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The multiplier approach to the projective Finsler metrizability problem

This paper is concerned with the problem of determining whether a projective-equivalence class of sprays is the geodesic class of a Finsler function. We address both the local and the global aspects of this problem. We present our results entirely in terms of a multiplier, that is, a type (0,2) tensor field along the tangent bundle projection. In the course of the analysis we consider several related issues of interest including the positivity and strong convexity of positively-homogeneous functions, the relation to the so-called Rapcs\'ak conditions, some peculiarities of the two-dimensional case, and geodesic convexity for sprays.


Introduction
A Finsler function -a smooth function on the slit tangent bundle τ : T • M → M of a smooth paracompact manifold M , which is positive, positively (but not necessarily absolutely) homogeneous, and strongly convex -determines through its Euler-Lagrange equations an equivalence class of sprays whose base integral curves are its geodesics, where the curves with given initial point and direction defined by two different members of the class are the same up to an orientation-preserving reparametrization. The geodesics of the class, in other words, define oriented paths in M . Such sprays are projectively equivalent. That is to say, if Γ is one such spray -a vector field on T • M such that τ * Γ (z,y) = y for any x ∈ M and y ∈ T x M , y = 0, and such that [∆, Γ] = Γ where ∆ is the Liouville field -then any other takes the form Γ − 2P ∆ for some function P on T • M which satisfies ∆(P ) = P (the choice of numerical coefficient is made for later convenience). A set of sprays related in this way is called a projective-equivalence class, or simply projective class, of sprays.
The projective metrizability problem is the corresponding inverse problem: given a projective class of sprays, does it come from a Finsler function in the way just described? There have been several publications on this question in the last few years: see for example [1,3,6,8,17,18]. It is a curious fact that none of these uses what is, in the context of the inverse problem of the calculus of variations in general, probably the most studied, and certainly the historically most significant, approach, namely that of the multiplier. The necessary and sufficient conditions for the existence of a Lagrangian for a given system of second-order ordinary differential equations, when expressed as conditions on a multiplier matrix, are known as the Helmholtz, or sometimes Helmholtz-Sonin, conditions. Such conditions for systems with two degrees of freedom were originally formulated by Douglas in 1941 in [9]. The subject was resurrected by Henneaux, and a more modern version of the Helmhotz conditions was given by Sarlet, both in 1982 (see [10] and [14]). For a recent survey see [12].
Our first aim in this paper is to formulate the necessary and sufficient conditions on a multiplier, conceived as a symmetric type (0, 2) tensor field along the tangent bundle projection τ , for it to be the Hessian (with respect to the fibre coordinates) of a positively-homogeneous function on T • M for which a given projective class of sprays satisfies the Euler-Lagrange equations. These necessary and sufficient conditions play the role of Helmholtz conditions for the projective Finsler metrizability problem. They differ from the Helmholtz conditions for the general inverse problem of the calculus of variations in several respects, arising from the homogeneity of the sought-for Finsler function and the fact that we work with a projective class of sprays rather than a single differential equation field.
We must make a point of clarification here. As well as the projective metrizability problem, there is another, related but different and more straightforward, inverse problem in Finsler geometry, that of determining whether a single given spray is the canonical geodesic spray of a yet unknown Finsler function. Given a Finsler function F and its corresponding energy E = 1 2 F 2 , there are actually two candidates for the role of a multiplier, namely the Hessians h and g of F and E respectively; they have significantly different properties in terms of homogeneity and regularity. The Helmholtz conditions for the canonical spray problem have been formulated previously (see [11,15]); the multiplier in this case is of the type of g. By constrast we use the Finsler function rather than the energy, and a multiplier of the type of h rather than g. We work with a projective class of sprays, and avoid reference to the canonical spray; moreover we express our results as far as possible in projectively-invariant terms.
We should also mention that when referring to a multiplier, and indeed more generally, we usually abbreviate the expression 'tensor field along the tangent bundle projection' to simply 'tensor field' or 'tensor'.
In addition to formulating the Helmholtz-like conditions for the projective metrizability problem, we analyse in detail the requirements on a Finsler function that it takes only positive values and is strictly convex.
Our first results on the metrizability problem, though global with respect to the fibres of τ , or y-global, are local with respect to M . As a third topic, we address the question of how these results may be made global in M . In the literature one will find mostly local results, but global questions are barely touched upon. We find that there is a cohomological obstruction on M to combining locally-defined Finsler functions into a global function; when this obstruction vanishes the resulting function satisfies all of the requirements of a Finsler function except positivity, and can be turned into a genuine Finsler function (by the addition of a total derivative) only locally. A well-known example of a spray which exhibits precisely this behaviour is Shen's circle example from [16].
Some authors, such asÁlvarez Paiva (in [1]), deal only with reversible paths, that is, paths which have no preferred orientation; we, on the other hand, cover the more general case of oriented paths, or sprays in the fully general sense. At the level of Finsler geometry this distinction corresponds to that between positively-homogeneous Finsler functions (the general case) and absolutely-homogeneous functions (the case discussed byÁlvarez Paiva). We do however specialise to the reversible case where appropriate, and we begin the discussion of this in Section 2.
One of the difficulties of working directly with the Finsler function, rather than with the energy, is that the conditions of positivity and strong convexity required of a Finsler function become somewhat tricky to deal with. We discuss this issue in Section 3. The argument is based on the theorem from [4] in which the triangle inequality and the fundamental inequality are established for Finsler functions: we prove a version which doesn't assume ab initio that the function in question is non-negative, and we use our result to show that any positivelyhomogeneous function whose Hessian is positive quasi-definite -as nearly positive definite as it can be -can be turned into a Finsler function locally by the addition of a total derivative. Sections 2 and 3 deal with preliminary matters. In Section 4 we turn to the projective metrizability problem, the main business of the paper, and discuss the Rapcsák conditions, which are essentially ways of formulating or reformulating the Euler-Lagrange equations. In Sections 5 and 6 we give the necessary and sufficient conditions for the existence of a Finsler function in terms of a multiplier. We discuss them initially in local terms: local, that is, relative to M . In fact throughout the paper, with the exception of Section 7, we deal only with objects which are defined and smooth as functions of y i for all y = 0, where y = (y i ) are the fibre coordinates on T M . That is to say, we deal almost entirely with objects which are y-global. We discuss in Section 6 the problem of extending our local results to results which are global in M .
In the main body of the paper we make the assumption that dim M ≥ 3. The twodimensional case has some special features which mean that our general results do not always apply. We make some remarks about the two-dimensional case in Section 7. The paper proper ends with the discussion of an illustrative example, and some concluding remarks.
Most of our local calculations are carried out in coordinates. We denote coordinates on M by (x i ), and the corresponding canonical coordinates on T M by (x i , y i ). We denote a spray Γ by The corresponding horizontal local vector fields are denoted by We write V i for ∂/∂y i when it is convenient to do so. The paper is written under the assumption that the reader is familiar with the differential geometry of sprays. However, we have given a number of basic results in an appendix, for reference and to fix notations. As a general reference for Finsler geometry we use the wellknown book by Bao, Chern and Shen, [4]. The standard modern reference for the geometry of sprays is Shen's book, [16]; for a survey using a coordinate-independent formalism see the paper by Bácsó and Z. Szilasi, [3].
A second appendix contains a discussion of the existence of geodesically convex sets for a spray, which can be used in the construction of so-called good open coverings, needed for the global results in Section 6. While this material on convex sets, which is of some interest in its own right, is not strictly speaking new, it seems to have been lost from view for some considerable time.

Reversible sprays
The geodesics of a spray are not in general reversible, even as paths. In this brief section we consider the special class of sprays for which the geodesics are reversible.
We denote by ρ the reflection map of T • M , namely (x, y) → (x, −y). For any spray Γ the vector fieldΓ = −ρ * Γ is also a spray, which we call the reverse of Γ. We say that Γ is reversible if it is projectively equivalent to its reverse, and strictly reversible if the two are equal. Then the geodesic paths of Γ are reversible if and only if Γ is reversible. Moreover, the geodesics are reversible as parametrized curves, which is to say that if γ is a geodesic so is t → γ(−t), if and only if Γ is strictly reversible.
If a spray is reversible so are all sprays projectively equivalent to it: that is to say, reversibility is a property of a projective class. We shall show that the projective class of a reversible spray contains a strictly-reversible spray. Proposition 1. If Γ is reversible then there is a projectively-equivalent spray which is strictly reversible.
Proof. LetΓ be the reverse of Γ, and setΓ ThenΓ is a spray, and ThusΓ is strictly reversible. ButΓ is projectively equivalent to Γ, from which it follows immediately thatΓ is projectively equivalent to Γ also.

Positivity and strong convexity
For a positively-homogeneous smooth function F on T • M to be a Finsler function it must be positive and strongly convex: that is to say, g ij = F ∂ 2 F ∂y i ∂y j + ∂F ∂y i ∂F ∂y j must be positive definite. The conditions for a projective class of sprays to be metrizable to be discussed below consist in the first instance of conditions for the existence of a positivelyhomogeneous function F whose Euler-Lagrange equations are satisfied by the given sprays, which we may describe as differential conditions. These differential conditions must then be supplemented by algebraic conditions which will ensure that there is such a function F which is positive and strongly convex. Formulating such algebraic conditions is somewhat tricky, for reasons we now explain.
It follows from the fact that F is homogeneous of degree 1 in the fibre variables that its Hessian satisfies y j ∂ 2 F ∂y i ∂y j (y) = 0.
At any point (x, y) ∈ T • M such that F (x, y) = 0, any vector u may be written uniquely as the sum of a scalar multiple of y and a vector v which is g-orthogonal to y, namely Then Thus provided that F (x, y) is positive, g is positive definite if and only if the Hessian is positive quasi-definite, in the sense that ∂ 2 F ∂y i ∂y j (y)v i v j ≥ 0 with equality only if v is a scalar multiple of y.
So for F to be a Finsler function it is necessary that its Hessian be positive quasi-definite; but this property is not sufficient. The problem is that we require F to be positive, but there is no way of ensuring this by imposing a condition on the Hessian. Note that if F is positively homogeneous so is any function obtained by adding a total derivative to it, that is, any function of the form F + y i ∂φ ∂x i where φ is a function on M ; the new function has the same Hessian, and its Euler-Lagrange equations are satisfied by the given sprays if those of F are. But adding a total derivative to a Finsler function may destroy the Finsler property.
Conversely, however, given a positively-homogeneous function F whose Hessian is positive quasi-definite in the sense specified above, it is always possible to modify F by the addition of a total derivative so as to obtain a local Finsler function, that is, a function on T • U for some open neighbourhood U of any point in M , which is positive, positively homogeneous and strongly convex. We shall prove this below.
It turns out, as will become apparent below, that there is no difficulty about positivity if F is absolutely rather than just positively homogeneous, that is if F (ky) = |k|F (y) for all k ∈ R.
The results in this section are not to be confused with that of Lovas [13], who showed in effect that assuming positivity one can prove strong convexity: he showed, that is, that for a positive, positively-homogeneous function F , if ∂ 2 F ∂y i ∂y j (y)v j = 0 only when v is a scalar multiple of y then the Hessian is positive quasi-definite (this formulation is to be found in [7]). Lovas's result is converse to ours.
For the first part of the argument we fix x ∈ M , that is, we essentially work in R n . The following result is Theorem 1.2.2 of [4], but without the assumption that F is non-negative.

Proposition 2.
Let F be C ∞ on R n − {0}, continuous on R n , positively homogeneous, and suppose that its Hessian is non-negative in the sense that for any w ∈ R n w i w j ∂ 2 F ∂y i ∂y j (y) ≥ 0.
Then for all y 1 , y 2 ∈ R n F (y 1 ) + F (y 2 ) ≥ F (y 1 + y 2 ) (the triangle inequality); and for all y, z ∈ R n with y = 0 F (z) ≥ z i ∂F ∂y i (y) (the fundamental inequality).
Proof. Provided that 0 does not lie in the line segment [y, y + w] we can apply the second mean value theorem to the function t → F (y + tw) to obtain for some ǫ with 0 < ǫ < 1. The final term is non-negative, so and similarly where we must now assume that 0 does not lie in the segment [y − w, y + w]. Summing, On setting y 1 = 1 2 (y + w) and y 2 = 1 2 (y − w) we obtain the triangle inequality, for all pairs of points y 1 , y 2 such that the segment [y 1 , y 2 ] does not contain the origin. Now by continuity F (0) = 0, so the triangle inequality holds self-evidently if either argument is 0. Suppose that 0 is an interior point of the segment [y 1 , y 2 ]. Consider first the case y 1 + y 2 = 0. For any y = 0 and any positive s, F (sy + y 1 ) + F (sy + y 2 ) ≥ 2sF (y), and on letting s tend to 0 we obtain F (y 1 ) + F (y 2 ) ≥ 0. Now suppose that y 2 = −ky 1 with k > 0, and furthermore suppose that k < 1 (otherwise reverse the roles of y 1 and y 2 ). Then y 1 + y 2 = (1 − k)y 1 , and Thus the triangle inequality holds for all y 1 , y 2 .
For the fundamental inequality replace y + w by z to obtain We have to exclude y = 0 since F won't be differentiable there. We also have to assume that the segment [y, z] does not include the origin, that is, that z is not a negative multiple of y. But if z = −ky with k > 0 then Note that by positive homogeneity F can only be constant if it is identically zero.

Corollary 1.
If F is not constant then it must take positive values.
Proof. If F is not constant there must be a point y at which the gradient of F is non-zero. Then for all z in an open half-space, z i ∂F/∂y i (y) > 0; and F (z) > 0 for all such z.
We assume for the rest of this section that the Hessian of F is positive quasi-definite.
Lemma 1. It cannot be the case that for fixed non-zero z, is constant (i.e. takes the same value for all y = 0).
Proof. If it were constant it would follow that for all y, which cannot be the case when the Hessian is positive quasi-definite if z = 0.
Lemma 2. Except when z is a positive scalar multiple of y, the fundamental inequality is strict: Proof. Evidently equality holds when z is a positive scalar multiple of y. Moreover, it follows from the second mean value theorem that the inequality is strict otherwise, except possibly when z is a negative scalar multiple of y. From the proof of Proposition 2 we know that if z = −ky with k > 0 then So the question is whether we can have F (−y) = −F (y) with y = 0. Now for every w = 0, But by Lemma 1 this cannot hold. Thus F (−y) > −F (y) and the inequality is strict in this case too.
Proposition 3. If F is absolutely homogeneous and its Hessian is positive quasi-definite then it is everywhere positive on R n − {0}.
Proof. It follows from the previous proof that if F (y) = 0 (where y = 0) then F (−y) > 0. But if F is absolutely homogeneous and F (y) = 0 then it would have to be the case that F (−y) = 0 also. So F can never vanish. We know that F must take positive values. So it must be positive everywhere on R n − {0}.
This result has appeared previously in [7].

Proposition 4.
If F is positively homogeneous and its Hessian is positive quasi-definite then there is a linear function y → α i y i such that ifF (y) = F (y) + α i y i thenF is everywhere positive on R n − {0}.
Proof. Fix z ∈ R n − {0} and defineF bȳ Then by Lemma 2,F > 0 on R n − {0}, except along the ray through z;F (z) = 0. Now consider the restriction ofF to the unit coordinate Euclidean sphere S. Denote by H z the closed hemisphere opposite the point where the ray through z intersects S, that is, the southern hemisphere whereẑ = z/|z| is the north pole. ThenF is positive on H z , which is compact, so there is some k > 0 such thatF (y) ≥ k for y ∈ H z . Now set ϕ(y) = 1 2 kẑ i y i whereẑ i = δ ijẑ j . Note that − 1 2 k ≤ ϕ(y) ≤ 1 2 k for y ∈ S: in fact ϕ takes its minimum value − 1 2 k at the south pole, is zero on the equator, and is positive on the open northern hemisphere S − H z . SetF =F + ϕ. ThenF is positive on S − H z , sinceF is non-negative and ϕ is positive; and also on H z , sincē F ≥ k and ϕ ≥ − 1 2 k. ThusF is positive everywhere on S, and so everywhere on R n − {0}; and for the chosen non-zero z.
Theorem 1. Let F be a positively homogeneous function on T • M such that for every x ∈ M the Hessian of F on T • x M is positive quasi-definite. Then for any x 0 ∈ M there is a neighbourhood U of x 0 in M and a functionF defined on T • U such thatF is a Finsler function which differs from F by a total derivative. If F is absolutely, rather than just positively, homogeneous then it itself is a Finsler function.
Proof. Take any point x 0 ∈ M and choose coordinates at x 0 . By the previous proposition there is a (constant) covector α such that ifF (x, y) = F (x, y) + α i y i thenF (x 0 , y) > 0 for y = 0. It follows that there is an open neighbourhood U of x 0 within the coordinate patch such that F (x, y) > 0 for all x ∈ U and y ∈ T • x M . (To see this, suppose the contrary. ConsiderF as function on P ×S where P is the coordinate patch and S, as before, is the unit Euclidean sphere. Then there is a sequence of points {(x r , y r )} in P × S with x r → x 0 such thatF (x r , y r ) ≤ 0 for all r. But {y r }, being a sequence in a compact set, has a convergent subsequence, which converges to y 0 say. ThenF (x 0 , y 0 ) ≤ 0, which is a contradiction.) SoF is a Finsler function on T • U , and it differs from F by α i y i which is the total derivative of the local function α i x i .
If F is absolutely homogeneous then it is everywhere positive by Proposition 3, and therefore a Finsler function.

The Euler-Lagrange equations and the Rapcsák conditions
For a function F on T • M which is positively homogeneous of degree 1, the Euler-Lagrange equations have the property that if there is a solution Γ which is a spray, then every spray projectively equivalent to Γ is also a solution. At a basic level therefore solving the projective metrizability problem for a given projective class of sprays involves finding a positively-homogeneous function F such that the Euler-Lagrange equations are satisfied for any spray Γ in the class. These equations, considered as conditions on F , constitute (in the terminology of [17,18]) the first set of Rapcsák conditions. There is a second equivalent set which is given in the next proposition. (As a matter of fact there are many versions of the Rapcsák conditions to be found in the literature: [3] contains seven. The two we give are the ones most useful for our purposes.) Proposition 5. The positively-homogeneous function F satisfies the first Rapcsák conditions if and only if Proof. The first Rapcsák conditions are On differentiating with respect to y j one obtains The part of this which is skew in i and j gives the required conditions. Conversely, if the given conditions hold and F is homogeneous then contraction with y j produces the first Rapcsák conditions.
It is of interest, though not of any significance here, that the second Rapcsák conditions follow from the first regardless of whether or not F is homogeneous; homogeneity is required for the converse step only.
The second Rapcsák conditions may be written The symmetric part of the equation whose skew part gives the second Rapcsák conditions may be written where ∇ is the dynamical covariant derivative operator associated with Γ. These conditions have important roles to play in the next section.

The Helmholtz-like conditions for a multiplier
This section is concerned with conditions on a multiplier necessary and sufficient for it to be the Hessian of a Finsler function for a given projective class of sprays.
Initially we work in a coordinate neighbourhood U of M . Without essential loss of generality we may and shall assume that the Poincaré Lemma holds on U : in fact we shall assume that U is contractible. We further assume, for reasons that will shortly become clear, that dim M ≥ 3; and indeed we make this assumption for most of the rest of this paper, until the discussion of the two-dimensional case in Section 7 in fact.
We denote a putative Hessian of a Finsler function by h, and its components by h ij , to distinguish it from the putative Hessian of an energy function, which is written g or g ij in the conventional way. We shall say that h is quasi-regular if h ij v j = 0 if and only if v i = ky i for some constant k. We shall call a positively-homogeneous function F whose Hessian is quasi-regular a pseudo-Finsler function. 2. there is a smooth function F on T • U , positively homogeneous of degree 1, such that 2. For each i we may consider h ij dy j as a 1-form on R n − {0}, n = dim M , with coordinates y i , where we regard the x i as parameters. The condition ∂h ij ∂y k = ∂h ik ∂y j says that this 1-form is closed. For dim M ≥ 3 it follows that h ij dy j is exact, that is, there are functionsF i depending smoothly on y i for y = 0, and smoothly also on the parameters x i , such that But by the symmetry of h ij ,F i dy i is also closed, and hence exact. By a similar argument there is a smooth functionF on T • U such that Then ∂ ∂y i y j ∂F ∂y j −F = h ij y j = 0, so that y i ∂F /∂y i −F is a function of x i alone, say f . Then F =F + f has the required properties; and so indeed has any function differing from F by a function linear in the y i .
The first result also follows from the second, of course, but the direct proof is available if one does not want to invoke the existence of F . Note that we require that dim M = n ≥ 3 to be able to conclude that a closed 1-form on R n − {0} is exact, and hence that F (x, y) is defined for all y = 0. It is for this reason that we make this assumption for most of the rest of the paper. Proof. WithΓ = Γ − 2P ∆ and P i = V i (P ) we have Thus ∇h is projectively invariant for a given projective class.
The following lemma involves the curvature tensors R i j and R i jk of a spray Γ.

Lemma 5.
If h ij satisfies the conditions of Lemma 3 then the following conditions are equivalent to each other: Proof. To obtain the second expression, differentiate h il R l j − h jl R l i with respect to y k to obtain and add the two similar expressions obtained by cyclically permuting i, j and k. Terms involving derivatives of the h ij cancel in pairs due to the symmetry condition they satisfy, while pairs of terms involving derivatives of the R l i give the R l ij . To obtain the first from the second, simply contract with y k .
This lemma is also to be found in [3].
The Weyl curvature W i j of a spray Γ is defined by The Weyl curvature is projectively invariant. See [16] for details.

Lemma 6.
If h ij satisfies the conditions of Lemma 3 then the condition h ik R k j = h jk R k i is equivalent to h ik W k j = h jk W k i , and in particular is projectively invariant.
Proof. We have Lemma 7. LetF be a smooth function on T • U which is positively homogeneous and satisfies Then there is a smooth positively-homogeneous function F on T • U , with the same Hessian as F , which satisfies the second Rapcsák conditions. Proof. We show first that H i (θ j ) − H j (θ i ) is independent of the y k , whereθ i = ∂F /∂y i . Now It is a simple and well-known consequence of the first assumption, together with the evident fact that is a basic 2-form, say χ. We show next that χ is closed. In computing dχ we may replace partial derivatives with respect to x k with H k . We have where ⊕ indicates the cyclic sum over i, j and k. But this vanishes if h ik W k j = h jk W k i by Lemmas 5 and 6. So χ is closed, and hence exact. Choose ψ = ψ i dx i such that χ = dψ, and set and F satisfies the second Rapcsák conditions. After all these preliminaries we can now give the main result of this section.
Theorem 2. Let U ⊂ M be any contractible open subset of a coordinate patch. Given a projective class of sprays, the following conditions are necessary and sufficient for the existence of a positively-homogeneous function F on T • U , such that every spray in the class satisfies the Euler-Lagrange equations for F : there is a tensor h whose components satisfy where ∇ is the dynamical covariant derivative operator of any spray in the class.
Proof. If such a function F exists then its Hessian satisfies these conditions. Conversely, from the first three conditions and Lemma 3 there is a smooth positively-homogeneous functionF on T • U whose Hessian is h. Take any Γ in the projective class. Then by Lemma 7 and the remaining conditions there is a smooth positively homogeneous function F on T • U , whose Hessian is also h, which satisfies the second Rapcsák conditions. By the equivalence of the first and second Rapcsák conditions Γ, and hence every spray in the class, satisfies the Euler-Lagrange equations for F . Proof. The functionF has the same Hessian as F and is positively homogeneous, soF = F +α i y i for some functions α i on U . If Γ satisfies the Euler-Lagrange equations for both F andF we must have y j ∂α i ∂x j − y j ∂α j ∂x i = 0, or equivalently α i dx i is closed and therefore exact; and if α i dx i = dφ thenF − F is the total derivative of φ.

Some global results
We now consider the problem of extending these results from coordinate neighbourhoods in M to the whole of M . We shall work with an open covering U = {U λ : λ ∈ Λ} of M by coordinate patches. We shall assume that U has the property that every U λ , and every nonempty intersection of finitely many of the U λ , is contractible. A covering with this property is known as a good covering; it can be shown (see Appendix B) that every manifold over which is defined a spray admits good open coverings by coordinate patches. Let us assume that there is a tensor field h which satisfies the conditions of Theorem 2, and in addition is everywhere quasi-regular. Then for each U λ there is a pseudo-Finsler function F λ defined on T • U λ ; and on U λ ∩ U µ , if it is non-empty, there is a function φ λµ , determined up to the addition of a constant, such that On U λ ∩ U µ ∩ U ν , if it is non-empty, φ µν − φ λν + φ λµ is a constant, say k λµν (add the expressions for F λ − F µ etc., and note that U λ ∩ U µ ∩ U ν , being contractible, is connected). Furthermore, for any four members U κ , U λ , U µ , U ν of U whose intersections in threes are non-empty That is to say, k satisfies a cocycle condition. If k is in fact a coboundary we can modify each φ λµ by the addition of a constant, so that (after modification) φ µν − φ λν + φ λµ = 0. We now show that if, for an open covering U of M by coordinate patches we can find functions φ λν which satisfy this cocycle condition then there is a globally-defined pseudo-Finsler function F on T • M for the given projective class of sprays.
Since by assumption M is paracompact the covering U of M admits a locally finite refinement V = {V α : α ∈ A}. (The covering V may not be good, but this will not matter for what follows.) There is a partition of unity subordinate to V, that is, for every α there is a smooth function f α such that supp(f α ) ⊂ V α , 0 ≤ f α ≤ 1 and α f α = 1. By local finiteness, for every x ∈ M , Since V is a refinement of U we can define a map σ : A → Λ such that V α ⊆ U σ(α) . For every α, β ∈ A such that V α ∩ V β is non-empty we define a real-valued function φ αβ by Then the φ αβ satisfy the cocycle condition, namely Proposition 6. Under these assumptions there is a globally-defined function F on T • M such that F α differs from F | τ −1 Vα by a total derivative.
Proof. Let V α be any member of the covering V. For x ∈ V α set Then for any two members of the covering V with non-empty intersection and we can define F consistently by where V α is any member of the covering containing x.
The first part of this proof is a simple special case of the argument used to show thatČech cohomology is a sheaf cohomology theory in [20]. So for dim M ≥ 3 the multiplier argument gives a global pseudo-Finsler function provided the cocycle k is a coboundary. Now k is in fact an element of theČech cochain complex for the covering U with values in the constant sheaf M × R. This cocycle will certainly be a coboundary if the corresponding cohomology group on M , namelyȞ 2 (U, R), is zero (we use the notation of [20]). It follows from the fact that U is a good open covering that this cohomology group is isomorphic to the de Rham cohomology group H 2 (M ). We have the following result. We give examples of sprays which admit a global pseudo-Finsler function but only local Finsler functions in Section 8, so the result above would appear to be the best possible.
The situation in the case of a reversible spray, or without loss of generality in the light of Proposition 1 a strictly reversible spray, is much more clear cut. We observe first that if a spray Γ satisfies the Euler-Lagrange equations of a positively-homogeneous function F , and we set F = ρ * F (where ρ is the reflection map), thenΓ, the reverse of Γ, satisfies the Euler-Lagrange equations of the (positively-homogeneous) functionF . So if Γ is strictly reversible, it satisfies the Euler-Lagrange equations for both F andF , and therefore for their sum. But if F andF are positively homogeneous, F +F is absolutely homogeneous: for if k < 0 F (ky) +F (ky) = F (−|k|y) +F (−|k|y) = |k|(F (−y) +F (−y)) = |k|(F (y) + F (y)).
So for a strictly reversible spray, on U λ there is a pseudo-Finsler function F λ which is absolutely homogeneous. Now two functions (such as F λ , F µ ) which are known to differ by a total derivative and are both absolutely homogeneous must be equal. In the light of these remarks and the final assertion of Theorem 1 we have the following result.

The two-dimensional case
It is well known that in two dimensions every spray is locally projectively metrizable (see for example [2,6]). 'Locally' here, however, refers to the fibre as well as to the base: the claim is only that a y-local Finsler function exists. We may seek to establish the existence of a Finsler, or at least pseudo-Finsler, function as follows. Thus by symmetry h has just one independent component, which we may take to be h 12 ; this must be non-zero except where y 1 = 0 or y 2 = 0 (separately) for h to be non-trivial. Moreover h 12 must satisfy the homogeneity condition y 1 ∂h 12 ∂y 1 + y 2 ∂h 12 ∂y 2 + h 12 = 0.
On differentiating the equation h 11 y 1 + h 12 y 2 = 0 with respect to y 2 we obtain y 1 ∂h 11 ∂y 2 + y 2 ∂h 12 ∂y 2 + h 12 = 0, whence by homogeneity of h 12 y 1 ∂h 11 ∂y 2 − ∂h 12 ∂y 1 = 0. Thus ∂h 11 ∂y 2 = ∂h 12 ∂y 1 except possible where y 1 = 0. But by continuity this must hold where y 1 = 0 (but y 2 = 0) also. This is the first of the two non-trivial cases of the equations it is required to prove; the other is proved similarly.
Lemma 9. Let U be a coordinate neighbourhood in a two-dimensional manifold on which is defined a spray Γ. Suppose that a symmetric tensor h ij satisfies (∇h) ij = 0. If η i = h ij y j then ∇η = 0, and if λ ij = ∆(h ij ) + h ij then ∇λ = 0.
Proof. The first follows immediately from the fact that ∇y = 0 (that is to say, the dynamical covariant derivative of the total derivative y i ∂/∂x i vanishes). For the second we use the facts that [∆, Γ] = Γ and that ∆( The Weyl tensor W i j vanishes when dim M = 2. In order for the Helmholtz conditions as given in Theorem 2 to be satisfied it is therefore enough that ∇h = 0 everywhere, and that η = 0 and λ = 0 on some cross-section of the flow of Γ, which we must assume to contain, for every y in it, the ray through y, for the latter condition to make sense. Indeed, we can always find a multiplier h locally in T • M by specifying its values on such a cross-section of the flow of Γ, arbitrarily subject to the conditions that η = 0 and λ = 0, and requiring that ∇h = 0: this is a first-order differential equation along each integral curve of Γ, and so the value of h along the curve is determined by the specified initial value. Moreover, η and λ satisfy similar first-order differential equations along the integral curves of Γ; and now we can use the fact that the equations are linear to show that since η = 0 initially, η = 0 everywhere, and likewise for λ. Since Γ never vanishes on T • M we can always find a local cross-section to its flow. Indeed, we can take a local two-dimensional submanifold of T • M which is transverse to Γ, and extend it to a local cross-section by including, for every y in it, the ray through y. This, together with the evident y-local version of Theorem 2, establishes the result stated above.
It is moreover easy to specify the freedom in the choice of multiplier.
Proposition 7. Suppose that in two dimensions h satisfies the Helmholtz conditions. Let f be a function on T • M such that Z(f ) = 0 for every vector field Z ∈ ∆, Γ (so that f is homogeneous of degree 0, and constant along the integral curves of any spray in the projective class). Then h = f h also satisfies the Helmholtz conditions. Conversely, any pair of tensors which satisfy the Helmholtz conditions are so related.
Proof. It is clear thatĥ satisfies the algebraic conditions and is homogeneous of degree −1. We have Conversely, if h ij y j = 0 =ĥ ij y j thenĥ must be a scalar multiple of h, sayĥ = f h. Then by homogeneity we must have ∆(f ) = 0, and if ∇ĥ = 0 = ∇h then Γ(f ) = 0 also.
The results above are y-local. But even if there is a y-global multiplier this by itself is not enough to guarantee the existence of a y-global positively-homogeneous function F on T • M of which it is the Hessian, because the Poincaré Lemma doesn't hold. One could imagine working with polar coordinates (r, θ) in each fibre. Each component h ij of h is periodic in θ; but because F is obtained by integrating (twice), there is no guarantee that an F can be found which is periodic.
Necessary and sufficient conditions for these two 1-forms to be exact are This doesn't completely answer the question, however, because there is another integration to carry out. It turns out to be better to start from scratch. Let us compute the Hessian of a positively-homogeneous function F in polar coordinates. We may set F (r, θ) = rϕ(θ), where ϕ is periodic. Then by straightforward calculations So ϕ ′′ + ϕ = τ , and the question is whether this equation has a periodic solution ϕ for given periodic τ . We expect that 2π 0 (τ sin θ)dθ = 2π 0 (τ cos θ)dθ = 0 should be necessary conditions for the existence of a periodic solution; and indeed and the integrals of these functions evidently vanish if ϕ is periodic. The conditions 2π 0 (τ sin θ)dθ = 2π 0 (τ cos θ)dθ = 0 are in fact sufficient, as can be seen as follows. Define u(θ), v(θ) by Then ϕ(θ) = u(θ) cos θ + v(θ) sin θ is a particular solution of the equation ϕ ′′ + ϕ = τ , as can easily be seen by direct calculation (it is in fact the solution obtained by the method of variation of parameters). Now so if these integrals vanish then the equation has periodic solutions. Notice that the addition to the particular solution of a term a cos θ + b sin θ (the complementary function) corresponds merely to the addition of a term linear in y to F . If τ = k 1 cos θ + k 2 sin θ there is no periodic F whose Hessian is h; and indeed this is the key case, since for any τ there are constants k 1 and k 2 (more exactly, functions on M ) such that τ − (k 1 cos θ + k 2 sin θ) does lead to a periodic F . We can nevertheless solve the equation ϕ ′′ + ϕ = k 1 cos θ + k 2 sin θ: a particular solution is The expressions for the corresponding h ij are When faced with a non-periodic F one possible course of action, which can certainly be carried out over a coordinate neighbourhood U ⊂ M , is to replace the fibre by its universal covering space (or in other words simply to ignore the fact that F is not periodic). It appears that in this way one would obtain, on restricting to a level set of F , an example of what Bryant calls a generalized Finsler structure (see [5]).

Example
The purpose of the example described below is to illustrate how a spray may belong to the geodesic class of a globally-defined pseudo-Finsler function, where the pseudo-Finsler function may be made into a Finsler function only locally.
Consider the projective class of the spray The geodesics of Γ are spirals with axis parallel to the z-axis, together with straight lines parallel to the z-axis and circles in the planes z = constant. To see this, note first that both √ u 2 + v 2 = µ and w are constant; and therefore (or directly) √ u 2 + v 2 + w 2 = λ is also constant. The geodesics are solutions of x = −λẏ,ÿ = λẋ,z = 0.
In fact Γ belongs to the geodesic class of the pseudo-Finsler function This is globally well defined but only locally a Finsler function: it is positive only for x 2 + y 2 < 4. It is globally pseudo-Finsler, however. To obtain a Finsler function in a neighbourhood of an arbitrary point (x 0 , y 0 , z 0 ) we can make a simple modification tõ this is positive for (x − x 0 ) 2 + (y − y 0 ) 2 < 4. Note that it differs from F by a total derivative. The planes z = constant have the property that a geodesic initially tangent to such a plane (so that w = 0 initially) remains always in the plane: that is to say, such planes are totallygeodesic submanifolds. The restriction of Γ to the submanifold z = 0, w = 0 of T • R 3 is the spray of Shen's circle example from [16]. We consider this as a spray defined on T • R 2 . It has for its geodesics all circles in R 2 of radius 1, traversed counter-clockwise. Again, this spray is locally projectively metrizable. One local Finsler function is the restriction of the one given for the spiral example, namely F (x, y, u, v) = u 2 + v 2 + 1 2 yu − 1 2 xv. Again, F is only locally defined as a Finsler function, though it is global as a pseudo-Finsler function.
Finally we shall compare the geodesics of the Finsler function of Randers type (to which class the examples above belong) with motion under the Lagrangian Here g is a Riemannian metric. In two-or three-dimensional flat space this is the Lagrangian for the motion of a classical charged particle, of unit charge, in the magnetic field determined by d(β i dx i ).
We have ∂L ∂y i = g ij y j + β i = y i + β i .
The energy E L is given by Thus motion under L is with constant Riemannian speed. The Euler-Lagrange field Γ L is determined by On the other hand ∂F ∂y i = and any geodesic spray Γ F of F satisfies (but is not determined by) We can fix the spray Γ F by the requirement that Γ F (α) = 0, so that motion is with constant Riemannian speed (this is not of course the canonical spray of F ). Then for motion with unit Riemannian speed, motion under the Lagrangian L is the same as motion along the geodesics of the fixed spray Γ F . (For another speed one would have to replace β by a constant multiple of it.) It has been claimed [19] that the derivation of the equations of motion from F rather than L is an example of Maupertuis's principle in action. Be that as it may, this analysis does reveal that the spiral example above can be related to the motion of a charged particle in a constant magnetic field along the z-axis. It is known that in this regime circular motion is possible in planes perpendicular to the direction of the magnetic field, the radius of such circles being a constant whose value depends on the strength of the field and the charge (amongst other things); it is known as the gyroradius, Larmor radius or cyclotron radius. This is of course Shen's circle example.

Concluding remarks
It is pointed out in [16] that there cannot be a globally-defined Finsler function for the circle example above, for the following reason. In a Finsler space whose geodesic spray is positively complete, that is, for which every geodesic is defined on [0, ∞), every pair of points in M can be joined by a geodesic. This is a conclusion of the Hopf-Rinow Theorem of Finsler geometry. In the circle example the geodesics are evidently positively complete, but equally evidently there are pairs of points which cannot be joined by a geodesic. The same is true of the spiral example.
This observation raises an interesting point about the different levels at which global questions enter the problem. We leave aside the two-dimensional case, which as we have seen is atypical. The differential conditions on a multiplier stated in Theorem 2 are merely the starting point for further analysis which generally proceeds as follows. These differential conditions are regarded as partial differential equations for the unknowns h ij . In principle these differential equations generate integrability conditions, which are further conditions on the coefficients Γ i and their derivatives. In favourable cases, for example when the spray is isotropic, such integrability conditions are satisfied, and one can assert the local existence of a multiplier satisfying the differential conditions. But there is no guarantee that such a multiplier is even defined y-globally.
If there is in fact a y-global multiplier, and it is everywhere positive quasi-definite, then there is always a Finsler function which is local in M , that is, defined y-globally over an open subset of M . There is a cohomological obstruction on M to combining such locally-defined Finsler functions; but even when this obstruction vanishes the result is globally only a pseudo-Finsler function in general, and can be turned into a genuine Finsler function (by addition of a total derivative) only locally. This is the situation exemplified in the previous section.
It seems that to make further progress one would have to impose some conditions on the global properties of the sprays of the projective class. The most important question is how one incorporates the observation above about completeness and the Hopf-Rinow property into the story -a question, we suspect, of some subtlety because ideally it should be answered in a projectively-invariant manner.
We denote by R j i the components of the Riemann curvature or Jacobi endomorphism of Γ. We have the following well-known formulas, which serve to define R j i : Of course [V i , V j ] = 0; we have We set then R i j = R i jk y k , and the bracket of horizontal fields is determined by Associated with any spray there is an operator ∇ on tensors along τ which is a form of covariant derivative, and indeed is often called the dynamical covariant derivative. We need it mainly for its action on tensors h with components h ij , when If Γ is a spray, another sprayΓ is projectively equivalent to Γ if there is a function P on T • M , necessarily of homogeneity degree 1, such thatΓ = Γ − 2P ∆. Theñ Γ i j = Γ i j + P δ i j + P j y i , P j = ∂P ∂y j .
in C for all s in some interval [0, t) for t sufficiently small. Secondly, G is open by continuity of γ(a, b, s). Thirdly, G is closed (as a subset of C × C). Consider a point (a, b) of C × C which lies in the relative closure of G; that is, (a, b) is the limit of a sequence (a n , b n ) of points of G. Each geodesic s → γ(a n , b n , s) lies in C for 0 ≤ s ≤ 1. There is a geodesic s → γ(a, b, s) from a to b, but we know only that it lies in C 1 (though of course its initial and final points a and b are in C). Now for each s with 0 ≤ s ≤ 1, γ(a, b, s) is the limit of the sequence γ(a n , b n , s), so γ(a, b, s) certainly lies inC for all s. But the geodesic s → γ(a, b, s) lies in C initially, and can neither meet B transversely, nor be tangent to it, without subsequently passing out ofC. So γ(a, b, s) lies in C for all s, 0 ≤ s ≤ 1, and (a, b) ∈ G. Since C is connected by assumption, so is C × C; G is a non-empty subset of C × C which is both open and closed, and so G = C × C. So if we can find a region C with these properties in C 2 it will be convex and simple. In fact the open Euclidean coordinate ball {(x i ) : δ ij x i x j < r 2 } for any sufficiently small positive r will do for C (we take x as origin of coordinates). Here B is the Euclidean coordinate sphere of radius r of course. For any positive r consider the function V r (x i ) = δ ij x i x j − r 2 . Consider a geodesic γ such that V r (γ(0)) = 0 and it is tangent to B at γ(0) ∈ B. We have We shall show that for r sufficiently small this is positive, which means that V r (γ(s)) > 0 for s in some open interval about 0, but s = 0, from which it will follow that any geodesic tangent to B locally lies outsideC. Without essential loss of generality we may assume that the first term is 2. For any r 0 > 0 the set {(x, y) : δ ij x i x j ≤ r 2 0 , δ ij y i y j = 1} is compact, so there is K > 0 such that |Γ i (x, y)| < K for all i and all (x, y) in that set. Take r ≤ r 0 . Then |γ i (0)| ≤ r and |Γ j (γ(0),γ(0))| < K, whence |δ ij γ i (0)Γ j (γ(0),γ(0))| < nrK.
So provided that r < 1/(2nK), d 2 ds 2 V r • γ s=0 > 0 as required. This argument is essentially the same as the one given by Whitehead in [22]. It actually shows that the final inequality holds for all geodesics, not just those tangent to B; and in fact Whitehead gives this as his requirement for C to be convex, though this seems to us to be a stronger condition than is really necessary. Be that as it may, we have shown that for every sufficiently small r the Euclidean coordinate ball of radius r about x is a simple, convex region.
A convex region C is path connected and so connected. It is also contractible on any point