We investigate the local metrizability of Finsler spaces with m-Kropina metric F = α1+mβ−m, where β is a closed null one-form. We show that such a space is of Berwald type if and only if the (pseudo-)Riemannian metric α and one-form β have a very specific form in certain coordinates. In particular, when the signature of α is Lorentzian, α belongs to a certain subclass of the Kundt class and β generates the corresponding null congruence, and this generalizes in a natural way to arbitrary signature. We use this result to prove that the affine connection on such an m-Kropina space is locally metrizable by a (pseudo-)Riemannian metric if and only if the Ricci tensor constructed from the affine connection is symmetric. In particular, we construct all counterexamples of this type to Szabo’s metrization theorem, which has only been proven for positive definite Finsler metrics that are regular on all of the slit tangent bundle.
I. INTRODUCTION
The study of differences and similarities between positive definite Finsler geometry and indefinite Finsler geometry is still in its beginnings and far from complete.1–3 The most prominent application of indefinite (to be precise Lorentzian) Finsler geometry is the one of Finsler spacetimes in classical and quantum gravitational physics,4–16 which recently put (pseudo-)Riemannian geometry and its applications into the focus of interest.17–25 Hence, a better understanding of the properties of indefinite Finsler geometry would be of great interest for physics as well as for mathematics.
Berwald spaces constitute an important class of Finsler spaces. They can be defined by the property that the canonical (Cartan) nonlinear connection reduces to a linear connection on the tangent bundle.26 It is natural to ask under what conditions this linear connection is (Riemann) metrizable, in the sense that there exists a (pseudo-)Riemannian metric that has the given linear connection as its Levi-Civita connection. In positive definite Finsler geometry, the answer to this question was given in 1988 by Szabo’s well-known metrization theorem,27 which guarantees that in this case the connection is always metrizable. In the more general context, where the fundamental tensor is allowed to have arbitrary, not necessarily positive definite, signature, the situation is more complex. It only became clear very recently that Szabo’s metrization theorem cannot be extended in general to arbitrary signatures.28 In other words, there exist Finsler metrics of Berwald type (most examples being not positive definite and not smooth on the entire slit tangent bundle) for which the affine connection is not metrizable by a (pseudo-)Riemannian metric.
It would be of great interest to know the precise conditions for metrizability in this more general context. As a first step in this direction, we investigate in this article the metrizability of a specific class of Finsler metrics, namely, m-Kropina metrics with a closed null one-form. The main result in this article, Theorem 6, states that the affine connection of such a space is metrizable if and only if the Ricci tensor constructed from the affine connection is symmetric and gives a second equivalent characterization in terms of the local expression of the defining (pseudo-)Riemannian metric and one-form, showing in particular that certainly not all such spaces are metrizable. This contrasts the situation for one-forms that are not null. In this case it is known that such an m-Kropina space is always metrizable by a metric conformal to α (see Ref. 49).
m-Kropina metrics, also called generalized Kropina metrics, were introduced by Hashiguchi et al. in Ref. 28 as a generalization of the standard Kropina metric.29 While the original Kropina metric has found a wide range of applications, m-Kropina metrics gained some popularity in the physics literature when it was discovered that they can be used to describe a modification of special relativity with local anisotropy,30,31 named very special relativity (VSR)32,33 and later generalized to Very General Relativity (VGR)34 or General Very Special Relativity (GVSR)35 in order to account for spacetime curvature, leading to physical predictions from curved m-Kropina spacetime geodesics36 and pp-waves.37
The structure of this article is as follows: We start in Sec. II by recalling the basic notions of Finsler geometry that are relevant for our purpose and Szabo’s metrization theorem for positive definite Berwald spaces. In Sec. III, we recall the definition of m-Kropina metrics and the precise necessary and sufficient condition under which they are of Berwald type (Sec. III A). In fact, we provide a new proof of this Berwald condition in the Appendix. Subsequently, in Sec. III B, we specialize to m-Kropina metrics constructed from a (pseudo-)Riemannian metric α and a one-form β that is null with respect to this metric and closed. We first prove Lemma 3, stating that such a space is of Berwald type if and only if α and β have a very specific form in local coordinates. In particular, when the signature of α is Lorentzian, α belongs to a certain subclass of the Kundt class and β generates the corresponding null congruence. This construction generalizes in a natural way to arbitrary signature. The coordinates introduced in this lemma allow us to find a simple expression for the linear connection coefficients and the skew-symmetric part of the affine Ricci tensor. We then prove our main result, Theorem 6, providing two equivalent necessary and sufficient conditions for metrizability: symmetry of the affine Ricci tensor and a local condition for the coordinate expressions of the (pseudo-)Riemannian metric α. We end with a conclusion and discussion of the work in Sec. IV.
II. FINSLER GEOMETRY
Finsler geometry is a natural extension of Riemannian geometry.39–40 Given the philosophy that the length of a curve is obtained by integrating the norm of the tangent vector along the curve, Finsler geometry provides the most general way of assigning, smoothly, a length to curves on a smooth manifold. While in Riemannian geometry the length of a tangent vector is given by a quadratic (metric-induced) norm, Finsler geometry relaxes this quadratic requirement.
First of all, some remarks about notation are in order. Throughout this work, we will usually work in local coordinates, i.e., given a smooth manifold M we assume that some chart is provided, and we identify any p ∈ U with its image . For p ∈ U, each Y ∈ TpM in the tangent space to M at p can be written as , where the tangent vectors furnish the chart-induced basis of TpM. This provides natural local coordinates on the tangent bundle TM via the chart
These local coordinates on TM in turn provide a natural basis of its tangent spaces T(x,y)TM,
A. Finsler spaces
For our purposes, a Finsler space is triple , where M is a smooth manifold, is a conic subbundle of TM\{0} (i.e., a non-empty open subset such that for any it follows that for any λ > 0) with non-empty fibers and F, the so-called Finsler metric, is a continuous map , smooth on , that satisfies the following axioms:
- F is positively homogeneous of degree one with respect to y,(2)
The fundamental tensor, with components , is nondegenerate on .
In the positive definite setting (meaning that gij is assumed to be positive definite), one usually requires that . In the more general setting, however, this would exclude almost all interesting examples that have been studied in the literature. In fact there is no consensus on a standard definition of Finsler space when the signature is indefinite (see, e.g., Refs. 6, 17, and 41–44). A fundamental result essential for doing computations in Finsler geometry is Euler’s theorem for homogeneous functions, which states that if a function f is positively homogeneous of degree r, i.e., f(λy) = λrf(y) for all λ > 0, then . In particular, this implies the identity
Finsler geometry reduces to (pseudo-)Riemannian geometry in the case that and F2 is quadratic in the fiber coordinates yi or equivalently when gij = gij(x) depends only on the base manifold. Then, gij is a (pseudo-)Riemannian metric on M. To avoid confusion, we stress again that, in contrast to (pseudo-)Riemannian geometry, the term Finsler metric refers to the scalar F and not the tensor gij. This is standard in most Finsler geometry literature. On the other hand, we use the term (pseudo-)Riemannian metric in the standard way of (pseudo-)Riemannian geometry, referring to a gij(x) that is independent of y.
The coefficients of the Cartan nonlinear connection, the unique homogeneous (nonlinear) connection on TM that is smooth on , torsion-free, and compatible with the Finsler metric can be expressed as
Torsion-freeness is the property that , and metric-compatibility is the property that δiF2 = 0, in terms of the horizontal derivative induced by the connection,
Alternatively, metric-compatibility can be defined as the property that , in terms of the so-called dynamical covariant derivative ∇. For torsion-free homogeneous connections, the latter definition of metric-compatibility is equivalent to the former. The curvature tensor, Finsler Ricci scalar, and Finsler Ricci tensor of (M, F) are defined, respectively, as
B. Berwald spaces
A Finsler space is said to be of Berwald type if the Cartan nonlinear connection defines a linear connection on TM or, in other words, an affine connection on the base manifold, in the sense that the connection coefficients are of the form
for a set of smooth functions . (See Ref. 45 for an overview of the various equivalent characterizations of Berwald spaces and Ref. 46 for a more recent one in terms of a first order partial differential equation.) From the transformation behavior of , it follows that the functions have the correct transformation behavior to be the connection coefficients of a (torsion-free) affine connection on M. We will refer to this affine connection as the associated affine connection or simply the affine connection on the Berwald space. In addition to the curvature tensors defined in Eqs. (6)–(8), one may define additional curvature tensors for Berwald spaces: the ones associated with the uniquely defined affine connection.
where we have employed the notation and for (anti-)symmetrization. We will refer to these as the affine curvature tensor and the affine Ricci tensor, respectively. We note that coincides (up to some reinterpretations) with the hh-curvature tensor of the Chern–Rund connection. A straightforward calculation reveals the following relation between the different curvature tensors:
It is appropriate to stress here that, although Rij and coincide in the positive definite setting and more generally whenever the Finsler metric is defined on all of ,28 this is not true in general, as need not be symmetric. As this distinction is essential for our results, we end this section with a schematic overview of some important properties of the two Ricci tensors.
Ricci tensors
The Finsler Ricci Tensor Rij is constructed from the canonical nonlinear connection associated with F according to Eqs. (6)–(8). The Finsler Ricci Tensor
always exists;
is symmetric, by definition; and
contains the same information as the Finsler Ricci scalar—more precisely, Ric = Rijyiyj and .
The affine Ricci Tensor is constructed from the affine connection associated with F according to Eq. (10). The affine Ricci Tensor
exists only for Berwald spaces because otherwise there is no uniquely defined affine connection;
coincides with the Ricci tensor constructed from any of the four well-known linear connections associated with F (Chern–Rund, Berwald, Cartan, Hashiguchi); and
is not necessarily symmetric (except in the positive definite case)—its symmetrization coincides with the Finsler Ricci Tensor, see Eq. (11).
In this work, we are primarily concerned with the affine Ricci tensor and in particular its property of being in general not symmetric as it can be used to characterize whether a given Finsler space is metrizable or not.
C. Szabo’s metrization theorem
Given a Finsler space of Berwald type, the Cartan nonlinear connection defines a linear connection on TM by definition. Hence, the natural question arises whether there exists a (pseudo-)Riemannian metric (desirably of the same signature) that has this connection as its Levi-Civita connection. Simply put, is every Berwald space metrizable? For positive definite Finsler spaces defined on all of TM\{0}, the answer is affirmative as proven by Szabo.27
(Szabo’s metrization theorem). Any positive definite Berwald space is metrizable by a Riemannian metric.
The proof of this theorem relies on averaging procedures47 for which it is essential that the Finsler metric F is defined everywhere on TM\{0}. In the case of Finsler spacetimes, however, the domain where F is defined is typically only a conic subset of TM\{0} and hence the classical proof does not extend to this case. It was indeed shown in Ref. 28 that Szabo’s metrization theorem is in general not valid for Finsler spacetimes. The culprit behind all counterexamples known to the authors is the fact that the affine Ricci tensor is in general not symmetric. Clearly, the property that the affine Ricci tensor be symmetric is a necessary condition for metrizability. We will see (Theorem 6) that for m-Kropina spacetimes with closed one-form, this is in fact also a sufficient condition at least locally.
III. m-Kropina metrics
An m-Kropina space (sometimes called generalized Kropina space) is a Finsler space of (α, β)-type with a Finsler metric of the form F = α1+mβ−m, where is constructed from a (pseudo-)Riemannian metric a = aij(x)dxidxj, β = biyi is constructed from a one-form b = bi(x)dxi, and m is a real parameter. By a slight abuse of terminology, one also refers to α and β simply as the (pseudo-)Riemannian metric and the one-form, respectively. We also introduce the notation b2 ≡ ‖b‖2 = aijbibj for the squared norm of β with respect to α. Throughout the remainder of this article, all indices are raised and lowered with aij.
In the physics literature, spacetimes with metric of m-Kropina type have been dubbed Very General Relativity (VGR) spacetimes34 or General Very Special Relativity (GVSR) spacetimes,35 introduced as generalizations of Very Special Relativity (VSR),32,33 which appears in the limiting case where α is flat. In the latter case, the corresponding m-Kropina metric is often referred to as the Bogoslovsky line element. When m = 1, the m-Kropina metric reduces to the standard Kropina metric29 F = α2/β.
A. The Berwald condition
The Berwald condition for m-Kropina spaces F = α1+mβ−m formulated by Matsumoto in Ref. 48 states that such a space is of Berwald type if and only if there exists a vector field fi on M such that
Here and throughout the remainder or the article, ∇ denotes the Levi-Civita connection corresponding to the (pseudo-)Riemannian metric α. In Ref. 48, the result is proven only for non-null one-forms β in Theorem 6.3.2.3 on p. 904, but as long as the dimension of the manifold is greater than 2 the proof is still completely valid for null one-forms also. Indeed, the only purpose of the assumption b2 ≠ 0 in the proof is to guarantee that α2 and β are co-prime as polynomials in y, i.e., that α2 is not the product of β with another polynomial, α2 = βγ. However, as long as dim M > 2, this is not possible anyway, irrespective of the value of b2. To see this, note that γ = ci(x)yi is necessarily a one-form due to homogeneity. Then, it follows by differentiating twice that , showing that aij has rank . However, since aij is assumed to be nondegenerate, it must have maximal rank, so this implies that dim M ≤ 2. Hence for dim M > 2 the assumption that b2=0 is not necessary and Eq. (12) is also valid when the one-form is null.
In the special case that β is a closed and hence locally exact one-form, any fk satisfying this condition can always be written as fk = cbk for some function c on the base manifold and the condition reduces to the simpler one obtained in Ref. 34, namely,
where we remark that our c is related to C(x) in Ref. 34 by C(x) = (1 + m)c/2 and that that our power m is related to the power n in Ref. 34 by n = −2m/(1 + m). To see this, assume that Matsumoto’s Berwald condition (12) holds. We have (db)(∂i, ∂j) = ∂ibj − ∂jbi = ∇ibj − ∇jbi = (1 + m)(fibj − fjbi), so if bi is locally exact then this expression vanishes and hence fibj = fjbi must hold for all i, j, which is only possible if fi is proportional to bi (this can be checked easily at any given point in M by choosing coordinates in which bi has only one nonvanishing component at that point). In other words, fk = cbk. In this case, (12) reduces to (13). Note that the opposite holds (trivially) as well: The latter condition implies that β is locally exact.
The fact that (12) and (13) do not agree for one-forms β that are not closed has recently caused some confusion in the literature as it was suggested in a recent article34 that the latter was the correct Berwald condition in full generality, i.e., also for non-closed one-forms. This, however, clearly contradicts the results obtained above. We have taken the opportunity here to resolve this issue. It turns out that the reason for the discrepancy is that the contribution of the antisymmetric part of the covariant derivative of β was overlooked in the proof given in Ref. 34. Indeed in the Appendix, we reproduce the argument from34 taking the antisymmetric part into account and we show that the resulting Berwald condition coincides with Matsumoto’s one, (12), as expected. Thus, we want to stress here again that (12) is the correct Berwald condition in general, whereas (13) only applies to the case in which the one-form β is closed.
Finally, as also proven in Ref. 48, whenever condition (16) is satisfied, the affine connection coefficients of the Berwald space can be expressed in terms of the Christoffel symbols for the Levi-Civita connection corresponding to α as
When the one-form β is closed, and we write fk = cbk as before, this reduces to
which agrees with the result obtained in Ref. 34.
For clarity, we summarize the preceding discussion with the following proposition:
Let F = α1+mβ−m be an m-Kropina metric on a manifold M with dimension greater than two.
- F is of Berwald type if and only if there exists a smooth vector field fi satisfyingIn this case, the affine connection coefficients of the Berwald space can be expressed in terms of the Christoffel symbols for the Levi-Civita connection corresponding to α as(16)(17)
- If the one-form β is closed, F is of Berwald type if and only if there exists a smooth function c ∈ C∞(M) satisfyingIn this case, the affine connection coefficients of can be expressed as(18)(19)
Conversely, Eq. (18) implies that β must be closed.
B. Metrizability of m-Kropina spaces with closed null one-form
From here onward, we will focus on m-Kropina metrics with closed null one-form and we will assume that n = dim M > 2. In other words, we will assume that db = 0 and b2 = aijbibj = 0. This will allow us to deduce the exact conditions for local metrizability. As a remark, we point out that the case m = 1 is excluded by our definition of Finsler space as can be seen from the expression of the determinant of the fundamental tensor,
which vanishes identically when m = 1 and b2 = 0.
The following lemma extends a result from Ref. 19. We will use the convention that indices a, b, c, … run from 3 to n, whereas indices i, j, k, … run from 1 to n, and we use the notation dudv for the symmetrized tensor product of one-forms, e.g., .
From here onward, we will assume our space is Berwald. Substituting the form of c into Eq. (19) and using that and consequently , we obtain the following.
We can use the preceding results to analyze the (deviation from the) symmetry of the affine Ricci tensor, which has a very simple expression in these coordinates, as the following result shows.
Let us now prove our main result.
Let (M, F = α1+mβ−m) be an m-Kropina space of Berwald type with closed null 1-form β and with dim M > 2. The following are equivalent:
The affine connection is locally metrizable by a (pseudo-)Riemannian metric.
The affine Ricci tensor is symmetric, .
- There exist local coordinates (u, v, x3, …, xn) such that b = du andwith h some (pseudo-)Riemannian metric of dimension n − 2.(37)
Before we present the proof, we want to point out two things. First, we note that if ϕ = 0 then , i.e., the affine connection is metrizable by the defining (pseudo-)Riemannian metric α. This was to be expected, since in that case the one-form β is parallel with respect to α. It is a well-known result that any (α, β)-metric for which β is parallel with respect to α is of Berwald type and that its affine connection coincides with the Levi-Civita connection of α. Second, since is conformally equivalent to a, the two metrics have identical causal structure and, moreover, their null geodesics coincide (as unparameterized curves). This implies that the null geodesics of any F satisfying any (and hence all) of the equivalent conditions of Theorem 6 coincide with the null geodesics of the defining (pseudo-)Riemannian metric α.
Theorem 6 provides necessary and sufficient conditions for an m-Kropina space with closed null one-form to be locally metrizable. In Sec. III C, we apply our results to an explicit example from the physics literature.
C. An explicit example: Finsler VSI spacetimes
In this section, we apply our results to the Finsler VSI spacetimes presented in Ref. 34, with the four-dimensional Finsler metric
By Lemma 2, this spacetime is of Berwald type. It is in general not metrizable since the corresponding affine Ricci tensor is not symmetric. By Theorem 6, the exact condition for metrizability in this case is that ∂aϕ = 0. The case ϕ = 0 provides a Finsler version of the gyratonic pp-wave metric,49,50 which according to Theorem 6 is metrizable by the Lorentzian gyratonic pp-wave metric.
A simple nontrivial locally metrizable example is provided by the case where , ϕ(u, x) = u, and Wa(u, x) = 0. If we relabel the coordinates xa as x and y, this leads to the Finsler metric
which has an affine connection given by following nonvanishing affine connection coefficients:
As indicated by Eq. (38) in Theorem 6, this connection is metrizable by the Lorentzian metric
IV. DISCUSSION
Recent developments around the non-metrizability of Berwald spaces of indefinite (in particular, Lorentzian) signature contrast the well-known metrizability theorem by Szabo for positive definite Berwald spaces. These findings inspired us to investigate the question of metrizability for m-Kropina Finsler metrics constructed from a (pseudo-)Riemannian metric and a closed null one-form in this article. While the analogous question for the case of not null (and not necessarily closed) one-forms is known to have a simple answer, namely that any such space is metrizable, the situation is different when null one-forms are considered. Our main result, Theorem 6, gives a necessary and sufficient condition for local metrizability: that the affine Ricci tensor—the Ricci tensor constructed from the affine connection, not to be confused with the more commonly discussed Finsler Ricci tensor—must be symmetric.
Moreover, in the coordinates introduced in Lemma 3, any Berwald m-Kropina metric attains a pretty simple form. It can then be seen at a glance whether a given geometry is locally metrizable or not. Moreover, in the metrizable case, our theorem gives the explicit form of a (nonunique) (pseudo-)Riemannian metric that “metrizes” the affine connection in terms of those coordinates.
The question of metrizability is not only a natural one from the mathematical point of view, but it is also of interest in the realm of physics, particularly in the field of Finsler gravity, which asserts that the spacetime geometry of our physical universe might be Finslerian. One of its postulates is that physical objects and light rays moving only under the influence of gravity follow Finslerian geodesics through spacetime. If the Finsler metric on spacetime were metrizable, this would imply that these trajectories reduce to the geodesics of a (pseudo-)Riemannian metric, precisely as is the case in Einstein gravity. Apart from obvious mathematical implications, it would be interesting to investigate the conceptual and physical consequences of this as well.
It would obviously be of great interest to have a generalization of Theorem 6 to arbitrary Finsler spaces of Berwald type. To this effect, we note that, curiously, all examples of non-metrizable Berwald spaces currently available in the literature, as well as all of the additional examples known privately to the authors, have an affine Ricci tensor that is not symmetric. Together with the results obtained in this article in the specific case of m-Kropina metrics, this leads us to hypothesize that perhaps a Berwald space is metrizable by a (pseudo-)Riemannian metric if and only if its affine Ricci tensor is symmetric. In fact, some general results about Riemann-metrizability of arbitrary symmetric affine connections are known.52–53 An affine connection is metrizable if and only if the holonomy group is a subgroup of the generalized (pseudo-)orthogonal group.53 Hence, a future project is to investigate the structure of the holonomy group of the affine connection corresponding to a Berwald space and how it relates to the geometry-defining Finsler metric.
ACKNOWLEDGMENTS
C.P. was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Project No. 420243324 and acknowledges support from cluster of excellence Quantum Frontiers funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC-2123 QuantumFrontiers—390837967. All of us would like to acknowledge networking support provided by the COST Action CA18108, supported by COST (European Cooperation in Science and Technology).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Sjors Heefer: Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Christian Pfeifer: Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Jorn van Voorthuizen: Formal analysis (supporting); Writing – review & editing (supporting). Andrea Fuster: Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
APPENDIX: PROOF OF THE BERWALD CONDITION FOR m-KROPINA METRICS
Here, we provide a proof of the Berwald condition (16) for m-Kropina spaces F = α1+mβ−m, which also serves as extension for the proof presented in Ref. 34, where it was overlooked that the one-form β need not be closed. The derivations in this section have been performed with the help of the xAct extension of Mathematica.54
The Finsler metric L for m-Kropina spaces is given by . Using the decomposition
where Aij = A[ij](x) is the antisymmetric and Sij = S(ij)(x) is the symmetric part of the covariant derivative, we find a geodesic spray of the form
Indices are raised and lowered with the components of the (pseudo-)Riemannian metric defining α. In order to be of Berwald type, the components Gk need to be quadratic functions of y. This is the case, since for a Berwald space, and so .
To reach this goal, the first term in (A4) must either cancel with one of the other terms appearing or the contraction must lead to a term proportional to β. Hence, the free index on must be on yk, bk, or Zk = Zk(x), where Zk are the components of another vector field Z on M, in the following way:
for T = T(x) being a function and Ui = Ui(x) being the components of a one-form on M. These are the only possible terms, since by construction, is a linear function in y, and so the RHS must be as well. Factoring the linear dependence in y on both sides of the equation leads to
which then implies by the antisymmetry Aij = A[ij] that
Defining fj = (Uj − Zj), we see that a necessary condition for a m-Kropina space to be Berwald is that the antisymmetric part of the covariant derivative of the one-form β is determined by bi and an additional one-form with components fj. Using this in (A1) we get
where the factor (m + 1) was added in front of the antisymmetric part to display the following expressions more compactly. For the geodesic spray, one finds
The use of the derived expression for the antisymmetric part of the covariant derivative (A7) ensures that the second term in the geodesic spray above is quadratic in y. To achieve this for the third term for the case m ≠ 1, let us investigate the structure of the y-dependence of this term. It is of the type
where each term X(y, y) = Xijyiyj, X = B, C, D, S, P, denotes a quadratic polynomial in y and B(y, y) = α2. In order for this function to be quadratic in y, it must satisfy
for some second order polynomial P(y, y). Since the left-hand side is a second order polynomial in y, the right-hand side must be. Assuming dim M > 2, it follows by the argument given right below Eq. (12) that B(y, y) = α2 is an irreducible quadratic polynomial in y. As long as m ≠ 1, D ≠ h(x)B(y, y). Thus, P(y, y) must satisfy P(y, y) = h(x)B(y, y), for a solution of the equation to exist. Hence, the fraction in the first term of line (A11) must be proportional to an arbitrary function h = h(x) on M. This yields the equation
Taking two derivatives with respect to y, we find
Redefining fi as and combining all expressions for the covariant derivative of β finally gives the desired expression as follows:
One can easily check that this condition on bj leads to a geodesic spray given by
which indeed is quadratic, and so the m-Kropina space subject to condition (16) is indeed Berwald.
For m = 1 and b2 ≠ 0, the first term in line (A11) is quadratic in y for any tensor components Sij and we must investigate the second term of that line, which becomes
and it can only be quadratic in y if and only if
for some one-form on M with components Qi = Qi(x). The only way to achieve this is if Sij = qaij for some function q = q(x) on M, which then must satisfy
Thus, for m = 1,
For m = 1 and b2 = 0, the determinant of the metric g vanishes globally, and hence this situation does not define a Finsler space or spacetime.