State-Oblivious Collapse in Kropina Geometry

• The paper introduces a natural construction where state-oblivious collapse produces Kropina pseudo-Finsler metrics whose geodesics align with chains in integrable LC structures.
• It employs a Fefferman-type construction to transform the properties of Lagrangian contact structures into explicit conformal metrics, enabling a Fermat principle for deriving geodesics.
• Rigidity results show that projectively equivalent Kropina metrics force LC structures sharing enough chains to coincide, distinguishing these metrics from affine geodesics.

This paper is highly relevant to Kropina metrics as geometric objects arising from integrable Lagrangian contact (LC) structures via a Fefferman-type construction, but it does not study Einstein-Kropina metrics in the usual Finsler sense. Its contribution is instead to construct a natural class of Kropina-type pseudo-Finsler metrics and to show that their geodesics are exactly the chains of an LC structure. For someone interested in “Einstein-Kropina metrics,” the paper is therefore useful mainly as a source of new geometric constructions of Kropina metrics, explicit formulas, and rigidity/correspondence results, rather than as a source of Ricci or Einstein curvature formulas.

The starting geometric structure is a Lagrangian contact structure on a manifold $M^{2n+1}$: a contact distribution

$D\subset TM$

together with a splitting

$D=E\oplus F$

into rank-$n$ Legendrian subbundles, each maximally isotropic for the Levi form. The paper works mostly with the half-integrable and especially the integrable case. In local coordinates $(x^i,u,p_i)$, where $F$ is tangent to the $p_i$-directions, the contact form and adapted coframe are

$$\sigma = du - p_i\,dx^i,\qquad \theta^i = dx^i,\qquad \pi_i = dp_i - f_{ij}dx^j,$$

with

$$E=\ker(\sigma,\pi_i),\qquad F=\ker(\sigma,\theta^i),$$

equivalently

$$E=\left\langle \frac{\partial}{\partial x^i}+p_i\frac{\partial}{\partial u}+f_{ij}\frac{\partial}{\partial p_j}\right\rangle,\qquad F=\left\langle \frac{\partial}{\partial p_i}\right\rangle.$$

The functions $D\subset TM$0 determine the LC structure locally. On $D\subset TM$1, the Levi form gives a conformal class of split-signature metrics

$D\subset TM$2

The Fefferman-type construction associates to an LC structure a conformal manifold $D\subset TM$3 (the paper writes the Fefferman space as $D\subset TM$4, but I will simply call it the Fefferman space when needed) fibering over $D\subset TM$5. At the homogeneous-model level, $D\subset TM$6 for $D\subset TM$7, while the Fefferman space is $D\subset TM$8 for $D\subset TM$9. In geometric terms, the LC model is the space of para-complex null lines in a para-Hermitian vector space $D=E\oplus F$0, and the Fefferman space is the real projectivization of the null cone. In the curved setting, if $D=E\oplus F$1 is the Cartan geometry of the LC structure, then the Fefferman space is

$D=E\oplus F$2

and it carries a conformal structure induced from the inclusion

$D=E\oplus F$3

A crucial ingredient is a distinguished nowhere-vanishing null conformal Killing field

$D=E\oplus F$4

on the Fefferman space, spanning the vertical bundle of the projection

$D=E\oplus F$5

In local coordinates $D=E\oplus F$6, where $D=E\oplus F$7 is the fiber coordinate along the Fefferman fibration, this field is simply

$D=E\oplus F$8

Its orthogonal complement projects to the contact distribution $D=E\oplus F$9.

For integrable LC structures, the paper gives an explicit representative of the Fefferman conformal class. This is one of the central formulas from which the Kropina metric is later derived. In the adapted coordinates above, with $n$0 the fiber coordinate, Theorem 3.3 states that a representative metric is

$n$1

where

$n$2

As printed in the paper, the displayed formula is

$n$3

with

$n$4

The paper emphasizes that this formula uses only the defining functions $n$5 of the LC structure and their first derivatives. Also, $n$6 is a true Killing field of this chosen representative.

The Kropina metric appears in Section 4.4 from the pair $n$7 through a Fermat principle. The general geometric mechanism is the same as in Lorentzian/Finslerian Fermat constructions: null geodesics of a pseudo-Riemannian metric possessing a null Killing field project to geodesics of a Kropina-type metric on a quotient or on a transverse section. Here the quotient is not taken globally; instead one chooses a local section

$n$8

of the Fefferman projection. Then the Kropina metric on $n$9 is defined by

$(x^i,u,p_i)$0

This is the key formula. It is defined precisely on vectors transverse to the contact distribution because $(x^i,u,p_i)$1 is the projection of $(x^i,u,p_i)$2, so the denominator vanishes exactly on $(x^i,u,p_i)$3. Thus the Kropina metric is not defined on all of $(x^i,u,p_i)$4, but on the open subset

$(x^i,u,p_i)$5

This is why the authors call it a pseudo-Finsler metric rather than an everywhere-defined Finsler metric.

In the standard Kropina notation $(x^i,u,p_i)$6, the paper does not explicitly rewrite $(x^i,u,p_i)$7 in that form, but (4.17) is exactly of that type: the numerator is a quadratic form in $(x^i,u,p_i)$8, namely the pullback of $(x^i,u,p_i)$9 to the section, while the denominator is a 1-form, namely the pullback of $F$0. So one may interpret

$F$1

and then

$F$2

Here $F$3 is generally indefinite because the Fefferman metric has split signature; hence the result is a pseudo-Finsler Kropina metric rather than a positive definite one.

The dependence on the choice of section is controlled very explicitly. If $F$4 is the fiber coordinate and $F$5 are two local sections, then with

$F$6

the corresponding Kropina metrics satisfy

$F$7

So changing the section adds an exact 1-form. Consequently the unparameterized geodesics are independent of the section. This is one of the reasons the construction is geometrically meaningful.

The variational principle is also stated. The geodesics of a Kropina metric are the stationary curves of the length functional

$F$8

for curves $F$9 transverse to $p_i$0. Since $p_i$1 is homogeneous of degree one in velocity, the functional is invariant under orientation-preserving reparametrization, so what is geometrically natural here are unparameterized geodesics. Their Euler–Lagrange equations are

$p_i$2

where $p_i$3 indexes coordinates on $p_i$4. The authors stress that this gives an efficient way to derive ODE systems for chains.

The relation between chains, null-chains, and Kropina geodesics is built in stages. In the Cartan-geometric language, chains are canonical curves of type $p_i$5, while null-chains are canonical curves of certain null directions in $p_i$6. The paper first analyzes the homogeneous model in detail. There chains are exactly para-complex projectivizations of intersections of the model null cone with non-degenerate para-complex planes; null-chains come from null para-complex planes. Then, via the development of curves and the Fefferman correspondence, the authors identify these canonical curves with projections of null-geodesics in the Fefferman space.

The precise correspondence is Theorem 4.6: if the LC structure is integrable and $p_i$7 is the null conformal Killing field on the Fefferman space, then

• chains are exactly projections of null-geodesics that are not perpendicular to $p_i$8,
• null-chains are exactly projections of null-geodesics that are perpendicular to $p_i$9.

This distinction is essential for the Kropina story, because the Fermat construction applies precisely to null-geodesics not perpendicular to $$\sigma = du - p_i\,dx^i,\qquad \theta^i = dx^i,\qquad \pi_i = dp_i - f_{ij}dx^j,$$0. Combining the Fermat principle with Theorem 4.6 gives the main Kropina theorem of the paper:

$$\sigma = du - p_i\,dx^i,\qquad \theta^i = dx^i,\qquad \pi_i = dp_i - f_{ij}dx^j,$$1

This is the central theorem linking Kropina metrics to LC geometry. The geodesics here are naturally understood as unparameterized geodesics, since both chains and Kropina geodesics come with projective/reparametrization freedom. The theorem requires the LC structure to be integrable, because then the Fefferman conformal Cartan connection is normal and the relevant null-geodesics are the actual null-geodesics of the Fefferman metric.

The paper gives a particularly concrete illustration in dimension three, where every LC structure is automatically integrable. Writing coordinates $$\sigma = du - p_i\,dx^i,\qquad \theta^i = dx^i,\qquad \pi_i = dp_i - f_{ij}dx^j,$$2 with

$$\sigma = du - p_i\,dx^i,\qquad \theta^i = dx^i,\qquad \pi_i = dp_i - f_{ij}dx^j,$$3

the Fefferman metric becomes

$$\sigma = du - p_i\,dx^i,\qquad \theta^i = dx^i,\qquad \pi_i = dp_i - f_{ij}dx^j,$$4

For the local section $$\sigma = du - p_i\,dx^i,\qquad \theta^i = dx^i,\qquad \pi_i = dp_i - f_{ij}dx^j,$$5, the induced Kropina metric evaluated on a curve

$$\sigma = du - p_i\,dx^i,\qquad \theta^i = dx^i,\qquad \pi_i = dp_i - f_{ij}dx^j,$$6

is

$$\sigma = du - p_i\,dx^i,\qquad \theta^i = dx^i,\qquad \pi_i = dp_i - f_{ij}dx^j,$$7

The corresponding Euler–Lagrange equations are written explicitly: $$\sigma = du - p_i\,dx^i,\qquad \theta^i = dx^i,\qquad \pi_i = dp_i - f_{ij}dx^j,$$8 and

$$\sigma = du - p_i\,dx^i,\qquad \theta^i = dx^i,\qquad \pi_i = dp_i - f_{ij}dx^j,$$9

By Theorem 4.7, these are exactly the ODEs for chains. The paper comments that this derivation via Kropina geometry is much more straightforward than alternative methods.

There is also important rigidity information. Because chains are Kropina geodesics, the authors import a recent rigidity theorem on projectively equivalent Kropina metrics from Cheng–Marugame–Matveev–Montgomery. They use the result that if two Kropina metrics of the form

$$E=\ker(\sigma,\pi_i),\qquad F=\ker(\sigma,\theta^i),$$0

are projectively equivalent on a sufficiently big family of curves, with non-integrable kernels $$E=\ker(\sigma,\pi_i),\qquad F=\ker(\sigma,\theta^i),$$1 and $$E=\ker(\sigma,\pi_i),\qquad F=\ker(\sigma,\theta^i),$$2 non-degenerate on $$E=\ker(\sigma,\pi_i),\qquad F=\ker(\sigma,\theta^i),$$3, then

$$E=\ker(\sigma,\pi_i),\qquad F=\ker(\sigma,\theta^i),$$4

where $$E=\ker(\sigma,\pi_i),\qquad F=\ker(\sigma,\theta^i),$$5 is constant and $$E=\ker(\sigma,\pi_i),\qquad F=\ker(\sigma,\theta^i),$$6 is a closed 1-form. Applied to the present construction, this yields Proposition 4.8: two integrable LC structures sharing a sufficiently big family of chains must coincide. Thus the Kropina realization is not merely formal; it yields structural rigidity.

Likewise, Proposition 4.9 says there is no affine connection having any sufficiently big family of chains among its geodesics. The argument compares the ODE form of pseudo-Finsler/Kropina geodesics,

$$E=\ker(\sigma,\pi_i),\qquad F=\ker(\sigma,\theta^i),$$7

with affine geodesic equations, up to reparametrization,

$$E=\ker(\sigma,\pi_i),\qquad F=\ker(\sigma,\theta^i),$$8

The conclusion is that chains genuinely belong to the pseudo-Finsler/Kropina world rather than affine geometry.

As for curvature and Einstein-type questions, the paper is explicit: it studies the conformal Fefferman metric, chains, null-geodesics, and Kropina realization, but it does not compute or discuss the Ricci tensor, scalar curvature, flag curvature, or Einstein condition of the associated Kropina metrics. There are no results here of the form “the Kropina metric is Einstein iff …”, nor any classification of Einstein-Kropina metrics. Similarly, the paper does not investigate whether the Fefferman metric itself is Einstein or conformally Einstein, except insofar as it is part of a conformal structure with a null Killing field. So for the query “Einstein-Kropina Metrics,” this paper supplies indirect geometric background, not direct Einstein theory.

Still, several features are potentially useful for future Einstein-Kropina study. First, the construction produces a large natural family of Kropina metrics from integrable LC structures. Second, the Kropina data are given explicitly in terms of the local functions $$E=\ker(\sigma,\pi_i),\qquad F=\ker(\sigma,\theta^i),$$9, hence one could in principle compute spray coefficients, flag curvature, or Ricci quantities from these formulas. Third, the section-change law

$$E=\left\langle \frac{\partial}{\partial x^i}+p_i\frac{\partial}{\partial u}+f_{ij}\frac{\partial}{\partial p_j}\right\rangle,\qquad F=\left\langle \frac{\partial}{\partial p_i}\right\rangle.$$0

means geodesics are insensitive to exact perturbations; for curvature questions one would need to understand which Finsler quantities are section-dependent and which are geometric. Fourth, the underlying pseudo-Riemannian origin via a null Killing field places these Kropina metrics in the broader framework of Fermat metrics, wind Finslerian structures, and Zermelo-type constructions in indefinite signature.

The literature cited by the paper helps situate this. The direct antecedent is [16], Cheng–Marugame–Matveev–Montgomery, where chains in CR geometry are realized as geodesics of a Kropina metric. The present paper is the LC analogue. The authors also cite Caponio–Javaloyes–Sánchez [23] on wind Finslerian structures and Fermat principles in spacetimes, as well as standard Finsler background [3] for pseudo-Finsler geodesic equations. The use of the term “Fermat principle” is exactly in this sense: null geodesics of a metric with a null Killing field project to geodesics of a Kropina metric. Thus the paper places its Kropina metrics squarely within the modern pseudo-Finsler/Fermat/Zermelo framework, even though it does not itself discuss Einstein conditions.

The projective-induced LC structures in Section 5 are also potentially relevant as a source of examples. There the LC structure need not be integrable in higher dimension, so Theorems 4.6 and 4.7 hold only in dimension two on the projective base or in the flat case. The Fefferman metric is still given explicitly: $$E=\left\langle \frac{\partial}{\partial x^i}+p_i\frac{\partial}{\partial u}+f_{ij}\frac{\partial}{\partial p_j}\right\rangle,\qquad F=\left\langle \frac{\partial}{\partial p_i}\right\rangle.$$1 and it is related to a Patterson–Walker metric by

$$E=\left\langle \frac{\partial}{\partial x^i}+p_i\frac{\partial}{\partial u}+f_{ij}\frac{\partial}{\partial p_j}\right\rangle,\qquad F=\left\langle \frac{\partial}{\partial p_i}\right\rangle.$$2

This gives another explicit ambient source from which Kropina-type metrics might be extracted when the hypotheses of the Fermat principle are satisfied. Again, however, no Einstein conclusions are drawn.

For theorem statements and proof ideas, the main Kropina-related chain of logic is as follows. Proposition 4.5 proves in the homogeneous model that null-geodesics in the Fefferman space project either to points, chains, or null-chains according to whether they are tangent to, transverse to, or perpendicular to the distinguished null field $$E=\left\langle \frac{\partial}{\partial x^i}+p_i\frac{\partial}{\partial u}+f_{ij}\frac{\partial}{\partial p_j}\right\rangle,\qquad F=\left\langle \frac{\partial}{\partial p_i}\right\rangle.$$3. The proof uses explicit ambient representatives of tangent vectors and null-geodesics in the real projectivized null cone. Theorem 4.6 transfers this to curved, integrable LC structures via Cartan development. Then Theorem 4.7 combines Theorem 4.6 with the Fermat principle: null-geodesics not perpendicular to $$E=\left\langle \frac{\partial}{\partial x^i}+p_i\frac{\partial}{\partial u}+f_{ij}\frac{\partial}{\partial p_j}\right\rangle,\qquad F=\left\langle \frac{\partial}{\partial p_i}\right\rangle.$$4 correspond to Kropina geodesics, hence chains are precisely Kropina geodesics.

The final assessment for the query “Einstein-Kropina Metrics” is therefore clear. This paper is not a paper about Einstein-Kropina metrics in the standard Finsler-curvature sense. It contains no Einstein equations, no Ricci computations for $$E=\left\langle \frac{\partial}{\partial x^i}+p_i\frac{\partial}{\partial u}+f_{ij}\frac{\partial}{\partial p_j}\right\rangle,\qquad F=\left\langle \frac{\partial}{\partial p_i}\right\rangle.$$5, and no characterization of when these Kropina metrics are Einstein. However, it is very useful if one wants to understand a sophisticated and explicit geometric source of Kropina metrics: namely, integrable Lagrangian contact structures via Fefferman spaces and null Killing fields. It gives exact formulas for the ambient conformal metric and the induced Kropina metric, identifies their geodesics with chains, supplies Euler–Lagrange equations, and derives rigidity consequences. To study actual Einstein-Kropina conditions, one would still need other papers from the Kropina/Finsler literature—especially those dealing with Ricci curvature of Kropina metrics, navigation constructions, and Einstein conditions for $$E=\left\langle \frac{\partial}{\partial x^i}+p_i\frac{\partial}{\partial u}+f_{ij}\frac{\partial}{\partial p_j}\right\rangle,\qquad F=\left\langle \frac{\partial}{\partial p_i}\right\rangle.$$6. But this paper provides a rich geometric framework and many explicit examples from which such Einstein-Kropina investigations could plausibly begin.