Conjugacy and Rigidity for Manifolds with a Parallel Vector

Field

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

June, 1992

Abstract

The main theorem in this paper is that any compact Riemannian manifold with geodesic flow isomorphic to the geodesic flow of a local Riemannian product M = (X × R)/Γ is isometric to M.

1 Introduction

In this paper we consider the question: Which compact Riemannian manifolds M are determined uniquely by their geodesic flows? To formulate this precisely we need a few definitions. If M and N are complete Riemannian manifolds then their geodesic flows are C^{0} conjugate if there is a homeomorphism F : SM ⟶ SN from the unit sphere bundle SM to the unit sphere bundle SN which commutes with the geodesic flows: F ∘ g_{M}^{t} = g_{N}^{t} ∘ F for all t ∈ R where g^{t} is the geodesic flow after time t. If 0 ≤ r ≤ ∞ then M and N have C^{r} conjugate geodesic flows if F can be chosen to be a C^{r} diffeomorphism. A complete Riemannian manifold M is C^{r} conjugacy rigid if any Riemannian manifold N whose geodesic flow is C^{r} conjugate to the geodesic flow of M is isometric to M. A more precise formulation of our question is then: Which compact Riemannian manifolds M are C^{r} conjugacy rigid?

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

†Supported by an NSF Postdoctoral Fellowship

It was pointed out by Weinstein (see [Be] sec.4F) that the geodesic flow of a Zoll surface is C^{∞} conjugate to the geodesic flow of a round sphere. Using a variation of this idea we show in section 6 that on any smooth manifold there are infinite dimensional families of pairwise non-isometric metrics with mutually C^{∞} conjugate geodesic flows. In particular, any Riemannian manifold containing an open subset isometric to a neighborhood of an equator S^{n−1}(1) ⊂ S^{n}(1) is not conjugacy rigid.

On the other hand surfaces of nonpositive curvature are C^{0} conjugacy rigid (see [C2] for the C^{1} case and [C-F-F] for the C^{0} case). When both M and N have negative curvature this question is closely related to the question of whether M and N must be isometric if they have the same marked length spectrum (see [O], [C2],[C-F-F]). In [C3] it was shown that if the geodesic flow of a compact n-dimensional Riemannian manifold N is C^{0} conjugate to the geodesic flow of a flat manifold M, and vol(N) = vol(M), then N is isometric to M. Finally, we mention that RP^{n} with its standard metric is C^{0} conjugacy rigid as follows from the Blaschke conjecture for spheres (proved in [Be] appendix D).

The rigidity result in this paper concerns a special class of compact Riemannian manifolds which includes Riemannian products X × S^{1} for any compact Riemannian manifold X, non-product flat tori T^{k}, and Riemannian products X × T^{k} where T^{k} is non-product flat torus. These are manifolds that have a parallel vector field. By the de Rham splitting theorem (see [K-N] p. 187) a complete Riemannian manifold M^{n} has a nontrivial parallel vector field if and only if it is of the form (X × R)/Γ where Γ ⊂ Isom(X) × Isom^{+}(R) ⊂ Isom(X × R), X is a simply connected complete Riemannian manifold, X × R has the product metric, and Isom^{+}(R) are the orientation preserving isometries. Our main result is that compact Riemannian manifolds with a parallel vector field are determined by their geodesic flows:

Theorem 1.1 Let M^{n} be a compact Riemannian manifold with a nontrivial parallel vector field. Then M^{n} is C^{1} conjugacy rigid.

An ingredient in the proof of the above theorem which is of independent interest is:

Proposition 1.2 If M and N are compact Riemannian manifolds with C^{1} conjugate geodesic flows then they have the same volume.

It should be pointed out that it is easy to construct C^{∞} self conjugacies of the geodesic flow of a round sphere that do not preserve the Liouville measure. Hence although C^{1} conjugacies do not have to be measure preserving, proposition 1.2 says that the total measure must be preserved. In particular the result of [C3] thus implies that flat manifolds are C^{1} conjugacy rigid.

In section 3 we study conjugacies between manifolds both of which are nontrivial Riemannian products. A key property for a conjugacy F : S(M_{1} × M_{2}) ⟶ S(N_{1} × N_{2}) between Riemannian products to have, is that it “preserve angles”, i.e. F preserves α (see the paragraph preceding proposition 2.2). One immediate consequence of proposition 3.5 (which was pointed out to us by Pat Eberlein) is

Proposition 1.3 Let M be a compact manifold of nonpositive curvature, and let N = X_{1} × X_{2} × ... × X_{k} where each X_{i} is a compact Riemannian manifold of negative curvature and k ≥ 2. If there is a C^{0} conjugacy F : SM → SN then M is isometric to N.

Outline of the proof of theorem 1.1: Suppose F : SM ⟶ SN is a C^{1} conjugacy of the geodesic flow of M^{n} to the geodesic flow of N^{n}. If vol_{SM} and vol_{SN} are the Liouville volume forms on SM and SN respectively, then F^{∗}vol_{SN} = ϕvol_{SM} for some ϕ ∈ C^{0}(SM). Our first step is to prove that if M has a unit parallel field S then ϕ = 1 on the “vertical” (i.e. {S(m)|m ∈ M}) directions in M (proposition 2.4). We then exploit an extremal property of the Jacobi equation along the vertical geodesics in M^{n} to show that vol(N^{n}) ≥ vol(M^{n}) with equality holding only if F (S) is a parallel vector field on N^{n}. By proposition 1.2 we have vol(M^{n}) = vol(N^{n}) and so F (S) is a parallel field in our situation (proposition 5.3). To complete the proof, we use the behavior of nearly vertical geodesics to see that M and N are isometric (corollary 3.3).

Remark: The conjugacy problem formulated above is very closely related to the boundary rigidity problem discussed in [C1]. A Riemannian manifold (M, ∂M, g_{0}) with boundary ∂M is called boundary rigid if every (N, ∂M, g_{1}) (with diffeomorphic boundary) with d_{g_{0}}(p, q) = d_{g_{1}}(p, q) for every p, q ∈ ∂M, must be isometric to M (d_{g} represents the distance in M between boundary points).

Theorem 1.1 along with the arguments in sections 5 and 7 of [C1], yield the fact that SGM subdomains of compact Riemannian manifolds with a parallel field are boundary rigid. The condition SGM is a condition on d_{g_{0}} (see [C1] sec.1) which says, loosely speaking, that every geodesic segment whose interior lies in the interior of M is the unique geodesic between its endpoints. We mention that Viktor Schroeder had independently noticed that the arguments of [C1] show that SGM subdomains of a manifold which is a product with an interval are boundary rigid.

The other known examples of boundary rigid manifolds are: subdomains M of an open hemisphere of a round sphere (see [Mi] and [Gr] sec.5.5B), compact M^{n} that can be isometrically immersed in R^{n} (see [Gr] sec.5.5B, [Mi], and [C1]), and any SGM surface (two dimensional) of nonpositive curvature [C2].

We would like to thank the referee for his suggestions for making this paper more readable.

2 The behavior of volume under C^{1} conjugacy

In this section we will often be dealing with differential forms θ with continuous coefficients whose exterior derivatives (defined weakly by Stokes’ theorem: ∫_{c} dθ = ∫_{∂c} θ for every smooth chain c) are also differential forms with continuous coefficients. We denote the space of such forms on a smooth manifold M by Ω_{C^{0}}^{∗}(M), and note that Ω_{C^{0}}^{∗}(M) is closed under exterior differentiation, pullback by C^{1} maps, and wedge products.

Lemma 2.1 Let M^{2n−1} be a compact manifold, and let X be a smooth vector field on M. Suppose θ_{0}, θ_{1} ∈ Ω_{C^{0}}^{1}(M) are both invariant under the flow of X and satisfy θ_{i}(X) ≡ 1, i = 1, 2. Then

∫_{M^{2n−1}} θ_{0} ∧ (dθ_{0})^{n−1} = ∫_{M^{2n−1}} θ_{1} ∧ (dθ_{1})^{n−1}

Proof: By Cartan’s identity we have i_{X}dθ_{i} = L_{X}θ_{i} − di_{X}θ_{i} = 0. (Note that each of the terms in the identity is well defined since dθ_{i} ∈ Ω_{C^{0}}^{2}(M), θ_{i} is invariant under the flow of X, and i_{X}θ_{i} ≡ 1; moreover the identity holds as one can see by integrating both sides over arbitrary 1-chains.) Let θ_{t} = (1 − t)θ_{0} + tθ_{1}, and θ̇_{t} = \frac{d}{dt}θ_{t}. Since i_{X} θ̇_{t} = 0 we have i_{X}(θ̇_{t} ∧ (dθ_{t})^{n−1}) = (i_{X} θ̇_{t})∧(dθ_{t})^{n−1}−(n−1)θ̇_{t}∧(i_{X}dθ_{t})∧(dθ_{t})^{n−2} = 0. Hence θ̇_{t}∧(dθ_{t})^{n−1} = 0 and

\frac{d}{dt} ∫_{M^{2n−1}} θ_{t} ∧ (dθ_{t})^{n−1}

= ∫_{M^{2n−1}} θ̇_{t} ∧ (dθ_{t})^{n−1} + (n − 1) ∫_{M^{2n−1}} θ_{t} ∧ (dθ̇_{t}) ∧ (dθ_{t})^{n−2}
= (n − 1) ∫_{M^{2n−1}}[−d(θ_{t} ∧ θ̇_{t} ∧ (dθ_{t})^{n−2}) + dθ_{t} ∧ θ̇_{t} ∧ (dθ_{t})^{n−2}]

which is 0 since M^{2n−1} is closed. Therefore ∫_{M^{2n−1}} θ_{0} ∧ (dθ_{0})^{n−1} = ∫_{M^{2n−1}} θ_{1} ∧ (dθ_{1})^{n−1}. □

We recall that the canonical contact form θ ∈ Ω^{1}(SM) is given by

θ(v)(ξ) = ⟨π_{∗}ξ, v⟩

where v ∈ SM, ξ ∈ T_{v}(SM), and π : SM ⟶ M is the bundle projection. For a discussion of the following facts, we refer the reader to [Be, Chapter 1]:

1. θ is invariant under the geodesic flow;

2. vol_{SM} = \frac{1}{(n−1)!}θ ∧ (dθ)^{n−1} where vol_{SM} is the canonical volume form on SM;

3. ∫_{SM} vol_{SM} = vol(S^{n−1}) ∫_{M} vol_{M} = vol(S^{n−1})vol(M) where vol(S^{n−1}) is the volume of the standard n − 1-sphere.

As a consequence of the preceding lemma, we get Proposition 1.2 of the introduction.

Proof of Proposition 1.2: If F : SM ⟶ SN is a C^{1} conjugacy of geodesic flows, and θ_{0}, θ_{1} are the contact forms on SM and SN respectively, then θ_{0} and F^{∗}θ_{1} ∈ Ω_{C^{0}}^{1}(M) both satisfy the conditions of the previous lemma. Hence volume(SM) = volume(SN) and so M and N have the same volume. □

If we have a Riemannian product M_{1} × ... × M_{k}, then we define α = (α_{1}, ..., α_{k}) : S(M_{1} × ... × M_{k}) ⟶ R^{k} by α_{i}(v) = |π_{i∗}(v)| where π_{i} : M_{1} × ... × M_{k} ⟶ M_{i} is projection onto the ith factor. If M = (M_{1} × ... × M_{k})/Γ where Γ ⊆ Isom(M_{1}) × ... × Isom(M_{k}) ⊆ Isom(M_{1} × ... × M_{k}), then α descends to a map defined on SM, which which we also denote by α.

Proposition 2.2 Let M be a compact Riemannian manifold of the form (M_{1}×...×M_{k})/Γ where Γ ⊆ Isom(M_{1})×...×Isom(M_{k}). If F : SM ⟶ SN is a C^{1} conjugacy for some compact Riemannian manifold N, then

∫_{α^{−1}(Ω)} vol_{SM} = ∫_{F (α^{−1}(Ω))} vol_{SN}

for every domain Ω ⊆ (R^{+})^{k}, where vol_{SM} and vol_{SN} are the Liouville volume forms on SM and SN respectively.

Remark: The proof shows in fact that

∫_{U} vol_{SM} = ∫_{F (U)} vol_{SN}

for U a connected component of α^{−1}(Ω).

Proof: It suffices to consider compact domains Ω ⊂ (0, ∞)×...×(0, ∞) ⊂ R^{k} with smooth boundary. Let θ_{0} be the canonical contact form on SM and let θ_{1} be the pullback of the contact form on SN by the conjugacy F. We want to show that

∫_{α^{−1}(Ω)} θ_{0} ∧ (dθ_{0})^{n−1} = ∫_{α^{−1}(Ω)} θ_{1} ∧ (dθ_{1})^{n−1}.

Let θ_{t} = (1 − t)θ_{0} + tθ_{1}, and θ̇_{t} = \frac{d}{dt}θ_{t}. Differentiating we get

\frac{d}{dt} ∫_{α^{−1}(Ω)} θ_{t} ∧ (dθ_{t})^{n−1}

= ∫_{α^{−1}(Ω)} (θ̇_{t} ∧ (dθ_{t})^{n−1} + (n − 1)θ_{t} ∧ dθ̇_{t} ∧ (dθ_{t})^{n−2})
= ∫_{α^{−1}(Ω)} (n(θ̇_{t} ∧ (dθ_{t})^{n−1}) − (n − 1)d(θ_{t} ∧ θ̇_{t} ∧ (dθ_{t})^{n−2}))
= ∫_{α^{−1}(Ω)} (n(θ̇_{t} ∧ (dθ_{t})^{n−1}) − (n − 1) ∫_{∂(α^{−1}(Ω))} θ_{t} ∧ θ̇_{t} ∧ (dθ_{t})^{n−2}.

We will show that both integrands are zero for small t. If X is the vector field generating the geodesic flow then i_{X} θ̇_{t} = 0 and i_{X}dθ_{t} = 0 so we have i_{X}(θ̇_{t} ∧ (dθ_{t})^{n−1}) = 0 and hence θ̇_{t} ∧ (dθ_{t})^{n−1} = 0. For small t, dθ_{t} is nondegenerate when restricted to Ker(θ_{t}), so for small t there exists a unique continuous vector field Y_{t} satisfying θ_{t}(Y_{t}) = 0 and i_{Y_{t}}dθ_{t} = θ̇_{t} (since the map Y_{t} ↦ i_{Y_{t}}dθ_{t} is a linear isomorphism from Ker(θ_{t}) to the annihilator of X). The uniqueness of Y_{t} together with the fact that θ̇_{t} and dθ_{t} are invariant under the geodesic flow of M imply that Y_{t} is invariant under the geodesic flow as well. The next lemma guarantees that Y_{t} is tangent to the level sets of α; in particular Y_{t} is tangent to ∂(α^{−1}(Ω)). This implies that the form θ_{t} ∧ θ̇_{t} ∧ (dθ_{t})^{n−2} restricts to zero on ∂(α^{−1}(Ω)) since for every x ∈ ∂(α^{−1}(Ω)) either Y_{t}(x) = 0, giving θ̇_{t}(x) = (i_{Y_{t}}dθ_{t})(x) = 0, or Y_{t}(x) ≠ 0 and i_{Y_{t}} (θ_{t} ∧ θ̇_{t} ∧ (dθ_{t})^{n−2}) = 0. We complete the proof of the proposition by noting that V ol_{t} = ∫_{α^{−1}(Ω)} θ_{t} ∧ (dθ_{t})^{n−1} is a polynomial in t since θ_{t} = (1 − t)θ_{0} + tθ_{1}; therefore it must be constant since it is constant for small t. □

Lemma 2.3 Let M be a compact Riemannian manifold of the form (M_{1} × ... × M_{k})/Γ where Γ ⊆ Isom(M_{1}) × ... × Isom(M_{k}). Then any continuous vector field Y on SM which is invariant under the geodesic flow must be tangent to the fibers of α : SM ⟶ R^{k}.

Proof: The geodesic flow shears the fibers of α, forcing invariant vector fields to be tangent to the fibers. To convey the idea of the proof in a simplified setting we give an analog for the lemma in the k = 1 case. Model. Let N be a compact Riemannian manifold, and let Z be a continuous vector field on T N which is invariant under the geodesic flow of N. Then α_{∗}(Z) = 0 where α : T N ⟶ R is defined as before, α(v) = |v|. Proof of Model: Let X and ξ be the geodesic spray and the homothetic vector field on T N, respectively. Let T_{0}N = {v ∈ T N||v| ≠ 0}, and let C be the distribution on T_{0}N whose restriction to each sphere bundle α^{−1}(r) is the canonical contact distribution (i.e. the kernel of the contact form θ). If Φ^{t} : T N ⟶ T N is the geodesic flow after time t, then Φ_{∗}^{t}(ξ) = ξ + tX, Φ_{∗}^{t}(C) = C, and we have a direct sum decomposition T T_{0}N = RX ⊕Rξ ⊕C. Now consider the projection of Z to the Φ^{t} invariant subbundle D = RX⊕Rξ. This is invariant under the geodesic flow, and so it must lie in RX since otherwise the shear Φ_{∗}^{t}(ξ) = ξ + tX will produce arbitrarily long vectors. Hence Z lies in RX ⊕ C, and α_{∗}(Z) = 0. Proof of the lemma: Let T_{0}M = {v ∈ T M||v| ≠ 0}. We extend Y to a flow invariant vector field on T_{0}M using the fiber homothety (i.e if S_{λ} : T M → T M denotes scalar multiplication by λ then Y_{v} = S_{|v|∗}Y_{v/|v|}), and denote the resulting vector field by the same name. For i = 1, ...k let T_{0}M_{i} = {v ∈ T M_{i}||v| ≠ 0}, X̄_{i} be the geodesic spray on T M_{i}, ξ̄_{i} be the homothetic vector field on T M_{i}, and let C̄_{i} be the distribution on T_{0}M_{i} whose restriction to α_{i}^{−1}(r) is the canonical contact distribution. If Φ_{i}^{t} is the geodesic flow on T M_{i} then Φ_{i∗}^{t}(ξ̄_{i}) = ξ̄_{i} + tX̄_{i}, C̄_{i} is invariant under Φ̄_{i}^{t}, and

T T_{0}M_{i} = RX̄_{i} ⊕ Rξ̄_{i} ⊕ C̄_{i}.

We now let X_{i}, ξ_{i}, and C_{i} be the corresponding objects on T_{0}M_{1}×...×T_{0}M_{k} ⊆ T M_{1}×...T M_{k} ≃ T (M_{1}×...×M_{k}). If Φ^{t} is the geodesic flow of T (M_{1}×...×M_{k}), and D_{i} = RX_{i} ⊕ Rξ_{i}, we have a Φ^{t} invariant decomposition

T (T_{0}M_{1} × ... × T_{0}M_{k}) = D_{1} ⊕ C_{1} ⊕ ... ⊕ D_{k} ⊕ C_{k}

and Φ_{∗}^{t}(ξ_{i}) = ξ_{i} +tX_{i}. Now observe that our flow invariant vector field Y lifts to a Φ^{t} invariant vector field on T_{0}M_{1}×...×T_{0}M_{k}, and so we may project this to each subbundle D_{i} to get Φ^{t} invariant sections Y_{i}. Now as in the Model, Φ^{t} shears ξ_{i} into ξ_{i} + tX_{i} and Y_{i} actually sits in RX_{i}; otherwise, by flowing Y_{i} under Φ^{t} we would get arbitrarily long vectors. But this says that Y is a section of RX_{1} ⊕ C_{1} ⊕ ... ⊕ RX_{k} ⊕ C_{k}, so it is tangent to the fibers of α. □

Proposition 2.4 When M = (X × R)/Γ for Γ ⊆ Isom(X) × Isom^{+}(R) is compact, and F : SM ⟶ SN is a C^{1} conjugacy of geodesic flows, then F^{∗}vol_{SN} = ϕvol_{SM} where ϕ(v) = 1 for every vertical vector v ∈ SM (i.e. every v ∈ SM tangent to the local R factor).

Proof: Since vol_{SM}, vol_{SN} are flow invariant and F is a C^{1}conjugacy, ϕ : SM ⟶ R is a continuous, flow invariant function. We will first show that ϕ|_{α_{1}^{−1}(0)} is constant on each of the two components (U^{+} the ‘upward’ component and U^{−} the ‘downward’ component) of α_{1}^{−1}(0). Pulling ϕ back to S(X×R) we get a continuous flow invariant function ϕ̂, and it is enough to see that ϕ̂|_{α_{1}^{−1}(0)} is constant on each component (Û^{+} and Û^{−}). Pick x_{1}, x_{2} ∈ X, and find a sequence of geodesics γ_{n} : R ⟶ X × R such that π_{1}(γ_{n}(0)) = x_{1}, π_{1}(γ_{n}(t_{n})) = x_{2} and α_{1}(γ′_{n}(0)) → 0, α_{1}(γ′_{n}(t_{n})) → 0. Assume that γ′_{n}(0) converges to a vector in Û^{+}. Then the uniform continuity of ϕ̂ implies that

ϕ̂|Û^{+}(x_{1} × R) = lim_{n→∞} ϕ̂(γ′_{n}(0)) = lim_{n→∞} ϕ̂(γ′_{n}(t_{n})) = ϕ̂|Û^{+}(x_{2} × R)

so ϕ̂|Û^{+} ≡ ϕ_{0}^{+} for some ϕ_{0}^{+} ∈ R. Similarly for Û^{−}.

The proposition now follows from proposition 2.2 by integrating ϕvol_{SM} over each component of α_{1}^{−1}([0, ϵ)) and letting ϵ → 0. More precisely, let U_{ϵ}^{+} be the ‘upward’ component of α_{1}^{−1}([0, ϵ)) and apply proposition 2.2 and the remark following it to get for every ϵ > 0,

∫_{U_{ϵ}^{+}} vol_{SM} = ∫_{F (U_{ϵ}^{+})} vol_{SN} = ∫_{U_{ϵ}^{+}} F^{∗}(vol_{SN})

= ∫_{U_{ϵ}^{+}} ϕvol_{SM}
= ϕ_{0} ∫_{U_{ϵ}^{+}} vol_{SM} + ∫_{U_{ϵ}^{+}}(ϕ − ϕ_{0})vol_{SM}.

Dividing by ∫_{U_{ϵ}^{+}} vol_{SM} and letting ϵ → 0 we deduce that ϕ|_{U^{+}} ≡ ϕ_{0}^{+} = 1. □

3 Conjugacy rigidity for products

By a uniform conjugacy F : SM → SN between the unit sphere bundles of complete Riemannian manifolds M and N we mean a C^{0} conjugacy where both F and F^{−1} are uniformly continuous.

Lemma 3.1 Let F : SM → SN be a uniform conjugacy between the unit sphere bundles of complete Riemannian manifolds M and N. Then there is a D > 0 such that for all v, w ∈ SM:

−2D ≤ d_{M}(π_{M}(v), π_{M}(w)) − d_{N}(π_{N}F (v), π_{N}F (w)) ≤ 2D

Proof: The uniform continuity of the conjugacy implies that F (resp. F^{−1}) carries fibers of π_{M} (resp. π_{N}) to sets of uniformly bounded diameter D in SN (resp. SM). For v, w ∈ SM, let τ : [0, l] → M be a minimizing geodesic from π_{M}(v) to π_{M}(w). Then both d_{N}(π_{N}F (v), π_{N}F (τ′(0))), and d_{N}(π_{N}F (w), π_{N}F (τ′(l))), are bounded by D and d_{N}(π_{N}F (τ′(0)), π_{N}F (τ′(l))) = d_{N}(π_{N}F (τ′(0)), π_{N}(g^{l}(F (τ′(0))))) ≤ l = d_{M}(π_{M}(v), π_{M}(w)). Hence the first inequality in the lemma follows from the triangle inequality. The second follows from the same argument applied to F^{−1}. □

Proposition 3.2 For i = 1, 2 let M_{i} = X_{i} × R be complete Riemannian manifolds, let π_{2} : M_{i} ⟶ R be projection onto the second factor, and let dx be the standard 1-form on R. Let U_{i} = {v ∈ SM_{i}|α_{1}(v) = 0, dx(π_{2∗}(v)) > 0}; U_{i} is the set of “upward pointing” elements of SM_{i}. Suppose F : SM_{1} ⟶ SM_{2} is a uniform conjugacy of geodesic flows satisfying F (U_{1}) = U_{2}, and define G : M_{1} ⟶ M_{2} by G(π(u)) = π(F (u)) for every u ∈ U_{1}. Then G is an isometry.

Proof: Our first step is to show that F preserves α. For x_{i} ∈ M_{i} let S_{i}(x_{i}) ∈ U_{i} be the “vertical” vector at x_{i}. Pick x_{i} ∈ M_{i} such that F (S_{1}(x_{1})) = S_{2}(x_{2}). Let h_{i} : M_{i} → R be the Busemann functions coming from the vertical geodesics γ_{i} (i.e. γ′_{i}(0) = S_{i}(x_{i}) and h_{i}(y) = lim_{t→∞} d(γ_{i}(t), y) − t). Let H_{i} : SM_{i} → R be the induced map (i.e. H_{i} = h_{i} ∘ π_{M_{i}}). Lemma 3.1 allows us to conclude that H_{2} ∘ F − H_{1} is bounded. This forces F to preserve α_{2} since for every v ∈ SM_{i}, α_{2}(v) is determined by the growth rate of h_{i} along the geodesic t ↦ exp(tv). Thus F preserves α.

Let Ḡ : X_{1} ⟶ X_{2} be the map induced by G : M_{1} ⟶ M_{2}, i.e. Ḡ(x) = π_{1}(G(x, t) for all (x, t) ∈ X_{1} × R = M_{1}. We first show that Ḡ has Lipschitz constant ≤ 1. Pick x, y ∈ X_{1} and a minimizing geodesic segment σ : [0, l] ⟶ X_{1} from x to y. Define η_{λ} : [0, \sqrt{l^{2} + λ^{2}}] ⟶ X_{1} × R = M_{1} by

η_{λ}(t) = (σ(\frac{tl}{\sqrt{l^{2} + λ^{2}}} ), \frac{tλ}{\sqrt{l^{2} + λ^{2}}} ) ,

so length(σ) = α_{1}(η′_{λ}(t))length(η_{λ}). Let η̄_{λ} : [0, \sqrt{l^{2} + λ^{2}}] ⟶ X_{2} × R be the geodesic in M_{2} corresponding to η_{λ}: η̄_{λ}(t) = π_{M_{2}} ∘F ∘η′_{λ}(t). Let σ̄_{λ} = π_{1} ∘η̄_{λ}. Then

length(σ̄_{λ}) = α_{1}(η̄′_{λ}(t))length(η̄_{λ})
= α_{1}(η′_{λ}(t))length(η_{λ})
= length(σ).

As λ goes to ∞, d(η̄′_{λ}(0), F (S_{1}(x, 0))) and d(η̄′_{λ}(\sqrt{l^{2} + λ^{2}}), F (S_{1}(y, λ))) go to 0. Hence σ̄_{λ}(0) → Ḡ(x) and σ̄_{λ}(\sqrt{l^{2} + λ^{2}}) → Ḡ(y). Hence we see that d(Ḡ(x), Ḡ(y)) ≤ d(x, y) for all x, y ∈ X_{1}. Applying the same reasoning to Ḡ^{−1} we conclude that Ḡ : X_{1} ⟶ X_{2} is an isometry.

To complete the proof that G is an isometry we will show that for every t ∈ R, the set G(X_{1} × {t}) lies at a constant height in X_{2} × R. Choose a geodesic γ : R ⟶ M_{1} of the form γ_{1}(t) = (x_{1}, t) for some x_{1} ∈ X_{1}, and let γ_{2} : R ⟶ M_{2} be the corresponding geodesic in M_{2}. We may assume that γ_{2}(0) = (x_{2}, 0) for some x_{2} ∈ X_{2} since otherwise we may arrange this by composing F with the differential of a translation. Pick m_{1} ∈ X_{1} × {0}, and let η_{λ} : [0, l_{λ}] ⟶ M_{1} be a minimizing geodesic segment from m_{1} to γ_{1}(λ). If η̄_{λ} is the corresponding family of segments in M_{2}, then η̄_{λ}(0) → G(m_{1}) and d(η̄_{λ}(l_{λ}), γ_{2}(λ)) → 0 as λ → ∞ by the uniform continuity of F. Hence

lim sup_{λ→∞} (d(G(m_{1}), γ_{2}(λ)) − length(η_{λ})) = lim sup_{λ→∞} (d(G(m_{1}), γ_{2}(λ)) − λ) ≤ 0.

This forces π_{2}(G(m_{1})) ∈ [0, ∞). Letting λ → −∞ instead of +∞ we get π_{2}(G(m_{1})) ∈ (−∞, 0]. Hence G(X_{1} × {0}) ⊆ X_{2} × {0}. □

Corollary 3.3 Let M_{i} be compact manifolds with unit parallel fields S_{i}. If there is a C^{0} conjugacy F : SM_{1} → SM_{2} such that F (S_{1}(M_{1})) = S_{2}(M_{2}) then M_{1} is isometric to M_{2}.

Proof: If n ≥ 3 then we can lift F to a conjugacy F̃ : SM̃_{1} ⟶ SM̃_{2}, and apply the preceding proposition to get a Γ equivariant isometry G̃ : M̃_{1} ⟶ M̃_{2}. This descends to the desired isometry G : M_{1} ⟶ M_{2}.

The case n = 2 (which concerns only flat tori and Klein bottles) was done in [C2]. □

Similar arguments give results for more general product manifolds.

Proposition 3.4 Let M = M_{1} × M_{2} and N = N_{1} × N_{2} be Riemannian products such that dim(M_{i}) ≠ 0. If there is a uniform conjugacy F : SM → SN which preserves α then M is isometric to N.

Proof: The proof is similar to the proof of proposition 3.2. We will show that M_{1} is isometric to N_{1}; the proof that M_{2} is isometric to N_{2} is the same. Let τ be any geodesic in M_{2}, then there is a unique unit vector field W along M_{1} × τ(0) such that π_{1∗}(W) = 0 and π_{2∗}(W) = τ′(0). We define a map G : M_{1} → N_{1} by G(x) = π_{1}(F (W (x, τ(0)))) which we will show is an isometry. Let x, y ∈ M_{1} and let σ : [0, l] → M_{1} be a minimizing geodesic segment from x to y. Define η_{λ} : [0, \sqrt{l^{2} + λ^{2}}] ⟶ M = M_{1} × M_{2} by

η_{λ}(t) = (σ(\frac{tl}{\sqrt{l^{2} + λ^{2}}} ), τ(\frac{tλ}{\sqrt{l^{2} + λ^{2}}} )) .

Let η̄_{λ} : [0, \sqrt{l^{2} + λ^{2}}] ⟶ N be the geodesic corresponding to η_{λ}: η̄_{λ}(t) = π_{N} ∘ F ∘ η′_{λ}(t). Let σ̄_{λ} = π_{1} ∘ η̄_{λ}. Then as in proposition 3.2 we see that length(σ̄_{λ}) = length(σ) and further that as λ → ∞ , σ̄_{λ} converges to a geodesic segment from G(x) to G(y) so G is distance nonincreasing. Reversing the roles of M and N we see that G is an isometry and the proposition follows. □

Although it is easy to find conjugacies of products where α is not preserved (for example, take F to be the identity on X_{1}×X_{2}×X_{3} where M_{1} = X_{1} while N_{1} = X_{1} × X_{2}) in the generic case one expects α to be preserved. A case in point is the following.

Proposition 3.5 Let M = M_{1} ×M_{2} be a compact Riemannian product such that dim(M_{i}) ≠ 0, and let N = X_{1} ×X_{2} ×...×X_{k} where each X_{i} is a compact Riemannian manifold of negative curvature (or more generally a manifold with Anosov geodesic flow). If there is a C^{0} conjugacy F : SM → SN then M is isometric to N.

Proof of proposition 3.5: We first remark that if Y is a Riemannian manifold of dimension ≥ 3 then the bundle projection SY ⟶ Y induces an isomorphism of fundamental groups (by the exact homotopy sequence of the fibration). Similarly, if [S^{1}, Z] denotes the set of homotopy classes of maps from S^{1} into a space Z, then the bundle projection induces a bijection [S^{1}, SY] ⟶ [S^{1}, Y]. In our situation dim(X_{i}) ≥ 2 for i = 1, ..., k, so dim(M) = dim(N) ≥ 4 and therefore the conjugacy F : SM ⟶ SN induces an isomorphism F_{♯} : π_{1}(M) ⟶ π_{1}(N) and a bijection F_{♭} : [S^{1}, M] ⟶ [S^{1}, N].

Lemma 3.6 For j = 1, 2 let ι_{j} : π_{1}(M_{j}) ⟶ π_{1}(M) be the inclusion of the j^{th} factor, and for i = 1, ..., k let p_{i} : π_{1}(N) ⟶ π_{1}(X_{i}) be the projection onto the i^{th} factor. Then for i = 1, ..., k either

1. p_{i} ∘ F_{♯} ∘ ι_{1} : π_{1}(M_{1}) ⟶ π_{1}(X_{i}) is onto, and p_{i} ∘ F_{♯} ∘ ι_{2} : π_{1}(M_{2}) ⟶ π_{1}(X_{i}) is trivial.

or

2. p_{i} ∘ F_{♯} ∘ ι_{1} : π_{1}(M_{1}) ⟶ π_{1}(X_{i}) is trivial, and p_{i} ∘ F_{♯} ∘ ι_{2} : π_{1}(M_{2}) ⟶ π_{1}(X_{i}) is onto.

Proof: For i = 1, ..., k, the subgroups H_{1} = p_{i} ∘ F_{♯} ∘ ι_{1}(π_{1}(M_{1})), H_{2} = p_{i} ∘ F_{♯} ∘ ι_{2}(π_{1}(M_{2})) ⊆ π_{1}(X_{i}) commute with each other (since their preimages ι_{1}(π_{1}(M_{1})) and ι_{2}(π_{1}(M_{2})) commute with each other) and together they generate π_{1}(X_{i}). But since X_{i} is a compact manifold with Anosov geodesic flow, this forces one of the two subgroups to be trivial and the other to be all of π_{1}(X_{i}). To see this, recall [Kl] that Centralizer(g) ≃ Z for every g ∈ π_{1}(X_{i}) − {e}. Hence if h_{1} ∈ H_{1} − {e}, then H_{2} ⊆ Centralizer(h_{1}) ≃ Z, so if H_{2} ≠ {e} we may pick h_{2} ∈ H_{2} − {e} such that h_{2} = h_{1}^{k} for some k ∈ Z. Then both H_{1} and H_{2} are contained in Centralizer(h_{2}) ≃ Z, which contadicts the fact that ⟨H_{1}, H_{2}⟩ = π_{1}(X_{i}) ≄ Z. □

By the lemma, we may renumber the factors X_{i} so that case 1 occurs for i = 1, ..., m and case 2 occurs for i = m + 1, ..., k. Now let N_{1} = ∏_{i=1}^{m} X_{i} and N_{2} = ∏_{i=m+1}^{k} X_{i}. We recall [Kl] that if X is a compact Riemannian manifold with Anosov geodesic flow, then each free homotopy class of closed curves in X contains a unique closed geodesic, and periodic orbits of the geodesic flow are dense in SX. Since N is a Riemannian product of such manifolds, freely homotopic closed geodesics in N have the same length, and periodic orbits of the geodesic flow of N are dense in SN. Therefore we may apply the following lemma to conclude the proof of proposition 3.5:

Lemma 3.7 Let Y = Y_{1} × Y_{2} and Ȳ = Ȳ_{1} × Ȳ_{2} be compact Riemannian manifolds of dimension ≥ 3, such that

1. Periodic orbits of the geodesic flow of Y are dense in SY.

2. Freely homotopic closed geodesics in Y have the same length.

Letting p_{i}’s denote projections onto i^{th} factors, assume that there is a conjugacy F : SY ⟶ SȲ such that the induced isomorphism F_{♯} : π_{1}(Y) ⟶ π_{1}(Ȳ) preserves the product structures of the fundamental groups; in other words, assume that for j = 1, 2 there are isomorphisms (F_{j})_{♯} : π_{1}(Y_{j}) ⟶ π_{1}(Ȳ_{j}) so that p_{j∗} ∘ F_{♯} = (F_{i})_{♯} ∘ p_{j∗}. Then F preserves α.

Proof: If Z is a compact Riemannian manifold satisfying 2, then there is a well-defined function L_{Z} : [S^{1}, Z] ⟶ R given by L_{Z}([γ]) = the length of any closed geodesic in [γ]. Since Y has the Riemannian product metric, Y_{1} and Y_{2} inherit property 2 and we have the relation

L_{Y} ([γ]) = length(γ) = \sqrt{length^{2}(p_{1} ∘ γ) + length^{2}(p_{2} ∘ γ)}

= \sqrt{(L_{Y_{1}} ∘ p_{1∗}([γ]))^{2} + (L_{Y_{2}} ∘ p_{2∗}([γ]))^{2}}

for every closed geodesic γ : S^{1} ⟶ Y. As Ȳ is conjugate to Y, it follows that Ȳ, Ȳ_{1}, and Ȳ_{2} possess property 2 as well; moreover we have the relation

LȲ = \sqrt{(LȲ_{1} ∘ p_{1∗})^{2} + (LȲ_{2} ∘ p_{2∗})^{2}}.

The existence of the conjugacy F : SY ⟶ SȲ implies that L_{Y} = LȲ ∘F_{♭}. If (F_{i})_{♭} : [S^{1}, Y_{i}] → [S^{1}, Ȳ_{i}] is the bijection induced by (F_{i})_{♯} : π_{1}(Y_{i}) → π_{1}(Ȳ_{i}), then p_{i∗} ∘ F_{♭} = (F_{i})_{♭} ∘ p_{i∗}, so we have

\sqrt{(L_{Y_{1}} ∘ p_{1∗})^{2} + (L_{Y_{2}} ∘ p_{2∗})^{2}} = L_{Y} = LȲ ∘ F_{♭}
= \sqrt{(LȲ_{1} ∘ p_{1∗} ∘ F_{♭})^{2} + (LȲ_{2} ∘ p_{2∗} ∘ F_{♭})^{2}}
= \sqrt{(LȲ_{1} ∘ (F_{1})_{♭} ∘ p_{1∗})^{2} + (LȲ_{2} ∘ (F_{2})_{♭} ∘ p_{2∗})^{2}}.

In fact, this implies that L_{Y_{i}} = LȲ_{i} ∘ (F_{i})_{♭} for i = 1, 2. To see this for i = 1, observe that if η ∈ [S^{1}, Y_{1}], then (η, const) ∈ [S^{1}, Y_{1} × Y_{2}] ≃ [S^{1}, Y], and

L_{Y_{1}}([η]) = L_{Y} ([(η, const)]) = LȲ_{1} ∘ (F_{1})_{♭}([η])

since LȲ_{2} ∘ (F_{2})_{♭} ∘ p_{2∗}([(η, const)]) = LȲ_{2} ∘ (F_{2})_{♭}([const]) = LȲ_{2}([const]) = 0. Now if γ : S^{1} → Y is a closed geodesic and γ̄ : S^{1} → Ȳ is the corresponding closed geodesic in Ȳ (i.e. γ̄′ = F ∘ γ′), then

α_{i}(γ′(t))

= \frac{length(p_{i} ∘ γ)}{length(γ)} = \frac{L_{Y_{i}} ∘ p_{i∗}([γ])}{L_{Y} ([γ])}

= \frac{LȲ_{i} ∘ (F_{i})_{♭} ∘ p_{i∗}([γ])}{LȲ ∘ F_{♭}([γ]))} = \frac{LȲ_{i} ∘ p_{i∗}([γ̄])}{LȲ ([γ̄])}

= \frac{length(p_{i} ∘ γ̄)}{length(γ̄)} = α_{i}(γ̄′(t)).

As periodic orbits of the geodesic flow are dense, this implies that α ∘ F = α.

□

Proof of proposition 1.3 using proposition 3.5: If the fundamental group of a compact manifold of nonpositive curvature has a trivial center, then the splitting theorem of Gromoll and Wolf (see [G-W]) implies that the manifold splits isometrically as a nontrivial product if and only if its fundamental group splits as a product. Since a conjugacy induces an isomorphism of fundamental groups in dimensions larger than two, and since the fundamental groups of compact manifolds of negative curvature have trivial center, proposition 3.5 implies proposition 1.3. □

There are other cases where proposition 3.4 may be applied. For example if the M_{i}’s are manifolds all of whose geodesics are closed of the same period then any conjugacy to a nontrivial product manifold must preserve α. We do not know in general what the optimal condition on the M_{i} would be.

4 Jacobi tensors and conjugacy

We begin by studying the image of minimizing geodesics under C^{0} conjugacies. A line in a Riemannian manifold is a geodesic which is the minimizing geodesic between any pair of points on it. A geodesic γ is called recurrent if there exists an increasing sequence t_{i} → ∞ such that γ′(t_{i}) → γ′(0). For a geodesic γ let γ̃ be a lift of γ to the universal cover M̃ of M then we define excess(γ) = excess(γ̃) = lim_{t→∞}t−dM̃(γ̃(0), γ̃(t)), which is possibly infinite. In fact:

Lemma 4.1 Each recurrent geodesic γ on a complete Riemannian manifold M either has excess(γ) = ∞ or it lifts to a line in the universal cover M̃.

Proof: We will show that excess(γ) is 0 or ∞. Thus if excess(γ) < ∞ then excess(γ) = 0. This clearly shows that γ̃ is a ray (i.e. γ̃ minimizes from γ̃(0) to γ̃(t) for all t > 0). But for all s > 0, γ_{s}(t) = γ(t − s) is also recurrent and hence has excess(γ_{s}) = 0. Thus γ̃ is a line.

Assume excess(γ) > 0. Then there is E > 0 and T such that for all t > T, dM̃(γ̃(0), γ̃(t)) ≤ t − E. Let ϵ = E/4. By the recurrence property of γ we can choose t_{i} > T such that d(γ(0), γ(t_{0})) < ϵ, t_{i} + t_{0} < t_{i+1}, and some lift γ̃_{i} of γ to M̃ has dM̃(γ̃_{i}(t_{i}), γ̃(0)) < ϵ and dM̃(γ̃_{i}(t_{i} + t_{0}), γ̃(t_{0})) < ϵ. Let τ̃ be a minimizing geodesic segment in M̃ from γ̃(0) to γ̃(t_{0}) and τ its projection to M. We know that the length L of τ satisfies L ≤ t_{0} − E. By the above we can choose lifts τ̃_{i} of τ such that dM̃(γ̃(t_{i}), τ̃_{i}(0)) < ϵ and dM̃(γ̃(t_{i} + t_{0}), τ̃_{i}(L)) < ϵ. Hence dM̃(γ̃(t_{i}), γ̃(t_{i} + t_{0})) < L + 2ϵ < t_{0} − E/2. Thus excess(γ) = ∞. □

For compact (or finite volume) M the recurrent geodesics are dense in SM (see [M1] theorem 2.3). Hence Lemma 4.1 has as an interesting consequence:

Corollary 4.2 If M is a compact manifold without conjugate points and N is a Riemannian manifold whose geodesic flow is C^{0} conjugate to that of M then N also has no conjugate points.

Proof: We first lift the conjugacy F to a uniform conjugacy between the universal covers F : SM̃ → SÑ. For n ≥ 3 it is clear that such a lift exists. For surfaces of genus greater than one this can also be done (see [C2]). For surfaces of genus one M is flat by Hopf’s theorem [H] and it was shown in [C2] that N must be flat. Hence we can assume we have such a lift.

Now lemma 3.1 tells us that for any geodesic γ, excessF (γ) = excessF (γ̃) is bounded. Now if γ is recurrent then F (γ) is also recurrent and hence F (γ̃) is a line. Thus N has no conjugate points since the recurrent geodesics are dense in SM. □

Let F : SM ⟶ SN be a C^{1} conjugacy. The space of Jacobi fields Ψ along a geodesic γ splits as Ψ = Ψ^{⊥} + Ψ^{t} + Ψ^{b}, where Ψ^{⊥} consists of those Jacobi fields that are perpendicular to γ, Ψ^{t} is spanned by γ′, and Ψ^{b} is spanned by tγ′. The Jacobi fields in Ψ^{⊥} + Ψ^{t} are the ones that come from variations of geodesics that are parameterized by arclength.

For a Jacobi field j in Ψ^{⊥} + Ψ^{t} we define a vector field T j along T γ in SM as the variation field of the variation T γ_{s}, where γ_{s} is a variation of parameterized geodesics whose variation field is j. T j is determined by the fact that π_{M∗}(T j) = j and that the vertical (with respect to the usual connection) component v(T j) of T j is equal to j′, the covariant derivative of j with respect to γ′ when v(T j) and j′ are thought of as tangent vectors perpendicular to γ′. As is easy to see, the T j’s are precisely those vector fields along T γ that are invariant under Dg^{t}. Since F takes parametrized geodesics to parametrized geodesics and is differentiable it induces a map Φ from Ψ^{⊥} + Ψ^{t} to the corresponding Jacobi fields along the image geodesic. More precisely we see F_{∗}(T j) = T Φ(j) and hence π_{N∗}(F_{∗}(T j)) = Φ(j). In particular Φ is a linear isomorphism. For j ∈ Ψ^{⊥} + Ψ^{t} we can write Φ(j) = Φ(j)^{⊥} + c(j)F (T γ) where c(j) is a constant. It is easy to see that Φ(j)^{⊥} is zero if and only if j ∈ Ψ^{t}.

Similar arguments apply to Jacobi tensors. These are tensors fields J of type 1,1 along a geodesic γ which when applied to parallel vector fields P yield a Jacobi field J(P). J will be called perpendicular if J(P) is perpendicular to γ whenever P is perpendicular to γ. We will let J^{⊥} be the corresponding perpendicular Jacobi tensor, i.e. for P perpendicular to γ, J^{⊥}(P) is the perpendicular part of J(P). If we fix a parallel orthonormal basis for γ^{′⊥} we can think of perpendicular Jacobi tensors as matrices whose column vectors represent Jacobi fields with respect to this basis. They are the solutions to J′′(t) + R(t)J(t) = 0 where R(t) is the curvature transformation (see [Grn] or [Be, p. 239] ). Along a geodesic γ_{v}(t) there are two perpendicular Jacobi tensors of particular interest to us. They are called I_{v}(t) and J_{v}(t) and are determined by : I_{v}(0) = Id, I′_{v}(0) = 0, J_{v}(0) = 0, J′_{v}(0) = Id. Let A_{v}(t) = Φ(I_{v})(t) and B_{v}(t) = Φ(J_{v})(t) be the corresponding (not necessarily perpendicular) Jacobi tensors along γ̄_{v} . Strictly speaking in order to define the tensors A_{v}(t) and B_{v}(t) we need to choose an isomorphism from γ_{v}^{′⊥}(0) to γ̄_{v}^{′⊥}(0). It is of course sufficient to choose parallel orthonormal bases along γ_{v} and γ̄_{v}. All equations will be written with respect to such fixed choices. Of course determinants and norms are well defined independent of such choices and vary continuously.

A nonsingular perpendicular Jacobi tensor J is Lagrangian if J(t)′J(t)^{−1} is symmetric. If A_{v}(t) is a nonsingular Lagrangian Jacobi tensor we define a new Lagrangian Jacobi tensor Z_{v}(t) by:

(4.1)  Z_{v}(t) = A_{v}(t) ∫_{0}^{t} A_{v}^{−1}(x)A_{v}^{−1∗}(x)dx.

Note that Z_{v}(0) = 0 and Z′_{v}(0) = A_{v}^{−1∗}(0) which is nonsingular. In which case, for dimension reasons, we can write our perpendicular Jacobi tensor B_{v}^{⊥} uniquely as:

(4.2)  B_{v}^{⊥}(t) = Z_{v}(t)C_{v} + A_{v}(t)D_{v}

where C_{v} and D_{v} are constant matrices.

Lemma 4.3 Let M be a compact Riemannian manifold and F : SM → SN a C^{1} conjugacy. Let v ∈ SM be such that A_{v}(t) = A_{v}^{⊥}(t) is nonsingular Lagrangian. Then det(C_{v}) = ϕ(v) where C_{v} is defined by equation 4.2 and ϕ(v) is defined by F^{∗}vol_{SN} = ϕvol_{SM}.

Proof: Fix v ∈ SM. DF_{v} is a linear isomorphism from T_{v}SM to T_{F (v)}SN which takes X_{M}(v) to X_{N}(v) (X_{M} and X_{N} are the vector fields generating the respective geodesic flows). Let DF_{v}^{⊥} be the linear isomorphism from X_{M}^{⊥}(v) to X_{N}^{⊥}(F (v)) gotten by restricting DF to X_{M}^{⊥} followed by projection to X_{N}^{⊥}. Then ϕ(v) = det(DF (v)) = det(DF^{⊥}(v)). Since the Liouville measure is the Riemannian measure with respect to the canonical Riemannian metric on SM and since we have fixed parallel orthonormal frames along γ_{v} (resp. γ̄_{v}) there is a canonical choice of orthonormal bases for X_{M}^{⊥}(v) (resp. X_{N}^{⊥}(F (v))) which consists of T J_{i}(0) where J_{i} is a Jacobi field with J′_{i}(0) = 0 and J_{i}(0) a basis vector or of the form J_{i}(0) = 0 and J′_{i}(0) is a basis vector. With respect to these bases it is easy to see that

DF_{v}^{⊥} = ( \begin{matrix}A_{v}(0) & B_{v}^{⊥}(0) \\ A′_{v}(0) & B_{v}^{⊥}′(0) \end{matrix} ) = ( \begin{matrix}A_{v}(0) & Z_{v}(0)C_{v} + A_{v}(0)D_{v} \\ A′_{v}(0) & Z′_{v}(0)C_{v} + A′_{v}(0)D_{v} \end{matrix} ) .

Since Z_{v}(0) = 0 and Z′_{v}(0) = A_{v}^{−1∗}(0) we see that:

ϕ(v) = det(DF_{v}^{⊥}) = det ( \begin{matrix}A_{v}(0) & 0 \\ A′_{v}(0) & A_{v}^{−1∗}(0) \end{matrix} ) ( \begin{matrix}I & D_{v} \\ 0 & C_{v} \end{matrix} ) = det(C_{v}).

□

5 Rigidity for manifolds with a parallel field

In this section we will complete the proof of theorem 1.1.

Let S be a unit length parallel field on a compact Riemannian manifold M. For a vertical vector v (i.e. v = S(x)) let γ_{v} be the corresponding geodesic (i.e. γ′_{v}(o) = v) and γ̄_{v} the image geodesic in N. The flow transformations of S are isometries, while the integral curves of S are the geodesics γ_{v} tangent to the local Euclidean de Rham factor. Since S induces a measure preserving flow on M, by the Poincaré recurrence theorem [M1, theorem 2.3] there is a dense set of γ_{v} which are recurrent and hence Lemmas 4.1 and 3.1 (as in the proof of Corollary 4.2) tell us that for all v, lifts of γ_{v} and γ̄_{v} are lines in the universal covers. Hence by Theorem 1 of [E-OS] no bounded Jacobi fields ever vanish along these geodesics. Because of the local product structure, if P is a parallel unit vector field along γ_{v} then i_{v}(t) = P (t), j_{v}(t) = t·P (t) and q_{v}^{s}(t) = si_{v}(t)+j_{v}(t) are Jacobi fields along γ_{v} where q_{v}^{s}(t) is just j_{g^{−s}v}(t + s) (note that the parameter shift has γ_{g^{−s}v}(t + s) = γ_{v}(t)). Since T i_{v} is a bounded vector field along T γ, we have T Φ(i_{v}) = F_{∗}(T i_{v}) is bounded and hence Φ(i_{v}) is bounded, and hence by the above, never vanishes. Also Φ(i_{g^{−s}v})(s + t) = Φ(i_{v})(t) since both are the image of P under F. We also have

sΦ(i_{v})^{⊥}(t) + Φ(j_{v})^{⊥}(t) + (sc(i_{v}) + c(j_{v}))γ̄′_{v}(t) = sΦ(i_{v})(t) + Φ(j_{v})(t) =

= Φ(q_{v}^{s})(t) = Φ(j_{g^{−s}v})(t + s) = Φ(j_{g^{−s}v})^{⊥}(t + s) + c(j_{g^{−s}v})γ̄′_{v}(t).

As w varies over SM and P varies over parallel unit vector fields along γ_{w}, c(j_{w}) varies continuously and hence is bounded. Therefore c(i_{v}) = 0, Φ(i_{v}) = Φ(i_{v})^{⊥}, and Φ(j_{g^{−s}v})^{⊥}(t + s) = Φ(j_{v})^{⊥}(t) + sΦ(i_{v})^{⊥}(t). In particular F maps parallel perpendicular Jacobi fields P along γ_{v} into perpendicular Jacobi fields along γ̄_{v} that never vanish.

If v = S(x) for a parallel vector field S, then using the fact that the metric is a product metric along γ_{v}, it is easy to see that I_{v}(t) = Id and J_{v}(t) = t · Id with respect to a parallel orthonormal frame. The above arguments show that

(5.1)  A_{g^{−s}v}(t + s) = A_{v}(t) , B_{g^{−s}v}^{⊥}(t + s) = sA_{v}^{⊥}(t) + B_{v}^{⊥}(t)

and A_{v}(t) is a nonsingular bounded perpendicular Jacobi tensor.

Lemma 5.1 Let M be a compact Riemannian manifold with a unit parallel field S and let F : SM → SN be a C^{1} conjugacy. Then the map G = π_{N} ∘ F ∘ S is a C^{1} diffeomorphism from M to N. Further if V ∈ T_{x}M is perpendicular to S(x) then DG(V) is perpendicular to DG(S(x)) = F (S(x)). Hence we can define a vector field F (S) on N by F (S)(G(x)) = F (S(x)).

Proof: The action of DG on S(x)^{⊥} is encoded in the tensor A_{S(x)}(t) (i.e. A_{S(x)}(t) represents DG with respect to our fixed choice of bases.) Thus DG takes S(x)^{⊥} to DG(S(x))^{⊥} and is nonsingular, so G : M ⟶ N is a covering map. If n ≥ 3 then π_{M} and π_{N} induce isomorphisms of fundamental groups (since the fibers are simply connected), so G is a composition of maps which are isomorphisms on π_{1}, and therefore G is a diffeomorphism in this case. The fact that DG(S(x)) = F (S(x)) follows from the fact that G takes γ_{S(x)} to F (γ_{S(x)}).

If n = 2 then M is a flat torus (or Klein bottle) and results in [C2] show that N must then be isometric to M. Although there are many self conjugacies of the geodesic flow of a flat two torus they all satisfy the lemma.

□

We note that if v = S(x) for some x ∈ M then A_{v}(t) is a nonsingular lagrangian tensor (i.e. A′_{v}(t)A_{v}^{−1}(t) is symmetric). This follows since A′_{v}(t)A_{v}^{−1}(t) is the second fundamental form of the image under G of a local horizontal slice of M (recall M is locally a product). (Although in general the image under G of a smooth submanifold need only be C^{1} in this case the normal is the image of S(x) and hence is differentiable.).

Lemma 5.2 If M is a compact manifold with a parallel unit field S and F : SM → SN is a C^{1} conjugacy then for all vertical v (i.e. v = S(m) for some m ∈ M) we have

lim_{s→∞} det^{−1/2}[1/s ∫_{−s}^{0} A_{v}^{−1}(x)A_{v}^{−1∗}(x)dx] = 1.

Furthermore the convergence is uniform in v.

Proof: For fixed t equations 4.2 and 5.1 yield

(5.2) Z_{g^{−s}v}(t + s)C_{g^{−s}v} + A_{g^{−s}v}(t + s)D_{g^{−s}v} = Z_{v}(t)C_{v} + A_{v}(t)D_{v} + sA_{v}(t).

Now A_{g^{−s}v}(t + s) = A_{v}(t) (equation 5.1) and

Z_{g^{−s}v}(t + s) = A_{v}(t){∫_{−s}^{t} A_{v}^{−1}(x)A_{v}^{−1∗}(x)dx} =

= A_{v}(t){∫_{−s}^{0} A_{v}^{−1}(x)A_{v}^{−1∗}(x)dx} + A_{v}(t){∫_{0}^{t} A_{v}^{−1}(x)A_{v}^{−1∗}(x)dx} =
= A_{v}(t){∫_{−s}^{0} A_{v}^{−1}(x)A_{v}^{−1∗}(x)dx} + Z_{v}(t)

Thus equation 5.2 yields:

A_{v}(t){∫_{−s}^{0} A_{v}^{−1}(x)A_{v}^{−1∗}(x)dx}C_{g^{−s}v} + Z_{v}(t)C_{g^{−s}v} + A_{v}(t)D_{g^{−s}v}
= Z_{v}(t)C_{v} + A_{v}(t)D_{v} + sA_{v}(t).

Since the norms of C_{v} and D_{v} vary continuously with vertical v, we see that the norms are uniformly bounded. Thus on each side of the equation there is only one term which is not uniformly bounded in s. Thus dividing by s and letting s go to ∞ we see that uniformly in v

lim_{s→∞} 1/sA_{v}(t){∫_{−s}^{0} A_{v}^{−1}(x)A_{v}^{−1∗}(x)dx}C_{g^{−s}v} = A_{v}(t).

Thus the nonsingularity of A_{v}(t) and the fact that det(C_{w}) = 1 for all vertical w (by lemma 4.3 and Proposition 2.4) allow us to conclude that

lim_{s→∞} det^{−1/2}[1/s ∫_{−s}^{0} A_{v}^{−1}(x)A_{v}^{−1∗}(x)dx] = 1.

□

In the next proof we will use the strict convexity of Ξ = det^{−1/2} on the space, SA, of positive definite self adjoint endomorphisms (see [B1] 11.8.9.5), which is an open, convex subset of the linear space of self-adjoint endomorphisms. For fixed S_{0} we define the linear part , L_{S_{0}}, of Ξ by L_{S_{0}}(S) = DΞ_{S_{0}}(S − S_{0}) + Ξ(S_{0}) and the remainder, R_{S_{0}}, of Ξ by Ξ(S) = L_{S_{0}}(S) + R_{S_{0}}(S). The strict convexity of Ξ implies that R_{S_{0}} is non-negative and strictly convex.

Let S be a measurable map from a space P, with probability measure dp, to SA. Integrating Ξ = L_{S_{0}} + R_{S_{0}} with S_{0} = ∫_{P} S(p)dp we see:

(5.3)  ∫_{P} Ξ(S(p))dp = Ξ(∫_{P} (S(p))dp) + ∫_{P} R_{∫_{P} S(p)dp}(S(p))dp.

Proposition 5.3 If M is a compact Riemannian manifold with a unit parallel field S and F : SM → SM is a C^{1} conjugacy then F (S) is a parallel vector field on N.

Proof: We will show that along any vertical geodesic γ̄_{v}, that A_{v}(t) is constant. This will imply that the Jacobi fields coming from variations of integral curves of F (S) are parallel. Now if V is a (locally defined) vector field invariant under the flow of F (S), then V |γ̄_{v} is such a Jacobi field along γ̄_{v} so

∇_{V} F (S) = ∇_{F (S)}V − [F (S), V] = ∇_{F (S)}V = 0.

Hence F (S) is parallel.

On M the vertical flow S_{t}(m) is measure preserving hence, using equation (5.3),

V ol(M) = V ol(N) = ∫_{M} det(DG(m))dm =

= 1/s ∫_{−s}^{0} ∫_{M} det(DG(S_{t}(m)))dmdt = ∫_{M} 1/s ∫_{−s}^{0} det(A_{S(m)}(t))dtdm =

= ∫_{M} det^{−1/2}[1/s ∫_{−s}^{0} A_{S(m)}^{−1}(t)A_{S(m)}^{−1∗}(t)dt]dm+
+ ∫_{M} 1/s ∫_{−s}^{0} R_{Q(v,s)}[A_{S(m)}^{−1}(t)A_{S(m)}^{−1∗}(t)]dtdm

where Q(v, s) = 1/s ∫_{−s}^{0} A_{S(m)}^{−1}(t)A_{S(m)}^{−1∗}(t)dt. But letting s go to infinity lemma 5.2 tells us that

lim_{s→∞} ∫_{M} 1/s ∫_{−s}^{0} R_{Q(v,s)}[A_{S(m)}^{−1}(t)A_{S(m)}^{−1∗}(t)]dtdm = 0.

Now let H(v, a, Q) = ∫_{a}^{a+1} R_{Q}[A_{v}^{−1}(t)A_{v}^{−1∗}(t)]dt. By change of variable H(v, a, Q) = H(g^{a}v, 0, Q). Hence if we let H(v, a) = inf{H(v, a, Q)|Q ∈ SA} ≥ 0 we see that H(v, a) = H(g^{a}v, 0). We note that H(v, a) = 0 if and only if A_{v}^{−1}(t)A_{v}^{−1∗}(t) is constant for t ∈ [a, a + 1]. We now have for N_{s} the greatest integer less than s ≥ 0

∫_{M} 1/s ∫_{−s}^{0} R_{Q(v,s)}[A_{S(m)}^{−1}(t)A_{S(m)}^{−1∗}(t)]dtdm ≥

≥ ∫_{M} 1/s ∑_{a=−N_{s}}^{−1} H(S(m), a)dm = 1/s ∑_{a=−N_{s}}^{−1} ∫_{M} H(g^{a}S(m), 0)dm =

= 1/s ∑_{a=−N_{s}}^{−1} ∫_{M} H(S(m), 0)dm = N_{s}/s ∫_{M} H(S(m), 0)dm.

(In the above we used the facts that dm is invariant under the flow S_{t} induced by S and that g^{a}S(m) = S(S_{a}m).) Thus letting s go to ∞ we see that ∫_{M} H(S(m), 0)dm = 0 and thus H(S(m), 0) = 0 for almost all m and A_{v}^{−1}(t)A_{v}^{−1∗}(t) is independent of t. (Use continuity to get all from almost all.)

Now differentiating the fact that A_{v}^{∗}(t)A_{v}(t) is a constant, we see that A′_{v}(t) = −A_{v}^{−1∗}(t)A_{v}^{′∗}(t)A_{v}(t) while the fact that A_{v}(t) is Lagrangian says that A′_{v}(t) = A_{v}^{−1∗}(t)A_{v}^{′∗}(t)A_{v}(t). Thus we see that A′_{v}(t) = 0 and hence A_{v}(t) is independent of t. □

Proof of Theorem 1.1 The theorem follows immediately from Proposition 5.3 and Corollary 3.3. □

6 Examples

In this section we first give a criterion for two surfaces of revolution (with boundary) to be conjugate, and then we apply this criterion to show that any smooth manifold M^{n} admits highly nonrigid metrics. We construct such examples by gluing in different surfaces of revolution which are isometric near their boundaries. Furthermore, these conjugacies can be arranged to preserve the contact forms, (see section 2 for definition). Let M^{2}(f, a) be the surface of revolution obtained by revolving about the x-axis the graph of the smooth positive function f(x) defined on an interval [0, a]. More generally let M^{n}(f, a) be the corresponding n-dimensional manifold of revolution.

Let Ω be the collection of pairs (f, a) of smooth positive functions f defined on some interval [0, a] such that f(a) = f(0), f has a maximum at some c ∈ (0, a), f′(x) > 0 for x ∈ [0, c), and f′(x) < 0 for x ∈ (c, a]. For y ∈ (f(0), f(c)) there are two points x_{−}^{y} ∈ (0, c), and x_{+}^{y} ∈ (c, a) in f^{−1}(y). Thus f ∈ Ω gives rise to a function

DL_{f}(y) = \frac{ds}{dy}(x_{−}^{y}) − \frac{ds}{dy}(x_{+}^{y}) for y ∈ (f(0), f(c))

where s(x) is the arc length along the graph of y = f(x). Notice that \frac{ds}{dy} is positive at x_{−}^{y} and negative at x_{+}^{y} and larger than 1 in absolute value at both points. We will call (g, b) ∈ Ω compatible with (f, a) ∈ Ω if g agrees with (f, a) in a small neighborhood of 0 and such that DL_{f} = DL_{g}; hence in particular f and g have the same maximum and minimum values. In this case, we see that there are canonical isometries I_{0} and I_{a} mapping a neighborhood of each of the two boundary components of M^{n}(f, a) to a corresponding neighborhood of M^{n}(g, b). We note that for a given (f, a) there are lots of compatible (g, b). In fact we can choose for the increasing part of g any smooth function increasing from f(0) to f(c) as long as \frac{ds}{dy} < DL_{f}(y)−1 and then define the decreasing part of g by DL_{g} = DL_{f} as long as we get smoothness at the maximum.

Lemma 6.1 Let (f, a) and (g, b) be two compatible elements of Ω. Then there is a smooth contact diffeomorphism F between the unit sphere bundles SM^{n}(f, a) and SM^{n}(g, b) which is equal to DI_{0} or DI_{a} near the two boundary components and F ∘ g^{t} = g^{t} ∘ F wherever they are defined.

Let M^{n} be a compact Riemannian manifold (without boundary) such that some M^{n}(f, a) for (f, a) ∈ Ω is a subdomain of M. Then for any compatible (g, b) ∈ Ω we can construct a new smooth Riemannian manifold N^{n} by replacing M^{n}(f, a) with M^{n}(g, b). The lemma tells us that the geodesic flows of M and N are C^{∞} contact conjugate (the conjugacy is the identity outside SM(f, a) ). Thus for any such M^{n} there is a large family of nonisometric N^{n} with geodesic flow C^{∞} contact conjugate to the geodesic flow of M^{n}. Also it is easy to see that on any differentiable manifold there are metrics such as above by, for example, choosing the metric in an open ball to be a metric ball of radius \frac{3}{4}π with constant curvature 1.

Proof of lemma: We will give the proof in 2 dimensions the extension to higher dimensions is clear.

For our surface M^{2}(f, a) (resp. M^{2}(g, b)) we will use coordinates (x, θ) for x ∈ [0, a] (resp. [0, b]) so the metric is (f^{′2}(x) + 1)dx^{2} + (f^{2}(x))dθ^{2}. For a unit vector V on M^{2} we will let ϕ(V) be the angle V makes with the curve x = const. so that Clairaut’s relation along a unit speed geodesic γ(t) says cos(ϕ(γ′(t)))f(x(γ(t))) is constant.

For a unit vector V in M^{2}(f, a) with base at (x_{0}, θ_{0}) such that x_{0} ∈ [0, c] and < V, \frac{d}{dx} >≥ 0 (or x_{0} ∈ [c, a] and < V, \frac{d}{dx} >≤ 0) then the geodesic γ_{V} (t) determined by V will have an increasing (decreasing) x coordinate for t ∈ [0, l_{0}] until the first value l_{0} of t such that f(γ_{V} (l_{0})) = f(x_{0}). Now if W is a similar unit vector in M^{2}(g, b) at (x_{1}, θ_{1}) where ϕ(V) = ϕ(W) and f(x) = g(x_{1}) then we shall show: Claim: l_{0} = l_{1} and θ(γ_{V} (l_{0})) − θ_{0} = θ(γ_{W} (l_{0})) − θ_{1}.

To see the claim it is sufficient (by integration over y) to see that for every y ∈ [f(x_{0}), f(c)] we have \frac{dθ}{dy}(x_{−}^{y}) − \frac{dθ}{dy}(x_{+}^{y}) and \frac{dt}{dy}(x_{−}^{y}) − \frac{dt}{dy}(x_{+}^{y}) are the same for γ_{V} and γ_{W}. By Clairaut’s relation for any y, \frac{dθ}{dt} and \frac{ds}{dt} are the same for γ_{V} and γ_{W} at points with the same y value. Now \frac{dθ}{dy} = \frac{dθ}{dt} \frac{dt}{ds} \frac{ds}{dy} hence combining the results at x_{−}^{y} and x_{+}^{y} and using the fact that (f, a) and (g, b) are compatible we get the result we need. Similarly using \frac{dt}{dy} = \frac{dt}{ds} \frac{ds}{dy} we get the other equality we need, and the claim follows.

We divide the geodesics and hence the unit sphere bundle into two sets: the interior ones (S_{i}M) are those unit vectors tangent to geodesics that never intersect the boundary and the exterior ones (S_{e}M) whose geodesics do intersect the boundary. Any vector V ∈ S_{e}M^{2}(f, a) can be uniquely written as g^{t}(V_{0}) where V_{0} is a unit vector at a point (0, θ) (note that t will be negative if V_{0} points outside M^{2}(f, a) and for the special case where V_{0} is tangent to the boundary take the smallest t in absolute value). Then let V_{1} be the corresponding unit vector at (0, θ) and the claim tells us that F (V) = g^{t}(V_{1}) is well defined and maps S_{e}M^{2}(f, a) smoothly onto S_{e}M^{2}(g, b) and is precisely DI_{0} or DI_{1} near the boundaries. For V ∈ S_{i}M^{2}(f, a) there is a unit vector V_{0} with g^{t}(V_{0}) = V, ϕ(V) = 0 (or π) and the base point (x_{0}, θ_{0}) has x_{0} ≤ c. We will let V_{1} be the corresponding unit vector at (x_{1}, θ_{0}) where f(x_{0}) = g(x_{1}), and define F (V) = g^{t}(V_{1}). Although there is some ambiguity in the choice of V_{0} (and t), the claim tells us that the map F is well defined. Further it is clear that F is C^{∞} and commutes with the geodesic flow. (Note that the geodesic flow can take vectors near \overline{S_{i}M} ∩S_{e}M to vectors in neighborhoods of the boundary where F is just DI_{0} and hence is smooth).

The only question left is whether the map F above preserves the contact form. Since the contact form is preserved by isometries and geodesic flows this is clear for V ∈ S_{e}M^{2}(f, a) and for V near the boundary of S_{i}M^{2}(f, a). Let γ be the geodesic in M^{2}(f, a) with γ(0) = (x_{0}, 0) and γ′(0) tangent to the curve x = x_{0}. Let γ_{1} be the image geodesic. By the S^{1} symmetry and the invariance of the contact form under the geodesic flows, it is sufficient to show that the contact form is preserved at γ′(0). To do this we need to find two independent Jacobi fields in Ψ^{⊥}(γ) that are mapped to Ψ^{⊥}(γ_{1}). One such Jacobi field is the variation field through the geodesics γ_{t} where γ_{t}(0) = (x_{0}+t, 0) and ϕ(γ′_{t}(0)) = 0. This variation (resp. the image variation) will consist of geodesics whose initial tangent vectors are perpendicular to the geodesic θ = 0 and hence the variation Jacobi field will have derivative 0 at t = 0 and hence remain perpendicular to γ (resp. γ_{1}). To find the other Jacobi field we use the S^{1} symmetry of the two spaces. The Killing field \frac{d}{dθ} is a Jacobi field along both γ and γ_{1} and F takes one to the other. The tangential components are easily seen to be |\frac{d}{dθ}(x_{0}, 0)|_{f}γ′(t) and |\frac{d}{dθ}(x_{1}, 0)|_{g}γ′_{1}(t). Since f(x_{0}) = g(x_{1}), the tangential components of \frac{d}{dθ} are taken to each other and hence so are the perpendicular components \frac{d}{dθ}^{⊥}. Except along the geodesic x = c and its image (which can be handled by continuity), \frac{d}{dθ}^{⊥} is a nontrivial Jacobi field which is independent from the other Jacobi field we considered since \frac{d}{dθ}^{⊥}(0) = 0 and thus F is contact. □

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