Einstein-Kropina Metrics in Finsler Geometry

• Einstein-Kropina metrics are singular Finsler metrics defined as F = α²/β (or L = A²/β²) that satisfy an Einstein condition via proportional Ricci curvature.
• They are characterized in positive-definite settings by navigation data (h, W) and in arbitrary signature by a pseudo-Riemannian metric a and a Killing field b.
• These metrics exhibit vanishing S-curvature and strong rigidity properties, with applications in homogeneous constructions, CR geometry, and Finsler gravity.

Einstein–Kropina metrics are Kropina metrics whose Finsler Ricci curvature satisfies an Einstein condition. In the standard positive-definite $(\alpha,\beta)$-notation, a Kropina metric is

$$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$

defined on the conic domain where $\beta>0$; in the arbitrary-signature pseudo-Finsler formulation, one works with the $2$-homogeneous Lagrangian

$L=\frac{A^2}{\beta^2},\qquad A(x,y)=a_x(y,y),$

on a conic subbundle of $\{ \beta\neq 0\}$ where the fundamental tensor is nondegenerate (Zhang et al., 2012, Heefer et al., 5 Jun 2026). The modern theory has two principal formulations. In the positive-definite setting, Einstein-Kropina metrics are characterized by navigation data $(h,W)$, where $h$ is a Riemannian metric and $W$ is a unit Killing vector field (Zhang et al., 2012). In arbitrary signature, they are characterized by a pseudo-Riemannian metric $a$ and a nowhere-null vector field $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$0, with the theorem that $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$1 is Einstein if and only if $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$2 is Einstein and $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$3 is Killing (Heefer et al., 5 Jun 2026).

1. Definitions, singularity, and normal forms

A Kropina metric is a singular $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$4-metric. Its singularity is intrinsic: because $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$5, the metric is only defined on a cone in each tangent space, and not on all of $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$6 (Zhang et al., 2012). In the positive-definite literature this is usually the region

$$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$7

while in the pseudo-Finsler formulation the maximal domain is

$$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$8

with nondegeneracy of the fundamental tensor imposing the further conditions $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$9 and $\beta>0$0 (Heefer et al., 5 Jun 2026).

The Einstein condition is formulated in two parallel ways. For the Finsler metric $\beta>0$1, one says that $\beta>0$2 is Einstein if

$\beta>0$3

for some scalar function $\beta>0$4 (Zhang et al., 2012). For the $\beta>0$5-homogeneous pseudo-Finsler Lagrangian $\beta>0$6, one says that $\beta>0$7 is Einstein if

$\beta>0$8

for some function $\beta>0$9 (Heefer et al., 5 Jun 2026). Ricci-flat metrics are included as Einstein metrics in both conventions.

Two notational tensors dominate the tensorial analysis of Kropina geometry. With $2$0 denoting Levi-Civita covariant differentiation with respect to $2$1, one sets

$2$2

Here $2$3 is the symmetric part of $2$4, while $2$5 is the antisymmetric part (Zhang et al., 2012, Liu et al., 2023). A $2$6-form $2$7 is a Killing form when $2$8, closed when $2$9, and a constant Killing form when $L=\frac{A^2}{\beta^2},\qquad A(x,y)=a_x(y,y),$0 and $L=\frac{A^2}{\beta^2},\qquad A(x,y)=a_x(y,y),$1 (Zhang et al., 2012).

In arbitrary signature, the pair $L=\frac{A^2}{\beta^2},\qquad A(x,y)=a_x(y,y),$2 representing $L=\frac{A^2}{\beta^2},\qquad A(x,y)=a_x(y,y),$3 is not unique. The same $L=\frac{A^2}{\beta^2},\qquad A(x,y)=a_x(y,y),$4 is unchanged under

$L=\frac{A^2}{\beta^2},\qquad A(x,y)=a_x(y,y),$5

for any nowhere-vanishing smooth function $L=\frac{A^2}{\beta^2},\qquad A(x,y)=a_x(y,y),$6. A standard normalization is therefore

$L=\frac{A^2}{\beta^2},\qquad A(x,y)=a_x(y,y),$7

whenever $L=\frac{A^2}{\beta^2},\qquad A(x,y)=a_x(y,y),$8 is genuinely pseudo-Finsler (Heefer et al., 5 Jun 2026).

2. Characterization theorems

The foundational positive-definite characterization is the navigation theorem. A non-Riemannian Kropina metric $L=\frac{A^2}{\beta^2},\qquad A(x,y)=a_x(y,y),$9 with navigation data $\{ \beta\neq 0\}$0 is Einstein if and only if $\{ \beta\neq 0\}$1 is an Einstein Riemannian metric and $\{ \beta\neq 0\}$2 is a unit Killing vector field with respect to $\{ \beta\neq 0\}$3 (Zhang et al., 2012). In that case, the Einstein scalar of $\{ \beta\neq 0\}$4 equals that of $\{ \beta\neq 0\}$5, and for $\{ \beta\neq 0\}$6, $\{ \beta\neq 0\}$7 is Ricci constant (Zhang et al., 2012).

The same paper gives a tensorial criterion in $\{ \beta\neq 0\}$8-language. A decisive consequence of the Einstein equation is

$\{ \beta\neq 0\}$9

so the symmetric part of $(h,W)$0 is forced to be pure trace (Zhang et al., 2012). In dimension $(h,W)$1, this combines with an additional first-order condition involving $(h,W)$2; in dimension $(h,W)$3, it combines with two scalar identities involving $(h,W)$4, $(h,W)$5, and their covariant derivatives (Zhang et al., 2012). A major special case is the constant Killing form case: if $(h,W)$6 is a constant Killing form, then a non-Riemannian Kropina metric $(h,W)$7 is Einstein if and only if $(h,W)$8 is Einstein (Zhang et al., 2012).

The arbitrary-signature extension replaces $(h,W)$9 by $h$0 and $h$1 by $h$2. With the normalization $h$3, the theorem is: $h$4 Moreover, if

$h$5

then

$h$6

This extends the positive-definite theorem of Zhang, Shen, and others to arbitrary signature, including Lorentzian signature (Heefer et al., 5 Jun 2026).

These two formulations are complementary rather than competing. The navigation formulation is especially effective in positive-definite and homogeneous settings, while the arbitrary-signature $h$7 formulation is adapted to pseudo-Finsler geometry and relativistic applications. A plausible implication is that the same structural rigidity survives across signatures because the Einstein condition continues to collapse the admissible background data to an Einstein metric plus a Killing direction.

3. Curvature consequences and refinements

Several consequences of the Einstein condition are now standard. Every Einstein Kropina metric has vanishing $h$8-curvature, with the explicit formula

$h$9

so the Einstein relation $W$0 forces $W$1 (Zhang et al., 2012). The same paper proves a conformal rigidity theorem: any conformal map between Einstein Kropina spaces must be homothetic (Zhang et al., 2012). It also proves that if a non-Riemannian Kropina metric is Ricci-flat, then it is Berwald (Zhang et al., 2012).

A related structural dichotomy comes from navigation geometry. For a Kropina space with navigation data $W$2, weakly-Berwaldness is equivalent to the condition that $W$3 is a unit Killing vector field; this is exactly the definition of a strong Kropina space. Berwaldness is equivalent to $W$4 being parallel with respect to the Levi-Civita connection of $W$5 (Yoshikawa et al., 2013). Constant flag curvature and $W$6-scalar flag curvature are likewise governed by the same navigation field: constant flag curvature occurs if and only if $W$7 is a unit Killing vector field and $W$8 has constant sectional curvature, while $W$9-scalar flag curvature occurs if and only if $a$0 is a unit Killing vector field and $a$1 has scalar sectional curvature $a$2 (Yoshikawa et al., 2013). These are not Einstein theorems, but they describe curvature regimes that frequently intersect Einstein-Kropina constructions.

A weighted generalization is developed for weakly weighted Einstein-Kropina metrics. For generalized weighted Ricci curvature

$a$3

and weight constants satisfying

$a$4

a weakly weighted Einstein-Kropina metric must have isotropic $a$5-curvature with respect to the Busemann–Hausdorff volume form. In navigation form, such metrics are characterized by a weighted Einstein equation on $a$6,

$a$7

together with the condition that $a$8 is Killing (Cheng et al., 2022). This extends the ordinary Einstein-Kropina pattern from Einstein $a$9 to weighted Einstein $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$00.

A later tensor-analytic development gives explicit formulas for the Ricci curvature, Ricci tensor, and scalar curvature of a Kropina metric, and characterizes isotropic scalar curvature by the condition

$$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$01

Under isotropic scalar curvature, the scalar curvature simplifies to

$$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$02

so the scalar curvature is governed entirely by the antisymmetric part $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$03 of $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$04 (Liu et al., 2023). This is not itself an Einstein classification, but it supplies explicit curvature data useful in Einstein-Kropina analysis.

4. Examples and homogeneous constructions

The direct arbitrary-signature theory produces several explicit families of Einstein-Kropina metrics. In odd dimension $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$05, the round sphere $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$06 with its standard Einstein metric and the canonical unit Killing field

$$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$07

yields an Einstein-Kropina metric $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$08 with Einstein coefficient $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$09. The same construction on odd-dimensional anti-de Sitter space $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$10 gives Lorentzian Einstein-Kropina metrics, described as the first known Lorentzian-signature examples in this theory (Heefer et al., 5 Jun 2026).

Product constructions enlarge the class. The $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$11-dimensional examples

$$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$12

carry Einstein-Kropina metrics obtained from Einstein products with equal Einstein constants and suitable unit Killing fields (Heefer et al., 5 Jun 2026). A further family comes from Einstein–Sasaki geometry: on $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$13, the paper constructs countably infinitely many explicit positive definite Einstein-Kropina metrics $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$14 using Einstein–Sasaki metrics $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$15 and their Reeb fields $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$16 (Heefer et al., 5 Jun 2026).

Low-dimensional behavior is especially rigid. In dimension $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$17, if $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$18 is Einstein-Kropina with $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$19, then $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$20 is flat, $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$21 is parallel, and $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$22 is locally trivial and Ricci-flat. In dimension $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$23, if $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$24 is Riemannian or Lorentzian and $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$25 is Einstein-Kropina, then $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$26 is locally isometric to $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$27, $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$28, or flat $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$29-space. In dimension $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$30, the existence of proper non-Ricci-flat Einstein-Kropina metrics is left open (Heefer et al., 5 Jun 2026).

Invariant and homogeneous constructions provide a parallel supply of examples. On a Lie group $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$31 with an Einstein left invariant Riemannian metric $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$32, any right invariant unit vector field $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$33 gives an Einstein non-Riemannian Kropina metric with the same Einstein scalar; if $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$34, it is Ricci constant (Hosseini et al., 2018). On compact semisimple Lie groups with the bi-invariant metric $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$35, this yields explicit examples, including a family on $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$36 (Hosseini et al., 2018). In dimension $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$37, all left invariant Einstein Kropina metrics on simply connected real Lie groups are classified: they occur precisely on $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$38, $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$39, and $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$40, with the specified invariant Killing fields (Hosseini et al., 2018). On homogeneous spaces, invariant Einstein Kropina metrics arise from invariant Einstein Riemannian metrics together with invariant Killing fields; the paper constructs such metrics on certain spheres and proves that projective spaces do not admit homogeneous non-Riemannian Einstein Kropina metrics (Hosseini et al., 2018).

5. Finsler gravity and rigidity

A major recent development is the interaction between Einstein-Kropina metrics and the Pfeifer–Wohlfarth vacuum equation in Finsler gravity. For $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$41, the $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$42-vacuum equation studied in this setting is

$$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$43

where $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$44 is the Landsberg scalar (Heefer et al., 5 Jun 2026). Within the Einstein-Kropina class, the resulting rigidity is exceptionally strong.

The classification theorem states that a Kropina metric

$$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$45

of arbitrary signature is an Einstein-type solution of the $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$46-vacuum equation if and only if all of the following hold:

1. $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$47 is Ricci-flat,
2. $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$48 is Killing with respect to $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$49,
3. $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$50,
4. $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$51. In that case,

$$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$52

so the metric is Ricci-flat and weakly weakly Landsberg (Heefer et al., 5 Jun 2026).

In Riemannian or Lorentzian signature, the nilpotence condition

$$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$53

forces $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$54 to be parallel. Equivalently, for Einstein-Kropina metrics in these signatures, the following are equivalent: solving the $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$55-vacuum equation, being weakly weakly Landsberg, being Berwald, and satisfying $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$56 (Heefer et al., 5 Jun 2026). The local normal form is then

$$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$57

with $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$58 Ricci-flat (Heefer et al., 5 Jun 2026).

This yields the dimension-dependent picture emphasized in the paper. In dimensions $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$59 and $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$60, every such Riemannian or Lorentzian solution is locally Euclidean or Minkowskian with $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$61 a constant translational vector field. In dimensions $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$62 and higher, nontrivial solutions appear precisely when $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$63 is a product of the real line with a Ricci-flat metric and $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$64 is the unique unit vector on the line factor (Heefer et al., 5 Jun 2026). The paper describes this as a surprising rigidity phenomenon: all Einstein-Kropina solutions of the $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$65-vacuum equation are Berwald and Ricci-flat, and the cosmological constant necessarily vanishes (Heefer et al., 5 Jun 2026).

6. Broader geometric context, related rigidity results, and scope

Einstein-Kropina geometry sits inside a wider Kropina literature whose strongest theorems often concern adjacent, but distinct, curvature regimes. A common source of confusion is the role of constant flag curvature, scalar flag curvature, projective flatness, and Douglasianity. In the singular Kropina and $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$66-Kropina setting, projectively flat metrics with constant flag curvature are often forced to be locally Minkowskian or Berwald, which is stronger than being Einstein in many settings (Yang, 2013, Yang, 2013). These rigidity theorems are structurally important, but they do not amount to a general Einstein classification.

The navigation viewpoint clarifies part of this hierarchy. Kropina metrics of constant flag curvature are governed by a Riemannian metric $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$67 of constant sectional curvature together with a unit Killing vector field $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$68, and globally defined constant-flag-curvature model spaces reduce, up to local isometry, to Euclidean space and odd-dimensional spheres (Yoshikawa et al., 2012). This background overlaps with, but does not exhaust, the Einstein-Kropina class.

Several papers construct natural Kropina metrics from parabolic and contact geometry without deriving Einstein criteria. In CR geometry, chains are geodesics of a Kropina metric obtained from the Fefferman metric and a null Killing field; under a pseudo-Einstein contact form with positive Tanaka–Webster scalar curvature, this construction becomes global (Cheng et al., 2018). An analogous Fefferman-type construction from integrable Lagrangian contact structures produces Kropina pseudo-Finsler metrics whose geodesics are the chains of the underlying contact geometry (Ma et al., 2023). These papers provide geometrically rich sources of Kropina metrics and strong projective-rigidity statements, but they do not compute Finsler Ricci curvature or classify when the resulting Kropina metrics are Einstein.

Generalized $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$69-Kropina metrics supply a different caution. For

$$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$70

the theorem that Einstein metrics with $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$71 must be Ricci-flat is a rationality-based obstruction, but it does not apply to classical Kropina metrics because ordinary Kropina corresponds to $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$72 (Taha, 26 Oct 2025). Thus the classical Einstein-Kropina problem remains distinct from the nonintegral $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$73-Kropina rigidity mechanism.

Taken together, these results give the present state of the subject. The direct Einstein theory is now structurally sharp: in positive-definite navigation language, Einstein-Kropina metrics are exactly those built from Einstein $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$74 and unit Killing $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$75; in arbitrary signature, they are exactly those built from Einstein $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$76 and Killing $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$77 (Zhang et al., 2012, Heefer et al., 5 Jun 2026). The main open territory lies not in the basic characterization, but in explicit classification beyond the known families, in low-dimensional exceptional behavior such as dimension $$F=\frac{\alpha^2}{\beta},\qquad \alpha=\sqrt{a_{ij}(x)y^iy^j},\qquad \beta=b_i(x)y^i,$$78, and in understanding how broader geometric constructions of Kropina metrics interact with Finsler Ricci curvature and Einstein conditions.