FINSLER STRUCTURES ON THE 2-SPHERE

SATISFYING K = 1

Robert L. Bryant

Duke University

July 9, 1995

Abstract. After recalling the structure equations of Finsler structures on surfaces, I define the notion of ‘generalized Finsler structure’ as a way of micro-localizing the problem of describing Finsler structures subject to differential geometric curvature conditions. I use this concept to study the geometry of Finsler structures with constant ‘Gaussian curvature’, showing that these can best be understood as more classical geometric structures on the space of geodesics of the given Finsler structures.

I use these ideas to construct examples of non-Riemannian Finsler structures on the 2-sphere whose ‘Gaussian curvature’ is a positive constant and of non-Riemannian complete Finsler structures on the plane whose ‘Gaussian curvature’ is a negative constant.

1. The Structure Equations of Finsler Surfaces

In this section, I will briefly review basic Finsler geometry and Cartan’s construction of the canonical coframing associated to a Finsler structure on a surface. This derivation has been treated in many places, cf. [BaChSh], [Ca], [Ch1], and [GaWi], to name just a few from different points of view and different eras. This section is mainly to fix notation and nomenclature. Throughout this discussion, M will denote a connected, smooth, oriented surface (i.e., 2-manifold).

A Finsler structure on a surface M is a smooth hypersurface Σ^{3} ⊂ T M for which the basepoint projection π : Σ → M is a surjective submersion having the property that for each x ∈ M, the π-fiber Σ_{x} = π^{−1}(x) = Σ∩T_{x}M is a smooth, closed, strictly convex curve enclosing the origin 0_{x} ∈ T_{x}M. If, for each x ∈ M, the curve Σ_{x} is symmetric about 0_{x}, I say that Σ is symmetric.

1991 Mathematics Subject Classification. 53B40, 53A55, 53C60, 53C22, 58G30, 58E25. Key words and phrases. Finsler geometry, differential invariants, geodesic flows. This work was begun during a visit to the Nankai Institute of Mathematics in May 1995.

My thinking on Finsler geometry has benefited enormously from conversations with Shiing-Shen Chern and Zhongmin Shen that took place during that visit. The support of the Nankai Institute and from the National Science Foundation in the form of Grant DMS-9205222 is hereby gratefully acknowledged.

A longer, more expository version of this manuscript dealing with several other issues involving prescribed curvature of Finsler structures on surfaces is available upon request or at the URL

file://www.duke.edu/~bryant/Finsler.dvi

Typeset by AMS-TEX

A differentiable curve γ : [a, b] → M will be said to be a Σ-curve if, for every s in the interval [a, b], the velocity vector γ′(s) lies in Σ. Every immersed curve in M can be reparametrized in an orientation preserving way so as to become a Σ-curve.

1.1. The canonical coframing. I now review Cartan’s natural parallelism on a 3-manifold Σ defining a Finsler structure on M. The following result is due to É. Cartan. I omit the proof, as it is well-known.

Proposition 1. Let Σ ⊂ T M be a Finsler structure on the oriented surface M, with basepoint projection π : Σ → M. Then there exists a unique coframing (ω, θ, η) on Σ with the properties:

(1) ω∧θ is a positive multiple of any π-pullback of a positive 2-form on M.

(2) The tangential lift γ′ of any Σ-curve satisfies (γ′)^{∗}θ = 0 and (γ′)^{∗}ω = dt.

(3) dω∧θ = 0

(4) ω∧dω = θ∧dθ

(5) dω = η∧θ and η∧dθ = 0 Moreover, there exist unique functions S, C and K on Σ so that

dω = η ∧ θ
(1)  dθ = −η ∧ (ω + S θ)
dη = −(K ω + C η) ∧ θ

Remarks. The 1-form ω is called Hilbert’s invariant integral. A Σ-curve γ is a geodesic of the Finsler structure if and only if its tangential lift satisfies (γ′)^{∗}η = 0. (Of course, by definition, (γ′)^{∗}θ = 0.)

The function S vanishes if and only if Σ is the unit circle bundle of a Riemannian metric on M, in which case, the function K becomes the π-pullback of the Gauss curvature of the underlying metric. The function C vanishes if and only if the Finsler structure is what is called Landsberg [BaChSh].

Let (W, T, H) be the triple of vector fields on Σ which is dual to the coframing (ω, θ, η). By the usual formulae, the duals of the structure equations are

(2)

[ H, W ] = T,
[ H, T ] = −W + S T + C H,
[ W, T ] = K H.

These equations will be useful below.

1.2. Generalized Finsler structures. I can now define what I mean by a generalized Finsler structure.

A generalized Finsler structure on a 3-manifold Σ is a coframing (ω, θ, η) which satisfies the structure equations (1). A generalized Finsler structure will be said to be amenable if the leaf space M of the codimension 2 foliation defined by the equations ω = θ = 0 can be given the structure of a smooth surface M in such a way that the natural projection π : Σ → M is a smooth submersion.

Every generalized Finsler structure is locally amenable in the sense that every point of Σ has a neighborhood to which the generalized Finsler structure restricts to be amenable. In fact, the next proposition shows that the difference between the concepts ‘Finsler structure’ and ‘generalized Finsler structure’ is global in nature; every generalized Finsler structure is locally diffeomorphic to a Finsler structure. The proof is routine, so it will be omitted.

Proposition 2. Let Σ be a 3-manifold endowed with an amenable generalized Finsler structure (ω, θ, η). Denote the ωθ-leaf projection by π : Σ → M and define a smooth map ν : Σ → T M by the rule ν(u) = π′(u)(W(u)). Then ν immerses each π-fiber Σ_{x} = π^{−1}(x) as a curve in T_{x}M which is strictly convex towards 0_{x}. Moreover, there is an orientation of M so that the ν-pullback of the canonical coframing induced on the ν-image of Σ coïncides with the given generalized Finsler structure. □

The reader may well wonder why anyone would bother with generalized Finsler structures since they are locally the same as Finsler structures. The reason is that in the study of Finsler structures defined by geometric conditions, such as conditions on the invariants S, C, and K, one is frequently led to solve differential equations in the larger class of generalized Finsler structures since it is this class that is locally defined. Then, as a separate step, one can determine the necessary conditions on a generalized Finsler structure that it actually be a Finsler structure. Thus, generalized Finsler structures provide a natural intermediate stage where problems can be localized and solved without the complication of global issues.

There is a simple necessary and sufficient test for a generalized Finsler structure to be a Finsler structure.

Proposition 3. A generalized Finsler structure (ω, θ, η) on a 3-manifold Σ is a Finsler structure if and only if its ωθ-leaves are compact, it is amenable, and the canonical immersion ν : Σ → T M is one-to-one on each π-fiber. □

The proof is straightforward. Note, however, that just having the ωθ-leaves be compact does not make the structure amenable since one could have a discrete subset of the leaves around which the foliation is not locally a product, the socalled ‘ramified’ circles. In this case, the leaf space will have the structure of a 2-dimensional orbifold near the ramified points. Moreover, even if the structure is amenable, the ν-image of each π-fiber Σ_{x} will in general be a closed, strictly convex curve in T_{x}M which winds around the origin μ times for some positive integer μ. This number μ, the multiplicity of the generalized Finsler structure, is equal to 1 if and only if each Σ_{x} is embedded via ν. Both sorts of phenomena can occur in practice.

1.4. The Bianchi identities. The structure equations of a generalized Finsler structure (ω, θ, η) on a 3-manifold Σ are

\begin{aligned}dω & = η ∧ θ \\ dθ & = −η ∧ (ω + S θ) \\ dη & = −(K ω + C η) ∧ θ \end{aligned}

where S, K, and C are smooth functions on Σ. Computing the exterior derivative of the second equation^{1} yields

0 = d(dθ) = −dη ∧ (ω + S θ) + η ∧ (dω + dS ∧ θ + S dθ)
= (K ω + C η) ∧ θ ∧ (ω + S θ) + η ∧ (η ∧ θ + dS ∧ θ − S η ∧ (ω + S θ))
= (dS − C ω) ∧ θ ∧ η,

which implies that there exist functions S_{2} and S_{3} so that

(2a)  dS = C ω + S_{2} θ + S_{3} η.

The derivative of the third equation yields, after some simplification and arrangement,

0 = −(dK ∧ ω + (dC + KS ω) ∧ η) ∧ θ,

which implies that there exist functions K_{1}, K_{2}, K_{3}, C_{2}, and C_{3} so that

(2 b,c)  \begin{matrix}dK = & K_{1} ω & + K_{2} θ & + K_{3} η \\ dC = & (K_{3} − KS) ω & + C_{2} θ & + C_{3} η \end{matrix}

The formulae (2a−c) constitute the Bianchi identities of the structure.

2. Geodesics and the Second Variation

Let (ω, θ, η) be a generalized Finsler structure on a 3-manifold Σ. The geodesics of the structure are its θη-integral curves. They define a foliation of Σ called the geodesic foliation. The (local) flow of W (the dual of ω) is the geodesic flow and a generalized Finsler structure is geodesically complete if W is complete on Σ, i.e., its flow exists for all time. The structure is geodesically amenable if the leaf space Λ of the geodesic foliation can be given the structure of a smooth surface in such a way that the natural projection ℓ : Σ → Λ is a smooth submersion.^{2}

When Σ ⊂ T M is a Finsler structure on a surface M, a ‘Σ-geodesic’ will mean an immersed curve γ : I → M (where I is an interval in ℝ) that satisfies the condition that, first, γ′(t) lie in Σ for all t ∈ I and, second, that the lifted curve γ′ : I → Σ be a geodesic of the generalized Finsler structure on Σ.

If γ : I → M is a Σ-geodesic and the interval I contains 0, then γ′(s) = exp_{sW}(u) where u = γ′(0). Thus, γ(s) = π(exp_{sW}(u)).

In the Riemannian geometry of surfaces, the Gaussian curvature plays an important role in the formula for the second variation of arc length. In the more general case of a Finsler structure, the function K plays the same role. More precisely, a compact Σ-geodesic γ : [a, b] → M will be a local minimum of the Σ-length functional if the quadratic form

Q_{γ}(f) = ∫_{a}^{b}((f′)^{2} − K_{γ} f^{2}) dt

1The derivative of the first equation is an identity.

2Every generalized Finsler structure is locally geodesically amenable.

has zero index and nullity on the space of smooth functions on [a, b] which vanish at the endpoints where K_{γ} = K∘γ′. This follows by an elementary calculation which I will omit.

In particular, note that if K is non-positive, then every geodesic segment is locally minimizing. On the other hand, if K ≥ a^{2} for some positive constant a, then, just as in the Riemannian case, no geodesic segment of length greater than π/a can be locally minimizing.

3. Structures satisfying K = 1

In this section, I want to study the (generalized) Finsler structures that satisfy K = 1 and construct examples of such on the standard 2-sphere.

Thus, let (ω, θ, η) be a generalized Finsler structure on a connected 3-manifold Σ that satisfies K ≡ 1. In this case, the structure equations become

dω = η ∧ θ
dθ = −η ∧ (ω + S θ)
dη = −(ω + C η) ∧ θ
dS = C ω + S_{2} θ + S_{3} η
dC = −S ω + C_{2} θ + C_{3} η

Consider the 1-form S θ + C η, the 2-form η∧θ, and the quadratic form η^{2} + θ^{2}. A computation from the structure equations yields

ℒ_{W}(S θ + C η) = ℒ_{W}(η ∧ θ) = ℒ_{W}(η^{2} + θ^{2}) = 0,

so that all of these quantities are invariant under the geodesic flow.

In fact, ℒ_{W} ω = 0, ℒ_{W} θ = η, and ℒ_{W} η = −θ, which implies

exp_{sW}^{∗} ω = ω
exp_{sW}^{∗} θ = cos s θ + sin s η
exp_{sW}^{∗} η = − sin s θ + cos s η

These formulae have some interesting consequences for geodesically complete structures with K ≡ 1. For example, set Φ = exp_{2πW}. Then Φ : Σ → Σ is a symmetry of the generalized Finsler structure. If any geodesic closes with length an integral multiple of 2π, then this implies that some power of Φ, say Φ^{k} has a fixed point. Since Φ^{k} is a symmetry of the coframing and Σ is connected, this implies that Φ^{k} is the identity map. Thus, having one closed geodesic of length an integer multiple of 2π implies that all of the geodesics are closed and have the same length.^{3}

Another interesting observation is that, if Σ is compact, then so is the group G of symmetries the generalized Finsler structure. It follows that either Φ^{k} is the identity for some k > 0, so that all of the geodesics close, or else the closure of the set {Φ^{k} | k ≥ 0} in G contains an abelian sub-group of positive dimension. Now, it can be shown that there cannot be a compact example with a 2-dimensional abelian subgroup of the group of symmetries, so the only possibilities for a compact example with positive dimensional symmetry group are the quotients of the standard Riemannian structure on the 3-sphere and examples with a 1-parameter symmetry group. I will construct examples of this latter type at the end of this section.

3Actually, if any geodesic on which J is non-zero closes, then it must have length an integral multiple of 2π. This follows from the formulae ℒ _{W} S = C and ℒ _{W} C = −S, which imply that S and C are periodic of period 2π with respect to arc length along any geodesic.

3.1. Canonical structures on the geodesic space. Suppose that U ⊂ Σ is an open set in which the geodesic foliation is amenable, i.e., the leaf space Λ of the integral curves of W in U carries the structure of a Hausdorff smooth manifold so that the quotient map ℓ : U → Λ is a smooth submersion. Then there exist a 1-form φ on Λ so that ℓ^{∗}φ = −S θ−C η; a 2-form dA on Λ so that ℓ^{∗}(dA) = η∧θ; and a positive definite quadratic form g on Λ so that ℓ^{∗}g = η^{2} + θ^{2}. In particular, when the space of geodesics is a manifold, it has a canonical metric g and orientation dA as well as a canonical 1-form φ, whose norm with respect to the metric g is the invariant J.

3.2. Recovering the generalized Finsler structure. In fact, starting with this data on a surface Λ, one can recover the generalized Finsler structure. To see this, let ν : F → Λ denote the oriented orthonormal frame bundle of Λ with respect to the metric g and orientation form dA. One can then define a map ℓ̂: U → F by the formula

ℓ̂(u) = (ℓ(u); ℓ_{∗}(H_{u}), ℓ_{∗}(T_{u})) ∈ F_{u} .

(The facts that (η∧θ)(H, T) ≡ 1 and that H and T are the duals to η and θ ensures that (ℓ_{∗}(H_{u}), ℓ_{∗}(T_{u})) is a dA-oriented, g-orthonormal frame at ℓ(u) ∈ Λ.) By the very definition of the canonical 1-forms α_{1} and α_{2} on F, it follows that ℓ̂^{∗}(α_{1}) = η and ℓ̂^{∗}(α_{2}) = θ. The defining equations of the Levi-Civita connection 1-form α_{21} on F are

dα_{1} = α_{21} ∧ α_{2} dα_{2} = −α_{21} ∧ α_{1}

so pulling these back via ℓ̂ yields

dη = ℓ̂^{∗}(α_{21}) ∧ θ dθ = −ℓ̂^{∗}(α_{21}) ∧ η.

Comparing this with the structure equations above then yields

ℓ̂^{∗}(α_{21}) = −(ω+Sθ+Cη) = −ω + ℓ^{∗}φ.

In particular, ℓ̂^{∗}(α_{1}∧α_{2}∧α_{21}) = −η∧θ∧ω ≠ 0, so ℓ̂ is a local diffeomorphism. If I denote the Gauss curvature of the metric g by R, then the usual structure equation yields dα_{21} = ν^{∗}(−R dA). In particular, computing the exterior derivative of the equation ℓ̂^{∗}(−α_{21} + ν^{∗}φ) = ω yields

ℓ^{∗}(dA) = η ∧ θ = dω = ℓ̂^{∗}(−dα_{21} + ν^{∗}(dφ)) = ℓ̂^{∗}ν^{∗}(R dA + dφ) = ℓ^{∗}(R dA + dφ),

so that dφ = (1−R) dA.

Conversely, suppose now that I start with an oriented surface Λ endowed with a Riemannian metric g with Gauss curvature R and a 1-form φ which satisfies dφ = (1−R) dA. Then on the orthonormal frame bundle ν : F → Λ with tautological 1-forms α_{1} and α_{2} and Levi-Civita connection 1-form α_{21} satisfying the standard structure equations

dα_{1} = α_{21} ∧ α_{2}
dα_{2} = −α_{21} ∧ α_{1}
dα_{21} = −ν^{∗}R α_{1} ∧ α_{2}

write ν^{∗}φ = −C α_{1} − S α_{2} for some functions S and C on F and consider the 1-forms

ω̄ = −α_{21} + ν^{∗}φ = −α_{21} − C α_{1} − S α_{2}
θ̄ = α_{2}
η̄ = α_{1}

Then these 1-forms and functions on F satisfy the structure equations

dω = η ∧ θ
dθ = −η ∧ (ω + S θ)
dη = −(ω + C η) ∧ θ

and so correspond to a generalized Finsler structure with K = 1. Thus, the given data (g, dA, φ) on the surface Λ suffice to determine a local solution to the K = 1 equation.

3.2.1. A normal form. This local picture is still not entirely satisfactory because of the need to solve the equation dφ = (1−R) dA. However, this last step can also be avoided in a more-or-less natural way. The metric g is conformal to a metric ḡ of constant Gauss curvature R̄, say g = e^{2u} ḡ. There is a natural map from the oriented orthonormal frame bundle F̄ of ḡ to F that simply scales a ḡ-orthonormal frame by e^{−u}. I will denote this map by μ : F̄ → F. Then the standard formulae yield

μ^{∗}α_{1} = e^{u} ᾱ_{1}
μ^{∗}α_{2} = e^{u} ᾱ_{2}
μ^{∗}α_{21} = ᾱ_{21} + ∗du

Pulling back the equation dα_{21} = ν^{∗}(dφ − dA) via μ then yields

d(ᾱ_{21} + ∗du) = ν̄^{∗}(dφ − e^{2u}dĀ),

so setting φ̄ = φ − ∗du, this can be written

dφ̄ + R̄ dĀ = e^{2u}dĀ.

This yields the following prescription: Start with a metric ḡ of constant curvature R̄ and choose any 1-form φ̄ satisfying the open condition dφ̄ + R̄ dĀ > 0. Then define the function u by the equation dφ̄ + R̄ dĀ = e^{2u}dĀ. Then the 1-forms

ω̄ = −ᾱ_{21} + ν^{∗}φ̄
θ̄ = e^{u} ᾱ_{2}
η̄ = e^{u} ᾱ_{1}

will satisfy the structure equations of a generalized Finsler structure with K = 1.

Explicitly, in the case R̄ = 0, let x and y be local coordinates on V ⊂ Λ so that, on V, one has ḡ = dx^{2} + dy^{2} and dĀ = dx∧dy. Choose a 1-form φ̄ = a dx + b dy satisfying the open condition (b_{x} − a_{y}) > 0. Then on V × S^{1} ≃ F̄_{V}, the 1-forms

ω̄ = −dϕ + a dx + b dy
θ̄ = \sqrt{b_{x}−a_{y}} (− sin ϕ dx + cos ϕ dy)
η̄ = \sqrt{b_{x}−a_{y}} ( cos ϕ dx + sin ϕ dy)

satisfy the structure equations of a generalized Finsler structure with K = 1 and every generalized Finsler structure with K = 1 is locally of this form. Thus, the general local solution is explicitly given in terms of two arbitrary functions a and b of two variables, as the general theory predicted.

3.3. Compact examples. I will now construct some examples of Finsler structures satisfying K = 1 on the 2-sphere.

3.3.1. The Riemannian case. The case where S (and hence, C) vanishes identically is the (generalized) Riemannian case, for which there is a unique local model. If Σ is compact, then the simply connected cover Σ̃ is diffeomorphic to the 3-sphere via a diffeomorphism which identifies the Finsler coframing to the standard leftinvariant coframing of S^{3} thought of as the Lie group SU(2). Thus Σ = Γ\ SU(2) where Γ ⊂ SU(2) is a finite subgroup.

3.3.2. The non-Riemannian case. For the rest of the discussion, I will suppose that S does not vanish identically.

Since dS ≡ Cω mod {θ, η} and dC ≡ −Sω mod {θ, η}, it follows that S cannot be constant. Thus, the group of symmetries of the coframing cannot act transitively. In particular, the dimension of this group must be at most 2. Recall that in the Riemannian case, the group of symmetries of a Riemannian metric may never be of dimension 2.

The first problem is how to describe the actual Finsler structures with K = 1. As will be seen, this is a more delicate matter than just solving the local generalized Finsler structure problem as was done above.

To see what the conditions should be, first suppose that Σ ⊂ T M is an actual Finsler structure satisfying K = 1 and let ω, θ, and η denote the canonical 1-forms satisfying the structure equations as above. Then the fibers of the projection π : Σ → M are smooth embedded circles and are the leaves of the system ω = θ = 0. Thus, an obvious necessary condition for a generalized Finsler structure to be a Finsler structure is that all the leaves of the system ω = θ = 0 be compact. This is not sufficient however because of two difficulties.

The first difficulty is that even though all of the leaves may be compact, so that the system ω = θ = 0 defines a foliation by circles, there may exist certain exceptional circles (necessarily isolated) around which the foliation is not locally product of a circle and a 2-disk. Such ‘multiply covered’ circles cause the leaf space M to have the structure of an orbifold. One can deal with this either by considering Finsler structures on orbifolds, which does not significantly change the local theory, or by simply deleting the offending circles, yielding a smooth quotient space.

The second difficulty is that even if the foliation is locally a product, with leaf space quotient π : Σ → M, the map ι : Σ → T M may immerse each fiber Σ_{x} into T_{x}M as a convex curve which winds around the origin more than once, say m > 1 times. If the images ι(Σ_{x}) ⊂ T_{x}M are all simple embedded curves, then one can divide Σ by an action of ℤ_{m} so as to get a new Σ′ which is a Finsler structure on M. However, there is no reason a priori for the images to be simple closed curves, even in the case where K = 1, as will be seen.

3.3.3. The local non-Riemannian case. Returning to the case of a Finsler structure Σ ⊂ T M, let D ⊂ M be a geodesically convex disk with smooth strictly convex boundary ∂D. (Such disks exist, for example, the geodesic ball B_{r}(x) satisfies this condition for sufficiently small r.) Then an (oriented) geodesic γ through any point in the interior meets the boundary ∂D in two distinct points γ_{−} (entering) and γ_{+} (exiting). Conversely, for any two distinct points p_{−} and p_{+} in ∂D, there is a unique geodesic segment γ lying in D and starting at p_{−} and ending at p_{+}. Thus, the space Λ_{D} of geodesic segments in D is topologically (in fact, smoothly) identifiable with (∂D × ∂D) \ Δ, a smooth surface diffeomorphic to a cylinder. Let Σ_{D} = π^{−1}(D). By the remarks just made, the geodesic foliation on Σ_{D} is amenable, with a smooth submersion ℓ : Σ_{D} → Λ_{D} having the leaves of this foliation as fibers. This yields the double fibration

Σ_{D}
ℓ↙  ↘π
Λ_{D}  D.

Note that for any x ∈ D, the set C_{x} = ℓ(π^{−1}(x)) ⊂ Λ_{D} consists of the oriented geodesics which pass through x, and the map ℓ : π^{−1}(x) → Λ_{D} is a smooth embedding of a circle into Λ_{D} which generates the fundamental group of Λ_{D} ≃ S^{1} ×(0, 1).

Now, assuming K ≡ 1 for the original Finsler structure, let g, dA and φ be the canonical metric, area form, and 1-form, respectively, constructed earlier on Λ_{D}. Identifying Σ_{D} with an open subset of F, the oriented orthonormal frame bundle of L_{D} via the map ℓ̂constructed earlier, the curves C_{x} are the images of the (closed) integral curves of the system α_{2} = α_{21} − ν^{∗}φ = 0.

The geometric interpretation of these integral curves is as follows: For any unit speed (oriented) curve γ : [a, b] → Λ_{D}, there are two natural functions associated with it. First, there is the geodesic curvature κ : [a, b] → ℝ and, second, there is the function f : [a, b] → ℝ defined by f(t) = φ(γ′(t)). I will say that γ is a φ-geodesic if κ = f. The notion of ‘λ-geodesic’ would make sense for any 1-form λ, and the 0-geodesics are, of course, the usual geodesics.^{4} Note that the λ-geodesics are solutions of a second order ODE and that there is exactly one oriented, unit-speed λ-geodesic in each tangent direction through each point of Λ_{D}.

By this definition, the curves C_{x} are φ-geodesics. Moreover, each closed φgeodesic C is the projection of a unique closed integral curve of the system α_{2} = α_{21} − ν^{∗}φ = 0 which must therefore constitute a single π-fiber, say π^{−1}(x). Of course, this implies that C = C_{x} and hence that C is embedded and represents a generator of the fundamental group of Λ_{D}. Note also that for each Finsler geodesic γ in D, the interval of points y in D which lie on γ give rise to an interval of closed φ-geodesics C_{y} which pass through γ when it is regarded as a point of Λ_{D}. Thus, each point of Λ_{D} has an open interval of tangent directions for which the corresponding φ-geodesic is closed and represents a generator of the fundamental group of Λ_{D}.

This leads to the following prescription for constructing local Finsler structures with K = 1. Since Λ_{D} is an annulus, the Uniformization Theorem implies that (Λ_{D}, g) is conformally equivalent to a standard annulus. In other words, there are coordinates r and ψ (well-defined modulo 2π) on Λ_{D} so that g = e^{2u} (dr^{2} + dψ^{2}) and dA = e^{2u} dr∧dψ and so that the image (r, ψ)(Λ_{D}) in ℝ × S^{1} is a standard cylinder of the form I × S^{1} where I ⊂ ℝ is an open interval. By the previous local discussion, I can then embed Σ_{D} as an open subset of I ×S^{1} ×S^{1} (with coordinates (r, ψ, ϕ)) in such a way that, for some functions a and b of r and ψ with (b_{r}−a_{ψ}) > 0 the following formulae hold,

ω = − dϕ + a dr + b dψ
θ = \sqrt{b_{r}−a_{ψ}} (− sin ϕ dr + cos ϕ dψ)
η = \sqrt{b_{r}−a_{ψ}} ( cos ϕ dr + sin ϕ dψ)

The functions a and b must then be chosen so that an open subset of the integral curves of the system θ = ω = 0 are closed and project to I×S^{1} to become generators of the fundamental group.

3.3.4. Rotationally invariant examples. To show that this is possible for something other than the Riemannian case, I am now going to construct some examples, adapting an idea found in Volume 3 of Darboux [Da] for constructing metrics on S^{2}, all of whose geodesics are closed. The simplest way to proceed to these examples is to try to find ones which are rotationally symmetric with respect to the cyclic coordinate ψ, i.e., so that a_{ψ} = b_{ψ} = 0. In fact, I am going to make the even more stringent assumption that a = 0, so that the forms involved simplify to

ω = − dϕ + b(r) dψ
θ = \sqrt{b′(r)} (− sin ϕ dr + cos ϕ dψ)
η = \sqrt{b′(r)} ( cos ϕ dr + sin ϕ dψ)

4I am told that the ‘physical interpretation’ of λ-geodesics is that of a particle moving under the influence of a ‘magnetic field’ specified by λ.

where, by assumption b′(r) > 0 for all r in the interval I, so I will write b = B′(r) where B is a strictly convex function on I. A computation shows that the invariant S is given by

S = −\frac{b′′(r) + 2b(r)b′(r)}{2(b′(r))^{3/2}} cos ϕ.

Thus, as long as b′ + b^{2} is not constant, the examples constructed will not be Riemannian.

Now, the system ω = θ = 0 has a first integral. To see this, note that

b(r) (− sin ϕ dr + cos ϕ dψ) − cos ϕ (−dϕ + b(r) dψ) = d(sin ϕ) − sin ϕ (b(r) dr)
= e^{B} d(e^{−B} sin ϕ).

Thus every integral curve of this system lies on a level surface of the function f = e^{−B} sin ϕ. In particular, the zero level surface is foliated by the curves defined by ψ ≡ ψ_{0} and ϕ ≡ 0 or π. None of these curves are closed, of course, so I must look at the other level sets. Since the system is invariant under the involution (r, ψ, ϕ) ↦ (r, −ψ, −ϕ), it suffices to look at the positive level surfaces of f.

First, suppose that b does not vanish on the interval I. Then B has a smooth inverse on I. Let C be an integral curve of θ and ω on which f is positive, say f ≡ e^{−B_{0}}. The equation e^{−B} sin ϕ = e^{−B_{0}} can be solved on C for r in the form

r = B^{−1}(e^{B_{0}} sin ϕ).

Since sin ϕ must be positive on C, it follows that 0 < ϕ < π on C. Moreover, I claim that ϕ must be monotone on C, which implies that it cannot be closed. To see this, note that because r is a function of ϕ on C and C is immersed, then at least one of dϕ or dψ is non-vanishing at every point. However, since b is non-vanishing and ω = −dϕ + b dψ, it follows that neither dϕ nor dψ can vanish. Thus, ϕ is monotone as claimed, and there are no closed integral curves.

Thus, from now on, I assume that b does vanish somewhere on I. Since b_{r} > 0, this zero must be unique. By translation in the variable r, I can assume that b(0) = 0 and can also make B unique by setting B(0) = 0. Since B′′ > 0, it follows that there exists a unique smooth function ρ of r so that ρ′ > 0, ρ(0) = 0, and e^{−B} = cos ρ. Note that ρ satisfies −π/2 < ρ < π/2. I will let ρ_{min} denote the infimum of ρ and ρ_{max} denote the supremum of ρ. I am now going to use ρ as the natural parameter on I, so that I can regard r as a function of ρ, instead of the other way around and write ṙ for dr/dρ. Thus, the system to be integrated now is of the form

−ṙ sin ϕ dρ + cos ϕ dψ = −ṙ cos ρ dϕ + sin ρ dψ = 0

In these coordinates, f = cos ρ sin ϕ ≤ 1. The level set f = 1 is just ρ = 0 and ϕ = π/2, which is a closed integral curve whose projection into I × S^{1} is a generator for the fundamental group. For any value ρ_{0} so that ρ_{min} < −ρ_{0} < 0 < ρ_{0} < ρ_{max}, the equation cos ρ sin ϕ = cos ρ_{0} > 0 is a closed curve C_{ρ_{0}} in I × S^{1}. In particular, since cos ρ and sin ϕ cannot vanish on C_{ρ_{0}}, it follows that on any integral curve of the system which lies in f = cos ρ_{0}, the equation can be written in the form

dρ = \frac{cos ϕ dψ}{ṙ sin ϕ} dϕ = \frac{sin ρ dψ}{ṙ cos ρ} ,

so that dψ cannot vanish on any such integral curve, so I can regard it as a parameter on the curve. In fact, these equations show that the point (ρ(ψ), ϕ(ψ)) traces around the curve C_{ρ_{0}} counterclockwise as ψ increases. Thus, the integral curve will close if and only if ψ increases by an rational multiple of π when this point makes a complete circuit around C_{ρ_{0}}. A simple calculation reveals that the increase of ψ corresponding to such a complete circuit is

P (ρ_{0}) = ∫_{−ρ_{0}}^{ρ_{0}} \frac{2 cos ρ_{0} ṙ(ρ)}{\sqrt{cos^{2} ρ − cos^{2} ρ_{0}}} dρ .

The condition on r needed to satisfy the π_{1}-generation condition is that P (ρ_{0}) ≡ 2π for all ρ_{0} in the range 0 < ρ_{0} < π/2.

Now, it is known that for ρ_{0} satisfying 0 < ρ_{0} < π/2

∫_{−ρ_{0}}^{ρ_{0}} \frac{2 cos ρ_{0}}{cos ρ \sqrt{cos^{2} ρ − cos^{2} ρ_{0}}} dρ ≡ 2π.

It follows that the necessary and sufficient condition that P (ρ_{0}) ≡ 2π for all ρ_{0} in the range 0 < ρ_{0} < π/2 is that ṙ be of the form ṙ = (1 + h(ρ))/(cos ρ) where h is an odd function defined on an interval symmetric about ρ = 0.

In order to have ṙ > 0 it will also be necessary to impose the condition h > −1. The identity b dr = b ṙ dρ = −e^{B} d(e^{−B}) = cot ρ dρ yields b = sin ρ/(1 + h(ρ)) and the condition b_{r} = db/dr > 0 becomes

1 + h(ρ) − h′(ρ) tan ρ > 0,

which I also impose. Conversely, if h is an odd function on the interval (−a, a) which satisfies these inequalities, the resulting generalized Finsler structure will have all of the ωθ-curves in the region f > cos a be closed.

3.3.5. Compact examples. Now, I further require that the odd function h be defined and smooth on all of ℝ, that it satisfy h(ρ + π) = −h(ρ) and that h and h′ vanish at ρ = ±π/2. This will ensure that the formulae above define a smooth metric on S^{2}, all of whose φ-geodesics are closed. By the general procedure described above, as long as h is non-zero, this will induce a non-Riemannian Finsler structure satisfying K ≡ 1 on the 2-sphere of oriented φ-geodesics. (Writing this Finsler structure out explicitly would require finding a second first integral of the φ-geodesic flow, a non-trivial task.)

4. Structures satisfying K = −1

In this final section, I want to briefly consider the corresponding problem of studying the generalized Finsler structures which satisfy K = −1.

Thus, let (ω, θ, η) be a generalized Finsler structure on a connected 3-manifold Σ that satisfies K ≡ −1. In this case, the structure equations become

dω = η ∧ θ
dθ = −η ∧ (ω + S θ)
dη = (ω − C η) ∧ θ
dS = C ω + S_{2} θ + S_{3} η
dC = S ω + C_{2} θ + C_{3} η

These structure equations already have global implications. For example, suppose that Σ were compact. Then it would necessarily be geodesically complete, but the structure equations above show that, along any integral curve of W, the function S satisfies an equation of the form S′′ − S = 0, where the prime denoted differentiation with respect to the flow parameter along the integral curve. However, the only solution of this equation which is bounded in both directions is the zero solution. Since Σ is supposed to be compact, S must be bounded on Σ and hence on every integral curve. Of course this implies that S (and hence C) must vanish identically, so that the structure is locally Riemannian. In particular any Finsler structure on a compact surface M which satisfies K ≡ −1 (or any negative constant, for that matter) must be a Riemannian metric, a result found in [Ak]

Note that J = −S^{2} + C^{2} is constant along the geodesic curves, i.e., the integral curves of W, and so is a first integral of the geodesic equations. Thus, in the non-Riemannian case, the geodesic flow must be completely integrable, again, a great contrast with the Riemannian case when K = −1.

4.1. Canonical structures on the geodesic space. Consider the 1-form S θ − C η, the 2-form η∧θ, and the quadratic form η^{2} − θ^{2}. A computation from the structure equations yields

ℒ_{W}(S θ − C η) = ℒ_{W}(η ∧ θ) = ℒ_{W}(η^{2} − θ^{2}) = 0,

so that all of these quantities are invariant under the geodesic flow.

If one assumes that the generalized Finsler structure is geodesically amenable, with geodesic projection ℓ : Σ → Λ, then it follows that there exist on Λ a 1-form φ so that ℓ^{∗}φ = −S θ + C η; a 2-form dA so that ℓ^{∗}(dA) = η∧θ; and a Lorentzian quadratic form g so that ℓ^{∗}g = η^{2} − θ^{2}.

Using the same sort of immersion of Σ into the ‘orthonormal frame bundle of g as I did in the K = 1 case, I can identify η and θ as the canonical forms on the Lorentzian orthonormal frame bundle of g and, due to the equations

dη = (ω + S θ − C η) ∧ θ
dθ = (ω + S θ − C η) ∧ η

one sees that ψ = (ω+S θ−C η) can be thought of as the Levi-Civita connection of this pseudo-metric. The curvature R of this metric is then defined by dψ = R η∧θ and is well-defined on Λ. Then, just as before, one derives

dφ = (1 − R) dA.

as the equation relating the 1-form φ with the oriented Lorentzian structure defined by g and the choice of oriented area form dA.

Conversely, starting with an oriented surface Λ endowed with a Lorentzian metric g of curvature R and area form dA and a 1-form φ which satisfies the equation dφ = (1−R) dA, one can define a generalized Finsler structure satisfying K = −1. I will now describe this construction. For simplicity, I am going to assume that Λ is also time-oriented.

Let ℓ : Σ → Λ be the bundle of oriented, time oriented g-frames on Λ of the form (p; e_{1}, e_{2}) where p is a point of Λ and (e_{1}, e_{2}) are an oriented, time oriented basis of T_{p}Λ which satisfies

1 = g(e_{1}, e_{1}) = −g(e_{2}, e_{2}), 0 = g(e_{1}, e_{2}).

Then Σ is an ℝ-bundle over Λ and the tautological 1-forms α_{1} and α_{2} satisfy

dα_{1} = ψ ∧ α_{2}
dα_{2} = ψ ∧ α_{1}

where ψ is the connection 1-form. It satisfies dψ = ℓ^{∗}R α_{1}∧α_{2}, where R is the function on Λ which represents the curvature of g.

Now, write ℓ^{∗}φ = C α_{1}−S α_{2} for some functions S and C on Σ, which can always be done. Then the hypothesis that dφ = (1 − R) dA ensures that the 1-forms

ω = ψ + ℓ^{∗}φ = ψ + C α_{1} − S α_{2}
θ = α_{2}
η = α_{1}

satisfy the structure equations of a generalized Finsler structure on Σ with K ≡ −1. Further details will be left to the reader.

4.2. Complete examples. To construct complete examples of actual Finsler structures on surfaces, it suffices to construct an example of a Λ endowed with the appropriate structures so that the positive φ-geodesics are are all closed and satisfy appropriate growth conditions.

Again, the method is to start with a normal form and assume a symmetry. The same sort of analysis as in the previous section leads to the following prescription: Let Λ be the cylinder (−π/2, π/2)×S^{1} with coordinates ρ and ϕ (which is periodic of period 2π). Let Σ = Λ×ℝ with coordinate z on the ℝ-factor. Let h : (−π/2, π/2) → (−1, 1) be an odd function of ρ which satisfies the inequality

1 + h(ρ) − h′(ρ) sin ρ cos ρ > 0.

Now consider the coframing on Σ defined by

ω = dz − (tan ρ)/(1 + h(ρ)) dϕ
θ = R(ρ)((1 + h(ρ)) cosh z dρ + sinh z dϕ)
η = R(ρ)((1 + h(ρ)) sinh z dρ + cosh z dϕ)

where the function R > 0 is defined by

(R(ρ))^{2} = \frac{1 + h(ρ) − h′(ρ) sin ρ cos ρ}{cos^{2} ρ (1 + h(ρ))^{3}}.

For any choice of h satisfying the above restrictions, this is a generalized Finsler structure satisfying K = −1. A computation shows that the invariant S vanishes if and only if h satisfies the differential equation

\frac{1}{(1 + h(ρ))^{2}} − \frac{h′(ρ) tan ρ}{(1 + h(ρ))^{3}} = c

for some constant c. (Note that h = 0 satisfies this condition.)

It is not hard to show that the ωθ integral curves are closed in Σ for any such choice of h. (Compare the argument from §3.) They foliate Σ and the leaf projection π : Σ → M induces a Finsler structure on M (topologically a disk) which satisfies K ≡ −1.

It is not hard to show that if there is a constant c < 1 so that |h(ρ)| ≤ c for all ρ, then the resulting Finsler structure on M is complete. Thus, these choices of h provide examples of complete non-Riemannian Finsler structures on the disk that satisfy K ≡ −1.

References

[Ak] H. Akbar-Zadeh, Sur les espaces de Finsler a courbures sectionnelles constantes, Acad. Roy. Belg. Bull. Cl. Sci. (5) 74 (1988), 281–322.

[BaChSh] D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Finsler Geometry, Springer-Verlag, New York (manuscript in preparation).

[Ca] É. Cartan, Les Espace Finsler, Exposés de Géometrie, t. 79, Hermann, Paris, 1934.

[Ch1] S.S. Chern, On the Euclidean connection in a Finsler space, Proc. Natl. Acad. Sci. USA 29 (1943), 33–37 (Reprinted in Shiing-shen Chern: Selected Papers, vol. II, Springer-Verlag, New York, 1989, pp. 107–111).

[Ch2] S.S. Chern, Local equivalence and Euclidean connections in Finsler spaces, Science Reports Tsing Hua Univ. 5 (1948), 95–121 (Reprinted in Shiing-shen Chern: Selected Papers, vol. II, Springer-Verlag, New York, 1989, pp. 194–212.).

[Da] G. Darboux, Leçons sur la Théorie générale des Surfaces, t. 3, Editions Chelsea, Third edition, 1972, Gauthier-Villars, Paris, 1894–1915.

[GaWi] R. Gardner and G. Wilkens, Classical geometries arising in feedback equivalence (1993), Fields Institute, Waterloo (preprint).

Department of Mathematics, Duke University, PO Box 90320, Durham, NC 27708-0320

E-mail address: bryant@math.duke.edu