |l_{i}^{s} − s| ≤ d(p, q). Since the inequality 8 is equivalent to the statement that exp_{p} : T_{p}T̃^{n} ⟶ T̃^{n} and exp_{p}^{−1} have Lipschitz constant ≤ C^{3} for every p ∈ T̃^{n}, we have

(v_{1}, v_{2}) = ^{}(γ′_{1}(0), γ′_{2}(0))

∼ lim inf_{s→∞} \frac{d(γ_{1}(s), γ_{2}(s))}{s} = lim inf_{s→∞} \frac{d(η_{1}^{s}(l_{1}^{s}), η_{2}^{s}(l_{2}^{s}))}{s}

∼ lim inf_{s→∞} ^{}((η_{1}^{s})′(0), (η_{2}^{s})′(0))

= ^{}(η′_{1}(0), η′_{2}(0)) = ^{}(w_{1}, w_{2})

where a ∼ b means that C_{1}^{−1}a ≤ b ≤ C_{1}a for some constant C_{1} depending only on the metric g. Since ^{}(v_{1}, v_{2}) ∼ ^{}(w_{1}, w_{2}), the holonomy maps S_{p}T^{n} ⟶ S_{q}T^{n} are uniformly Lipschitz. □

6 Concluding remarks

It is very tempting to try to prove a version of proposition 1.5 without the hypothesis that F is Lipschitz. This leads one to a modified version of proposition 5.2 where M^{n} is diffeomorphic to T^{n} and the map Ψ : M^{n} × D^{n} ⟶ T^{∗}M^{n} is a (possibly non-Lipschitz) homeomorphism onto its image. In this case the map Φ : D^{n} ⟶ H^{1}(M^{n}; R) is also a homeomorphism, so one can still make sense of both sides of equation 5 as a statement about the measures of two sets. In fact, such a version of proposition 5.2 can be proved when n = 2 using an elementary smoothing argument. This leads to a new proof of E. Hopf’s theorem (see also [CF]). However, the smoothing argument breaks down when n ≥ 3. The problem is that one can’t control the boundary behavior of smooth maps Ψ_{k} approximating Ψ, and so there is no guarantee that ∫_{M^{n}×D^{n}} Ψ_{k}^{∗}vol_{T^{∗}T^{n}} converges to the measure of Ψ(M^{n} × D^{n}) as k → ∞. An elementary argument shows that the modified proposition 1.5 would follow if one can find Lipschitz homeomorphisms Ψ_{k} : M^{n} × D^{n} ⟶ T^{∗}M^{n} converging uniformly to Ψ : M^{n} × D^{n} ⟶ T^{∗}M, such that Ψ_{k}(M^{n} × {v}) is the image of a weakly closed Lipschitz one-form for every v ∈ D^{n}.

Another approach to finishing the Hopf conjecture would be to study the regularity of the foliation F. All examples (known to the authors) of Finsler tori without conjugate points have geodesic flow smoothly conjugate to the geodesic flow of a flat Finsler torus; in particular, for such tori the foliation F is C^{∞}.

References

[Av] A. Avez, Variétés riemanniennes sans points focaux, C. R. Acad. Sci. Paris, Sér. A-B, 270, (1970), A188-A191.

[Ba] W. Ballmann, Nonpositively curved manifolds of higher rank, Ann. of Math., 122, (1985), 597-609.

[BBB] W. Ballmann, M. Brin, K. Burns, On surfaces with no conjugate points, J. Diff. Geom., 25, (1987), no. 2, 249-273.

[BGS] W. Ballmann, M. Gromov, V. Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics vol. 61, Birkhäuser, 1985.

[BuKn] K. Burns, G. Knieper, Rigidity of surfaces with no conjugate points, J. Diff. Geom., 34, (1991), no. 3, 623-650.

[BuSp] K. Burns, R. Spatzier, Manifolds of nonpositive curvature and their buildings, Pub. I.H.E.S., (1987), no. 65, 35-59.

[Bus] H. Busemann, The Geometry of Geodesics, Academic Press, 1955.

[ChEb] J. Cheeger, D. Ebin, Comparison theorems in Riemannian geometry, North-Holland, 1975.

[Cr1] C. Croke, Rigidity and the distance between boundary points, J. Diff. Geom., 33, (1991), no. 2, 445-464.

[Cr2] C. Croke, Simply connected manifolds with no conjugate points which are flat outside a compact set, Proc. Amer. Math. Soc., 111, (1991), no. 1, 297-298.

[Cr3] C. Croke, Volumes of balls in manifolds without conjugate points, Int. J. Math., 3, no. 4 (1992), 455-467.

[CF] C. Croke, A. Fathi, An inequality between energy and intersection, Bull. Lond. Math. Soc., 22, (1990), 489-494.

[CrKl] C. Croke, B. Kleiner, Gap theorems for manifolds without conjugate points, in preparation.

[CS] C. Croke, V. Schroeder, The fundamental group of compact manifolds without conjugate points, Comment. Math. Helv., 61, (1986), 161-175.

[E] J. Eschenburg, Horospheres and the stable part of the geodesic flow, Math. Zeit, 153, (1977), 237-251.

[EsOS] J. Eschenburg, J. O’Sullivan, Growth of Jacobi fields and divergence of geodesics, Math. Z., 150, (1976), no. 3, 221-237.

[FrMa] A. Freire, R. Mané, On the entropy of the geodesic flow in manifolds without conjugate points, Invent. Math., 69, (1982), no. 3, 375-392.

[G] L. Green, A theorem of E. Hopf, Mich. Math. J., 5, (1958), 31-34.

[GrGu] L. Green, R. Gulliver, Planes without conjugate points, J. Diff. Geom., 22, (1985), no. 1, 43-47.

[GrWu] R. Greene, H. H. Wu, On a new gap phenomenon in Riemannian Geometry, Proc. Nat. Acad. Sci., 79, (1982), no. 2, part 1, 714-715.

[GrWo] D. Gromoll, J. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bull. A.M.S., 77, (1971), 545-552.

[Grm] M. Gromov, Volume and bounded cohomology, Pub. I.H.E.S., no. 56, (1982), 5-99.

[He] J. Heber, On the geodesic flow of tori without conjugate points, (to appear in Math Z).

[HeMo] G. Hedlund, M. Morse, Manifolds without conjugate points, Trans. Amer. Math. Soc., 51, (1942), 362-386.

[H] E. Hopf, Closed surfaces without conjugate points, Proc. Nat. Acad. Sci., 34, (1948), 47-51.

[Kl] B. Kleiner, (in preparation).

[Klbg] W. Klingenberg, Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math., (2), 99 (1974), 1-13.

[K] A. Knauf, Closed orbits and converse KAM theory, Nonlinearity, 3, (1990), 961-973.

[LY] H. B. Lawson, S. T. Yau, Compact manifolds of nonpositive curvature, J. Diff. Geom., 7, (1972), 211-228.

[Man] R. Mané, Ergodic theory and Differentiable Dynamics, Springer-Verlag, 1987.

[Mos] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120, (1965), 286-294.

[OS] J. O’Sullivan, Manifolds without conjugate points, Math. Ann., 210, (1974), 295-311.

[PdM] J. Palis, W. de Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, 1982.

[San] L. Santaló, Un invariante afin para los cuerpos convexos del espacio de n dimensions, Portugal. Math., 8, Fasc. 4, (1949), 155-161.

[Van] W. Vannini, A note on manifolds without conjugate points of cohomogeneity two, Indiana Univ. Math. J. 38 (1989)no.1, 137-140.