---
title: Einstein-Kropina Metrics in Finsler Geometry
url: https://www.emergentmind.com/topics/einstein-kropina-metrics
type: topic
---

# Einstein-Kropina Metrics in Finsler Geometry

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 [1207.1944], [2606.07121]. 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 [1207.1944]. In arbitrary signature, they are characterized by a pseudo-Riemannian metric \(a\) and a nowhere-null vector field \(b\), with the theorem that \(L\) is Einstein if and only if \(a\) is Einstein and \(b\) is Killing [2606.07121].

## 1. Definitions, singularity, and normal forms

A Kropina metric is a singular \((\alpha,\beta)\)-metric. Its singularity is intrinsic: because \(F=\alpha^2/\beta\), the metric is only defined on a cone in each tangent space, and not on all of \(TM\setminus 0\) [1207.1944]. In the positive-definite literature this is usually the region
\[
A=\{(x,y)\in TM\mid \beta(x,y)>0\},
\]
while in the pseudo-Finsler formulation the maximal domain is
\[
\mathcal A^{\max}=\{(x,y)\in TM:\beta(x,y)\neq 0\},
\]
with nondegeneracy of the fundamental tensor imposing the further conditions \(A(x,y)\neq 0\) and \(a_x(b_x,b_x)\neq 0\) [2606.07121].

The Einstein condition is formulated in two parallel ways. For the Finsler metric \(F\), one says that \(F\) is Einstein if
\[
\operatorname{Ric}=\sigma F^2
\]
for some scalar function \(\sigma(x)\) [1207.1944]. For the \(2\)-homogeneous pseudo-Finsler Lagrangian \(L\), one says that \(L\) is Einstein if
\[
\mathrm{Ric}=\lambda\,L
\]
for some function \(\lambda\) [2606.07121]. Ricci-flat metrics are included as Einstein metrics in both conventions.

Two notational tensors dominate the tensorial analysis of Kropina geometry. With \(b_{i|j}\) denoting Levi-Civita covariant differentiation with respect to \(\alpha\), one sets
\[
r_{ij}:=\frac12(b_{i|j}+b_{j|i}),\qquad s_{ij}:=\frac12(b_{i|j}-b_{j|i}).
\]
Here \(r_{ij}\) is the symmetric part of \(\nabla b\), while \(s_{ij}\) is the antisymmetric part [1207.1944], [2308.08349]. A \(1\)-form \(\beta\) is a Killing form when \(r_{ij}=0\), closed when \(s_{ij}=0\), and a constant Killing form when \(r_{ij}=0\) and \(s_i=0\) [1207.1944].

In arbitrary signature, the pair \((a,b)\) representing \(L=L_{a,b}\) is not unique. The same \(L\) is unchanged under
\[
(a,b)\mapsto (a,-b),\qquad (a,b)\mapsto (f a,b),
\]
for any nowhere-vanishing smooth function \(f\). A standard normalization is therefore
\[
a(b,b)=1,
\]
whenever \(L\) is genuinely pseudo-Finsler [2606.07121].

## 2. Characterization theorems

The foundational positive-definite characterization is the navigation theorem. A non-Riemannian Kropina metric \(F=\alpha^2/\beta\) with navigation data \((h,W)\) is Einstein if and only if \(h\) is an Einstein Riemannian metric and \(W\) is a unit Killing vector field with respect to \(h\) [1207.1944]. In that case, the Einstein scalar of \(F\) equals that of \(h\), and for \(n\ge 3\), \(F\) is Ricci constant [1207.1944].

The same paper gives a tensorial criterion in \((\alpha,\beta)\)-language. A decisive consequence of the Einstein equation is
\[
r_{00}=c(x)\alpha^2,
\]
so the symmetric part of \(\nabla b\) is forced to be pure trace [1207.1944]. In dimension \(2\), this combines with an additional first-order condition involving \(s_{ij}\); in dimension \(n\ge 3\), it combines with two scalar identities involving \(\overline{\operatorname{Ric}}\), \(s_{ij}\), and their covariant derivatives [1207.1944]. A major special case is the constant Killing form case: if \(\beta\) is a constant Killing form, then a non-Riemannian Kropina metric \(F=\alpha^2/\beta\) is Einstein if and only if \(\alpha\) is Einstein [1207.1944].

The arbitrary-signature extension replaces \((\alpha,\beta)\) by \((a,b)\) and \(F\) by \(L=A^2/\beta^2\). With the normalization \(a(b,b)=1\), the theorem is:
\[
L\text{ is Einstein} \iff \bigl(a\text{ is Einstein and }b\text{ is Killing}\bigr).
\]
Moreover, if
\[
\mathring R_{ij}=\kappa\,a_{ij},
\]
then
\[
\mathrm{Ric}=\frac{\kappa}{4}L.
\]
This extends the positive-definite theorem of Zhang, Shen, and others to arbitrary signature, including Lorentzian signature [2606.07121].

These two formulations are complementary rather than competing. The navigation formulation is especially effective in positive-definite and homogeneous settings, while the arbitrary-signature \(L_{a,b}\) 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 \(S\)-curvature, with the explicit formula
\[
S(x,y)=\frac{n+1}{b^2}\left(r_0-\frac{\beta}{\alpha^2}r_{00}\right),
\]
so the Einstein relation \(r_{00}=c\alpha^2\) forces \(S=0\) [1207.1944]. The same paper proves a conformal rigidity theorem: any conformal map between Einstein Kropina spaces must be homothetic [1207.1944]. It also proves that if a non-Riemannian Kropina metric is Ricci-flat, then it is Berwald [1207.1944].

A related structural dichotomy comes from navigation geometry. For a Kropina space with navigation data \((h,W)\), weakly-Berwaldness is equivalent to the condition that \(W\) is a unit Killing vector field; this is exactly the definition of a strong Kropina space. Berwaldness is equivalent to \(W\) being parallel with respect to the Levi-Civita connection of \(h\) [1305.2683]. Constant flag curvature and \(p\)-scalar flag curvature are likewise governed by the same navigation field: constant flag curvature occurs if and only if \(W\) is a unit Killing vector field and \((M,h)\) has constant sectional curvature, while \(p\)-scalar flag curvature occurs if and only if \(W\) is a unit Killing vector field and \((M,h)\) has scalar sectional curvature \(K(x)\ge 0\) [1305.2683]. 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
\[
Ric_{a,c}=Ric+a\dot S-cS^2,
\]
and weight constants satisfying
\[
\nu=3(n-1)-4a(n+1)-c(n+1)^2=0,
\]
a weakly weighted Einstein-Kropina metric must have isotropic \(S\)-curvature with respect to the Busemann–Hausdorff volume form. In navigation form, such metrics are characterized by a weighted Einstein equation on \(h\),
\[
Ric_h+a(n+1)\operatorname{Hess}_h f-c(n+1)(df\otimes df)=(n-1)\mu h,
\]
together with the condition that \(W\) is Killing [2205.13854]. This extends the ordinary Einstein-Kropina pattern from Einstein \(h\) to weighted Einstein \(h\).

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
\[
r_{00}=c(x)\alpha^2.
\]
Under isotropic scalar curvature, the scalar curvature simplifies to
\[
R=-\frac{n}{4b^2}\left(2s^m s_m+b^2 s^m{}_t s^t{}_m\right),
\]
so the scalar curvature is governed entirely by the antisymmetric part \(s_{ij}\) of \(\nabla b\) [2308.08349]. 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 \(n=2k+1\), the round sphere \(S^n\) with its standard Einstein metric and the canonical unit Killing field
\[
b_{S^n}=\sum_{j=1}^{(n+1)/2}\left(x^{2j-1}\partial_{x^{2j}}-x^{2j}\partial_{x^{2j-1}}\right)
\]
yields an Einstein-Kropina metric \(L_{S^n}\) with Einstein coefficient \((n-1)/4\). The same construction on odd-dimensional anti-de Sitter space \(AdS_n\) gives Lorentzian Einstein-Kropina metrics, described as the first known Lorentzian-signature examples in this theory [2606.07121].

Product constructions enlarge the class. The \(5\)-dimensional examples
\[
S^3\times \widetilde S^2,\qquad AdS_3\times \widetilde H^2
\]
carry Einstein-Kropina metrics obtained from Einstein products with equal Einstein constants and suitable unit Killing fields [2606.07121]. A further family comes from Einstein–Sasaki geometry: on \(S^2\times S^3\), the paper constructs countably infinitely many explicit positive definite Einstein-Kropina metrics \(L_{p,q}\) using Einstein–Sasaki metrics \(a_{p,q}\) and their Reeb fields \(b_{p,q}\) [2606.07121].

Low-dimensional behavior is especially rigid. In dimension \(2\), if \(L=A^2/\beta^2\) is Einstein-Kropina with \(a(b,b)=1\), then \(a\) is flat, \(b\) is parallel, and \(L\) is locally trivial and Ricci-flat. In dimension \(3\), if \(a\) is Riemannian or Lorentzian and \(L\) is Einstein-Kropina, then \(a\) is locally isometric to \(S^3\), \(AdS_3\), or flat \(3\)-space. In dimension \(4\), the existence of proper non-Ricci-flat Einstein-Kropina metrics is left open [2606.07121].

Invariant and homogeneous constructions provide a parallel supply of examples. On a Lie group \(G\) with an Einstein left invariant Riemannian metric \(h\), any right invariant unit vector field \(W\) gives an Einstein non-Riemannian Kropina metric with the same Einstein scalar; if \(n\ge 3\), it is Ricci constant [1807.10666]. On compact semisimple Lie groups with the bi-invariant metric \(h=-B\), this yields explicit examples, including a family on \(SO(n)\) [1807.10666]. In dimension \(3\), all left invariant Einstein Kropina metrics on simply connected real Lie groups are classified: they occur precisely on \(\mathbb R^3\), \(\widetilde E_0(2)\), and \(SU(2)\), with the specified invariant Killing fields [1807.10666]. 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 [1807.10666].

## 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 \(n>2\), the \(\Lambda\)-vacuum equation studied in this setting is
\[
(n+2)\mathrm{Ric}-g^{ij}\bar\partial_i\bar\partial_j\mathrm{Ric}\,L-2\mathcal P\,L+2\Lambda\,L=0,
\]
where \(\mathcal P\) is the Landsberg scalar [2606.07121]. Within the Einstein-Kropina class, the resulting rigidity is exceptionally strong.

The classification theorem states that a Kropina metric
\[
L=\left(\frac{A}{\beta}\right)^2
\]
of arbitrary signature is an Einstein-type solution of the \(\Lambda\)-vacuum equation if and only if all of the following hold:
1. \(a\) is Ricci-flat,
2. \(b\) is Killing with respect to \(a\),
3. \(\mathring\nabla_j b^k\,\mathring\nabla_k b^i=0\),
4. \(\Lambda=0\).
In that case,
\[
\mathrm{Ric}=0,\qquad \mathcal P=0,
\]
so the metric is Ricci-flat and weakly weakly Landsberg [2606.07121].

In Riemannian or Lorentzian signature, the nilpotence condition
\[
\mathring\nabla_j b^k\,\mathring\nabla_k b^i=0
\]
forces \(b\) to be parallel. Equivalently, for Einstein-Kropina metrics in these signatures, the following are equivalent: solving the \(\Lambda\)-vacuum equation, being weakly weakly Landsberg, being Berwald, and satisfying \(\mathring\nabla_j b^i=0\) [2606.07121]. The local normal form is then
\[
a=(dx^1)^2+\sum_{k,\ell=2}^n \bar a_{k\ell}(x^2,\dots,x^n)\,dx^k\otimes dx^\ell,\qquad b=\partial_{x^1},
\]
with \(\bar a\) Ricci-flat [2606.07121].

This yields the dimension-dependent picture emphasized in the paper. In dimensions \(3\) and \(4\), every such Riemannian or Lorentzian solution is locally Euclidean or Minkowskian with \(b\) a constant translational vector field. In dimensions \(5\) and higher, nontrivial solutions appear precisely when \(a\) is a product of the real line with a Ricci-flat metric and \(b\) is the unique unit vector on the line factor [2606.07121]. The paper describes this as a surprising rigidity phenomenon: all Einstein-Kropina solutions of the \(\Lambda\)-vacuum equation are Berwald and Ricci-flat, and the cosmological constant necessarily vanishes [2606.07121].

## 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 \(m\)-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 [1302.3303], [1302.4119]. 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 \(h\) of constant sectional curvature together with a unit Killing vector field \(W\), and globally defined constant-flag-curvature model spaces reduce, up to local isometry, to Euclidean space and odd-dimensional spheres [1209.0340]. This background overlaps with, but does not exhaust, the Einstein-Kropina class.

Several recent 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 [1806.01877]. 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 [2301.09907]. 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 \(m\)-Kropina metrics supply a different caution. For
\[
F=\pm \beta^{-m}(c\alpha^2+r\beta^2)^{\frac{1+m}{2}},
\]
the theorem that Einstein metrics with \(m\notin\mathbb Z\) must be Ricci-flat is a rationality-based obstruction, but it does not apply to classical Kropina metrics because ordinary Kropina corresponds to \(m=1\) [2510.22466]. Thus the classical Einstein-Kropina problem remains distinct from the nonintegral \(m\)-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 \(h\) and unit Killing \(W\); in arbitrary signature, they are exactly those built from Einstein \(a\) and Killing \(b\) [1207.1944], [2606.07121]. 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 \(4\), and in understanding how broader geometric constructions of Kropina metrics interact with Finsler Ricci curvature and Einstein conditions.

Source: https://www.emergentmind.com/topics/einstein-kropina-metrics