GEOMETRY AND REAL-ANALYTIC INTEGRABILITY

LEO T. BUTLER

Abstract. This note constructs a compact, real-analytic, riemannian 4-manifold (Σ, g) with the properties that: (1) its geodesic flow is completely integrable with smooth but not real-analytic integrals; (2) Σ is diffeomorphic to T^{2} × S^{2}; and (3) the limit set of the geodesic flow on the universal cover is dense. This shows there are obstructions to realanalytic integrability beyond the topology of the configuration space.

1. Introduction

Taĭmanov proves that if (Σ, g) is a compact real-analytic manifold whose geodesic flow is integrable with real-analytic first integrals, then there is an invariant torus T in the unit-tangent sphere bundle SΣ such that π_{1}(T) → π_{1}(Σ) is almost surjective [5]. The example in [2] shows that this is false in the smooth category. In that example (Σ, g) is a compact real-analytic riemannian 3-manifold with a nilpotent π_{1} that is not almost abelian.

To state the present note’s results: Let E be the total space of a C^{1}- vector bundle over T^{2} with even Euler number. The inclusion of C into S^{2} = C ∪ {∞} induces a compactification of E into an S^{2}-bundle Σ over T^{2} with a natural action of SO_{2}R on its fibres. Define a metric g on Σ by SO_{2}R-equivariantly identifying T Σ as the orthogonal direct sum of the sub-bundle V ≃ T S^{2} of vertical fibres and a horizontal sub-bundle H ≃ T T^{2} and equip each fibre of V (resp. H) with the metric of the round sphere of unit radius (resp. standard flat metric). There are natural identifications of H and V that allow g to be uniquely defined (see sections 3.3–3.4).

Theorem 1.1. The compact, real-analytic, riemannian 4-manifold (Σ, g) has the following properties

(1) its geodesic flow, φ : R × SΣ → SΣ, is completely (resp. noncommutatively) integrable with smooth but not real-analytic integrals;

(2) Σ is bundle isomorphic to T^{2} × S^{2};

(3) the limit set of its geodesic flow on the universal cover is dense; and (4) its geodesic flow has zero topological entropy.

Since the geodesic flow of (T^{2} × S^{2}, g_{flat} × g_{round}) is real-analytically integrable, the topology of Σ does not preclude real-analytically integrable geodesic flows. How does one know that the first integrals in Theorem 1.1 cannot be real analytic? It is shown that if T ⊂ SΣ is a regular φ-invariant torus, then the subgroup im (π_{1}(T) → π_{1}(Σ)) has rank at most 1. By Taĭmanov’s result, mentioned above, this precludes real-analytic integrability (see sections 3.5–3.6).

Date: November 17, 2006.

2000 Mathematics Subject Classification. 37J30, 37K10, 53C22, 53D25.

Key words and phrases. geodesic flows, integrable systems, momentum map, realanalytic integrability.

The author thanks Alexei Bolsinov and two anonymous referees for their comments.

The geodesic flow φ is more than just smoothly integrable. Let us recall some terminology to describe a type of topologically-tame integrability [4]: A C^{r} (1 ≤ r ≤ ∞) action ϕ : R^{s} × M → M is integrable if there is an open, dense subset R that is covered by angle-action charts (θ, I) : U → T^{k} × R^{l} which conjugate ϕ_{t} (t ∈ R^{s}) with a translation-type map (θ, I) ↦ (θ + ω(I)t, I) where ω : R^{l} → Hom(R^{s}, R^{k}) is a smooth map. Evidently, there is an open dense subset L ⊂ R fibred by ϕ-invariant tori [1]. Let f : L → B be the C^{r} fibration which quotients L by these invariant tori and let Γ = M −L be the singular set. If Γ is a tamely-embedded polyhedron, then ϕ is called k-semisimple with respect to (f, L, B).

Theorem 1.2. The geodesic flow φ is 4-semisimple with respect to an (f, L, B) such that f has non-trivial monodromy and a trivial Chern class; it is also 3-semisimple with respect to an (f′, L′, B′) such that f′ has trivial monodromy and a non-trivial Chern class. The fibres of f′ are contractible in SΣ.

Remarks. (1) The contractibility of the fibres of f′ and the minimality of φ restricted to a generic fibre of f′ implies that φ is not real-analytically integrable (see sections 3.5–3.6). The contractibility of the fibres of f′ also implies the density of the geodesic flow’s limit set on the universal cover.

(2) If Σ is constructed from E with an odd Euler number, then parts 1,3,4 of Theorem 1.1 and Theorem 1.2 are true. The cohomology and homotopy rings of Σ are isomorphic to those of T^{2} × S^{2}, but Σ is not isomorphic to T^{2} ×S^{2}. (3) The example of Theorem 1.1 generalizes via the non-degenerate 2-step nilmanifolds of [3]. These examples are multiply-connected. Are there simply-connected examples? (4) Theorem 1.1 arose from an attempt to understand whether a hyperbolic manifold admits a semisimple geodesic flow. If so, then its invariant tori are contractible and property 3 of Theorem 1.1 is true — morally, at least. See Theorems 6–7 and Question C of [4] for precise statements. The examples of the present note suggest that there may exist a semisimple geodesic flow on a hyperbolic manifold.

2. The topology of Σ

Terminology: Let G be a group. The anti-diagonal subgroup is the subgroup {(g, g^{−1}) : g ∈ G} of G × G. If X and Y are G-spaces, then define X ×_{G} Y to be the set of orbits of the anti-diagonal subgroup.

Since Σ is an S^{2}-bundle over T^{2}, there is a canonical principal SO_{3}Rbundle SO_{3}R ↪ P → T^{2} such that Σ = P ×_{SO_{3}R} S^{2} is an associated bundle.

Lemma 2.1. The Euler number of E is even iff P (hence Σ) is trivial.

Proof. It suffices to show that P admits a section under the stated condition. To do this, let us recall some obstruction theory. Let G be a compact, connected Lie group and let G ↪ B → X be a principal G-bundle over a CW complex X. Let us try to construct a section of B. Let X_{p} be the pskeleton of X and let D_{p} be X’s cellular chain group in dimension p. Assume that s_{p} : X_{p} → B|X_{p} is a section of B|X_{p}. The obstruction to extending s_{p} to a section s_{p+1} over X_{p+1} is a cochain θ ∈ Hom(D_{p+1}; π_{p}(G)) – the coefficients are untwisted because B is a principal fibre bundle.

The cochain θ can be defined on the generators of D_{p+1} as follows. Let f_{i} : (D^{p+1}, S^{p}) → (X_{p+1}, X_{p}) be the attaching map of a p-cell and τ_{i} = f_{i,∗}[D^{p+1}, S^{p}] be the induced cellular chain (orient D^{p+1} arbitrarily). The pullback bundle f_{i}^{∗}B is trivial since D^{p+1} is contractible, so let μ_{i} : f_{i}^{∗}B → G be a principal fibre-bundle map. Define ⟨θ, τ_{i}⟩ ∈ π_{p}(G) to be the homotopy class of the map

\xymatrix{S^{p} \ar[r]^{f_{i}} & X_{p} \ar[r]^{s_{p}} & B|X_{p} \ar[r]^{incl.} & f_{i}^{∗}B \ar[r]^{μ_{i}} & G.}

The obstruction cochain θ measures the compatibility of the trivialization of B|X_{p} induced by s_{p} and the trivialization of f_{i}^{∗}B induced by the contractibility of D^{p+1}.

B|X_{1} admits a section s_{1} since π_{1}(B) acts trivially on π_{∗}(G). Assume that X_{2} contains a single 2-cell with attaching map f. The obstruction cochain is the homotopy class of

\xymatrix{S^{1} \ar[r]^{f} & X_{1} \ar[r]^{s_{1}} & B|X_{1} \ar[r]^{incl.} & f^{∗}B \ar[r]^{μ} & G.}

Let E_{1} ⊂ E be the unit-circle bundle, which is also a principal SO_{2}Rbundle. When (G, B, X) = (SO_{2}R, E_{1}, T^{2}), the above discussion shows that the obstruction cochain θ lies in Hom(Z; π_{1}(SO_{2}R)) = Z and it can be identified with the Euler class of E. When (G, B, X) = (SO_{3}R, P, T^{2}), the obstruction cochain θ′ lies in Hom(Z; π_{1}(SO_{3}R)) = Z_{2}. Moreover, it is clear that the map π_{1}(SO_{2}R) → π_{1}(SO_{3}R) maps θ → θ′, i.e. the reduction of θ mod 2 is θ′. This proves the lemma. □

3. The metrics

The metric g is constructed from an SO_{2}R × SO_{2}R-invariant metric on E_{1} × S^{2} using the representation

Σ = E_{1} ×_{SO_{2}R} S^{2}.

We construct an SO_{2}R-invariant metric on E_{1}, first.

3.1. E_{1}. The unit-circle bundle E_{1} is diffeomorphic to the nilmanifold

Γ\N,

where N = Nil is the unipotent group 3 × 3 upper triangular matrices and

Γ = ⟨α = ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , β = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) , γ = ( \begin{matrix}1 & 0 & \frac{1}{k} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) ⟩ ,

where k ≠ 0 is the Euler number of E_{1}. Define a riemmannian metric g_{1} on E_{1} by declaring that log α, log β, log γ is an orthonormal, left-invariant frame on N. Let x : Γ\N → S^{1} (resp. y : Γ\N → S^{1}) be the map induced by g ∈ N ↦ ⟨log g, log α⟩ (resp. by β). The map (x, y) : E_{1} → T^{2} is a non-canonical form of the fibre bundle map E_{1} → T^{2}.

The metric g_{1} has the explicit form at Γg

g_{1} = dx^{2} + dy^{2} + (dz − \frac{1}{2}(xdy − ydx))^{2},

where log g = x log α + y log β + z log γ.

The left-trivialization of T^{∗}N induces a trivialization of T^{∗}E_{1} = Γ\N ×𝔫^{∗}, where 𝔫^{∗} is the dual space of N’s Lie algebra. Let a : 𝔫^{∗} → R (resp. b, c) be the linear form induced by log α (resp. log β, log γ). The hamiltonian of g_{1} is

H_{1} = \frac{1}{2}(a^{2} + b^{2} + c^{2}),

which has smooth first integrals

f_{1} = c, f_{2} = exp(−c^{−2}) sin 2π (\frac{a}{c} + y) , f_{3} = exp(−c^{−2}) sin 2π (\frac{b}{c} − x) .

The centre of N is Z(N) = exp(R log γ) while the centre of Γ is Z(Γ) = exp(Z log γ). Z(N)/Z(Γ) acts freely on Γ\N. From the identification SO_{2}R = Z(N)/Z(Γ), we see that the fibres of E_{1} are just the orbits of Z(N)/Z(Γ). Moreover, the momentum map of this SO_{2}R-action on T^{∗}E_{1} is

Ψ_{1}(Γg, μ) = c(μ),

for all g ∈ N, μ ∈ 𝔫^{∗}.

3.2. S^{2}. Let S^{2} = {ξ ∈ R^{3} : |ξ| = 1} and let T^{∗}S^{2} = {(ξ, p) ∈ R^{3} × R^{3} : |ξ| = 1, ⟨ξ, p⟩ = 0} where ⟨, ⟩ is the euclidean metric on R^{3}. Let g_{2} be the round metric on S^{2} induced by the euclidean metric. Let μ_{i} be the standard orthonormal basis of so_{3}R and define the 1-form α_{i} at ξ to be α_{i}(•) = ⟨•, μ_{i}ξ⟩. The round metric is expressed as

g_{2} = α_{1}^{2} + α_{2}^{2} + α_{3}^{3} = dr^{2} + 4π^{2} sin^{2} r dϕ^{2},

where (r, ϕ mod 1) are radial coordinates in which the vector field μ_{1}ξ equals \frac{1}{2π} \frac{∂}{∂ϕ}.

The momentum map of the SO_{2}R action on T^{∗}S^{2} is

Ψ_{2}(ξ, p) = ⟨p, μ_{1}ξ⟩ = \frac{1}{2π} p_{ϕ},

while the hamiltonian of the metric g_{2} is

H_{2} = \frac{1}{2} (⟨p, μ_{1}x⟩^{2} + ⟨p, μ_{2}x⟩^{2} + ⟨p, μ_{3}x⟩^{2}) = \frac{1}{2} (p_{r}^{2} + (2π sin r)^{−2} p_{ϕ}^{2}) .

3.3. E_{1}×S^{2} and Σ. The riemannian metric g_{1}×g_{2} has hamiltonian H_{1}+H_{2} and first integrals f_{1}, f_{2}, f_{3} & Ψ_{1}, Ψ_{2} [N.B. f_{1} = Ψ_{1}]. The metric is also SO_{2}R ×SO_{2}R-invariant, hence it is invariant under the anti-diagonal action of SO_{2}R. Therefore, there is a well-defined submersion metric g on Σ = E_{1} ×_{SO_{2}R} S^{2}. The momentum map of the anti-diagonal SO_{2}R-action is Ψ = Ψ_{1} − Ψ_{2}.

It is well-known that T^{∗}Σ is symplectomorphic to Ψ^{−1}(0)/SO_{2}R. It is trivial to verify that

Lemma 3.1. H_{1}, H_{2}, f_{1}, f_{2}, f_{3} are functionally independent on an open dense subset of Ψ^{−1}(0). Moreover, H_{1}, H_{2}, f_{1}, f_{2} Poisson commute while H_{1}, H_{2}, f_{1} Poisson commute with f_{2}, f_{3}.

This Lemma proves that the geodesic flow of (Σ, g) is completely integrable with the integrals induced by H_{1}, H_{2}, f_{1}, f_{2}; and it is non-commutatively integrable with the integrals induced by H_{1}, H_{2}, f_{1}, f_{2}, f_{3}. Note that f_{2} and f_{3} are only smooth.

3.4. An explicit expression for g. Let Ê_{1} = N/Z(Γ), the universal abelian covering space of E_{1}, and let Σ̂ = Ê_{1} ×_{SO_{2}R} S^{2} be the universal covering space of Σ. There is the commutative diagram of riemannian im/submersions

\xymatrix{(R^{2} × S^{1} × S^{2}, K̂) \ar@{.>}[d]^{ρ̄} \ar@{.>}[r]^{W} & (Ê_{1} × S^{2}, ĝ_{1} + g_{2}) \ar[r] \ar[d] & (Ê_{1} × S^{2}, g_{1} + g_{2}) \ar[d] \\ (R^{2} × S^{2}, k̂) \ar@{.>}[r]^{w} & (Σ̂, ĝ) \ar[r] & (Σ, g),}

where dotted arrows indicate maps that remain to be defined. An explicit formula for g in local coordinates is provided by k̂.

Let (x, y, z mod 1) be coordinates on Ê_{1} = N/Z(Γ) and let (r, ϕ mod 1) be polar coordinates on S^{2} (section 3.1–3.2). Identify SO_{2}R with R/Z. The anti-diagonal action of SO_{2}R on Ê_{1} × S^{2} is

θ ∗ (x, y, z mod 1, r, ϕ mod 1) = (x, y, z + θ mod 1, r, ϕ − θ mod 1),

for all θ in SO_{2}R. Let s = z + ϕ and t = z, so that (x, y, r, s, t) is a system of local coordinates on Ê_{1} × S^{2}. This defines W; since the SO_{2}Rorbits are the circles (x, y, r, s) = constant, this also defines ρ̄ and w. Since K̂ = W^{∗}(ĝ_{1} + g_{2}),

K̂ = dx^{2} + dy^{2} + (dt − \frac{1}{2}(xdy − ydx))^{2} +
dr^{2} + 4π^{2} sin^{2} r (ds − dt)^{2}.

The frame \{\frac{∂}{∂x} − \frac{1}{2}y \frac{∂}{∂s}, \frac{∂}{∂y} + \frac{1}{2}x\frac{∂}{∂s}, \frac{∂}{∂r}, \frac{∂}{∂s}\} on R^{2} × S^{2} horizontally lifts to

the frame \{X = \frac{∂}{∂x} − \frac{1}{2}y (\frac{∂}{∂s} + \frac{∂}{∂t}) , Y = \frac{∂}{∂y} + \frac{1}{2}x (\frac{∂}{∂s} + \frac{∂}{∂t}) , R = \frac{∂}{∂r},

S = \frac{∂}{∂s} + \frac{4π^{2} sin^{2} r}{1+4π^{2} sin^{2} r}\frac{∂}{∂t}\} on R^{2} × S^{1} × S^{2}. Along with T = \frac{∂}{∂t}, this forms a

K̂-orthogonal frame. A simple calculation shows that X , Y and R have unit norm and the norm of S is \sqrt{\frac{4π^{2} sin^{2} r}{1+4π^{2} sin^{2} r}}. Therefore

k̂ = dx^{2} + dy^{2} + dr^{2} + \frac{4π^{2} sin^{2} r}{1 + 4π^{2} sin^{2} r} (ds − \frac{1}{2}(ydx − xdy))^{2}.

Remarks. (1) Clearly k̂|_{\{x=const.,y=const.\}} equals dr^{2} + \frac{4π^{2} sin^{2} r}{1+4π^{2} sin^{2} r} ds^{2}, which

is a non-round SO_{2}R-invariant metric on S^{2}. In addition, it is clear that these fibres are totally geodesic. (2) If u is a smooth function which vanishes at r = 0, π, has u′(0) = 2π, u′(π) = −2π and the even derivatives of u vanish at r = 0, π, then

k̂_{u} = dx^{2} + dy^{2} + dr^{2} + u(r)^{2} (ds − \frac{1}{2}(ydx − xdy))^{2}

defines a smooth metric on R^{2} × S^{2}. A simple computation shows that k̂_{u} is invariant under the deck-transformation group and so induces a metric g_{u} on Σ. Theorem 1.1 is true for (Σ, g_{u}), and if u is analytic, then Theorem 1.2 is true also. If u(r) = 2π sin r, then {x = const., y = const.} is a totally geodesic round S^{2} but (Σ, g_{u}) is not isometric to (T^{2} × S^{2}, g_{flat} + g_{round}).

3.5. The limit set. Say that a point is recurrent for a flow if its orbit visits arbitrarily small neighbourhoods of itself in both forward and backward time. The limit set is the closure of the set of recurrent points.

Let (Σ̃, g̃) be the universal riemannian cover of (Σ, g) and let φ̃ be the geodesic flow of (Σ̃, g̃). The functions x, y, a, b, c induce well-defined smooth functions on T^{∗}Σ and hence on T^{∗}Σ̃. Let x̃ (ỹ) be the single-valued function on T^{∗}Σ̃ induced by x (y). The map

ν = (a + cỹ, b − cx̃, c), ν : T^{∗}Σ̃ → R^{3}

is a first integral of φ̃ (it is a coordinatized incarnation of the momentum map of N’s left action on T^{∗}Σ̃). For ν_{o} = (ν_{1}, ν_{2}, ν_{3}) ∈ R^{3} such that ν_{3} ≠ 0, it is apparent that if ν(P) = ν_{o}, then

|x̃(P)| ≤ |\frac{b(P) + ν_{2}}{ν_{3}} | ,  |ỹ(P)| ≤ |\frac{a(P) + ν_{1}}{ν_{3}} | .  (∗)

On SΣ̃ the functions a and b are bounded above by unity, so

Lemma 3.2. The map ν|{c ≠ 0} ∩ SΣ̃ is proper.

Let S ⊂ SΣ be a regular invariant 3-torus and let S̃ ⊂ SΣ̃ be a lift of this torus. Since S̃ is regular it lies in {c ≠ 0}, and therefore it is a closed subset of a fibre of ν. Hence S̃ is compact. But φ̃|S̃ is a translation-type flow, so its limit set is S̃. Since the union of regular tori is dense, this proves that the limit set is dense. Since the limit set is closed, it is SΣ̃. This proves part 3 of Theorem 1.1.

3.6. Real-analytic non-integrability. Let’s complete the proof of part 1 of Theorem 1.1. For the next seven paragraphs inclusive (Σ, g) is a compact real-analytic riemannian manifold with geodesic flow φ and first Betti number q. Assume that φ is real-analytically integrable (or more generally, geometrically simple [5]) and let T be the induced singular fibration of SΣ. A regular fibre of T is an isotropic torus. Let the lift of • on SΣ to the universal cover SΣ̃ be denoted by ˜•.

Lemma 3.3. There is an open set U of fibres of T such that for each T ∈ U, T̃ is diffeomorphic to a cylinder T^{r} × R^{q} where r + q ≤ dim Σ.

Proof. Taĭmanov’s Theorem says that there is an open set U ⊂ T of regular invariant tori such that for each T ∈ U the π_{1}-image of T → Σ has finite index. Therefore, there is a splitting T = T^{r} × T^{q} where π_{1}(T) → π_{1}(Σ) factors through an injection π_{1}(T^{q}) → π_{1}(Σ). □

Remark. It is clear from the construction of action-angle variables that the splitting of T̃ can be made compatible with its tautological affine structure.

Let us continue with the notation and hypotheses of Lemma 3.3. In addition, Lemma 3.4. Let S be a singular fibration of SΣ whose regular fibres are s-dimensional φ-invariant tori. If there is a dense set of fibres of S̃ on which φ̃ is minimal, then s ≤ r.

Proof. Recall that a flow is minimal if every orbit is dense. Let v ∈ SΣ̃ be a regular point of each singular fibration and let S̃ ∈ S̃ (resp. T̃ ∈ T̃) be the regular S̃-fibre (resp. T̃-fibre) through v. It can be assumed that v is chosen so that φ̃|S̃ is minimal and T̃ ∈ Ũ. Since φ̃|S̃ is minimal, v is a recurrent point for φ̃|T̃. Therefore, all points of T̃ are recurrent. Let T̃ = T^{r} × R^{q} be a splitting from Lemma 3.3 which is compatible with the tautological affine structure on T̃ and let v = (v′, v′′) relative to this splitting. Since v is a recurrent point for φ̃|T̃, the orbit closure \overline{φ̃_{R}(v)} must be contained in T^{r} × \{v′′\}. Since φ̃|S̃ is minimal, \overline{φ̃_{R}(v)} = S̃. Therefore S̃ ⊂ T^{r} × \{v′′\}. Thus s ≤ r. □

Proof of Part 1 of Theorem 1.1. Let us return to the (Σ, g) constructed in sections 3.1–3.4. In this case q = 2 and S is the singular fibration of SΣ by the connected components of the common level sets of H_{1}, H_{2}, f_{1}, f_{2} and f_{3}, so s = 3. The inequalities (*) and the subsequent argument shows that every regular fibre of S̃ is a 3-torus. The geodesic flow φ is minimal on a dense set of regular fibres of S, hence S̃. This follows from equation (9) of [2], which constructs action-angle variables for the geodesic flow of (E_{1}, g_{1}). By Lemma 3.3, if φ is real-analytically integrable, then r ≥ s = 3. Since r + q ≤ 4, q ≤ 1. As q = 2, this is absurd. Therefore, φ is not real-analytically integrable in either the commutative or non-commutative sense. This proves part 1 of Theorem 1.1.

Remark. The fact that φ is non-commutatively integrable appears to be important for the above proof of real-analytic non-integrability. This is mistaken. The hamiltonian H′_{1} = (2 + sin 2πy)H_{1} on T^{∗}E_{1} plus H_{2} on T^{∗}S^{2} induces a real-analytic metric g′ on Σ whose geodesic flow φ′ is completely integrable with integrals H′_{1}, H_{2}, f_{1} and f_{2}. Construction of action-angle variables for the induced singular toral fibration S shows that φ′ is minimal on a generic fibre of S. The inequality (*) for ỹ shows that the regular fibres of S̃ can only go to infinity in the x̃-direction, so they are either compact T^{4} or cylinders T^{3} × R^{1}.

On the other hand, assume that φ′ is real-analytically integrable with induced singular fibration T . Let v ∈ SΣ be a regular point of both singular fibrations. Let S ∈ S and T ∈ T be the respective fibres through v. By the density of minimal fibres, it can be assumed that φ′|S is minimal and that T ∈ U. Since T is a φ′-invariant torus, minimality implies S ⊂ T. Since T is isotropic, it is a lagrangian torus of the same dimension of S. Therefore, S is open and closed in T, hence S = T. By Lemma 3.3, T̃ splits as T^{2} × R^{2}. By the above comments S̃ = T^{3} × R^{1} or T^{4}. But T̃ is homeomorphic to S̃. Absurd.

4. Semisimplicity

Let Γ′_{1} = {p ∈ T^{∗}Σ : c(2H_{1} − c^{2})(2H_{2} − c^{2}) = 0}, L′_{1} = T^{∗}Σ − Γ′_{1} and B′_{1} = T^{2} × R^{∗} × R^{+} × R^{+}. Define the map f′_{1} : L′_{1} → B′_{1} by

f′_{1} = (\frac{a}{c} + y mod 1, \frac{b}{c} − x mod 1, c, 2H_{2} − c^{2}, 2H_{1} − c^{2}).

This map is a proper submersion whose fibres are isotropic, φ-invariant 3tori. The singular set Γ′_{1} is a real-analytic set, hence a tamely-embedded polyhedron.

Similarly, let Γ_{1} = Γ′_{1}, L_{1} = L′_{1} and B_{1} = T^{1} × R^{∗} × R^{+} × R^{+}. Define the map f_{1} : L_{1} → B_{1} by

f_{1} = (\frac{a}{c} + y mod 1, c, 2H_{2} − c^{2}, 2H_{1} − c^{2}).

This map is a proper submersion whose fibres are isotropic, φ-invariant 4tori. These two constructions, along with the arguments in [4], imply all but the final sentence of Theorem 1.2.

The map f′_{1} extends to a map f′_{2} : L′_{2} = {c ≠ 0} → B′_{2} = T^{2} × R^{∗} × R^{≥0}×R^{≥0}. By the homotopy-lifting theorem, the inclusion of a fibre of f′_{1} is homotopic to the inclusion of the fibre T_{s,t} over (0 mod Z^{2}, 1, s, t) (s, t > 0). As T_{0,0} is an elliptic critical fibre for f′_{2}, and an SO_{2}R-orbit, it follows that T_{s,t} is contractible in L′_{2}. This suffices to complete the proof of Theorem 1.2.

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[4] L. T. Butler. Invariant fibrations of geodesic flows. Topology 44:4 (2005) 769–789.

[5] I. A. Taĭmanov. Topological obstructions to the integrability of geodesic flows on nonsimply connected manifolds. Izv. Akad. Nauk SSSR Ser. Mat., 51:2 (1987) 429– 435, 448.

School of Mathematics, The University of Edinburgh, 6214 James Clerk Maxwell Building, Edinburgh, UK, EH9 3JZ

   E-mail address: l.butler@ed.ac.uk