Pseudo-Finsler Metrics: Concepts & Applications

• Pseudo-Finsler metrics are direction-dependent structures defined via a homogeneous function on a conic tangent bundle that generalizes both Finsler and pseudo-Riemannian metrics.
• They are characterized by a fundamental tensor obtained as the fiberwise Hessian, whose indefinite signature influences null structures and geodesic behavior.
• Applications span pseudo-Finsler spacetimes, anisotropic connections, and modified gravity theories, with explicit models like Randers-type, Kropina, and Finsleroid metrics.

Pseudo-Finsler metrics are direction-dependent metric structures that generalize both ordinary Finsler metrics and pseudo-Riemannian metrics by allowing the fundamental tensor to have arbitrary signature on a conic domain of the tangent bundle. In the supplied literature, they are described either through a positively $1$-homogeneous function $F$ on a conic subset of $TM\setminus\{0\}$, or through a positively $2$-homogeneous Lagrangian $L$, with the fundamental tensor obtained as a fiberwise Hessian. This framework includes standard Finsler geometry when the Hessian is positive definite, pseudo-Riemannian geometry when it is independent of direction, and Lorentz-Finsler or Finsler-spacetime models when the Hessian has Lorentzian signature on a timelike cone (Javaloyes et al., 2014, Javaloyes et al., 2011, Javaloyes et al., 2021).

1. Foundational definitions and conventions

A recurrent convention is to start from a function $F(x,v)$ on the tangent bundle, positively homogeneous of degree $1$,

$F(x,Kv)=K\,F(x,v),$

so that the length functional

$\int F\!\left(x(t),\frac{dx}{dt}\right)\,dt$

is independent of reparametrization. The associated Finsler metric is then

$$g_{ab}(x,v)=\frac{1}{2}\,\frac{\partial^2}{\partial v^a\partial v^b}\big[F^2(x,v)\big].$$

An equivalent convention in several of the supplied papers uses a $F$0-homogeneous function $F$1 on a conic domain $F$2,

$F$3

with

$F$4

In both conventions, the fundamental tensor is direction-dependent and homogeneous of degree $F$5 in the velocity variables; standard identities include

$F$6

If $F$7 is positive definite on all $F$8, one recovers an ordinary Finsler metric; if $F$9 is independent of $TM\setminus\{0\}$0, one recovers a pseudo-Riemannian metric; if the index is $TM\setminus\{0\}$1, one obtains Lorentzian Finsler geometry or a Finsler spacetime (Javaloyes et al., 2014, Torromé et al., 2013, Tomasiello, 2024).

The supplied papers also distinguish several notions that are often conflated. A pseudo-Finsler metric may be defined on all of $TM\setminus\{0\}$2 while allowing the fundamental tensor to be indefinite or degenerate. A conic Finsler metric retains positive definiteness but is defined only on an open conic domain $TM\setminus\{0\}$3. A conic pseudo-Finsler metric allows both restrictions simultaneously. This distinction is structurally important because many classical examples, such as Kropina and Matsumoto metrics, live naturally on proper conic domains rather than on the full tangent bundle (Javaloyes et al., 2011).

2. Conic domains, signature, indicatrices, and null structure

The natural domain of a pseudo-Finsler metric is frequently a conic subset $TM\setminus\{0\}$4, fiberwise open and stable under positive rescaling. The corresponding indicatrix

$TM\setminus\{0\}$5

or $TM\setminus\{0\}$6 encodes the local causal or norm geometry, while the null set

$TM\setminus\{0\}$7

plays the role of the light cone in Lorentz-Finsler settings. In Finsler spacetime terminology, one works with a connected conic subbundle $TM\setminus\{0\}$8 of future-pointing timelike vectors such that $TM\setminus\{0\}$9 on each $2$0, $2$1 has Lorentzian signature $2$2, and $2$3 extends continuously to $2$4 on $2$5; the papers emphasize that each $2$6 is a convex cone (Voicu et al., 2023, Javaloyes et al., 2021).

A central technical difference from positive-definite Finsler geometry is the existence of nonzero null vectors. In Euclidean-signature Finsler geometry, the main singularity is usually confined to the zero section, so one works on the slit tangent bundle. In Lorentzian signature, null vectors are nonzero, and this shifts the singular behavior to the entire null cone. Skakala and Visser show this sharply in the bi-metric setting: given two Lorentzian metrics $2$7 and $2$8, with elementary norms

$2$9

the natural combined pseudo-Finsler norm is

$L$0

This unifies the two signal cones because $L$1 whenever either $L$2 or $L$3. However, the derived metric contains factors such as $L$4 and $L$5, so it becomes singular on either null cone. The paper’s conclusion is that, in Lorentzian signature, a pseudo-Finsler norm may be perfectly reasonable while the associated pseudo-Finsler metric is generically ill-defined on the null cones (Skakala et al., 2010).

This Lorentzian obstruction is not a marginal pathology. It means that many standard theorems from positive-definite Finsler geometry cannot be transferred unchanged to pseudo-Finsler settings, especially in multi-cone or signal-propagation problems. A common misconception is that Lorentzian pseudo-Finsler geometry can be treated as a straightforward analytic continuation of Euclidean Finsler geometry; the supplied literature explicitly warns against this (Skakala et al., 2010, Javaloyes et al., 2011).

3. Connections, geodesics, Jacobi fields, and isometries

Pseudo-Finsler geometry combines a canonical nonlinear connection, derived from the geodesic spray, with linear or anisotropic connections encoding covariant differentiation. In coordinates, the spray takes the form

$L$6

and the associated nonlinear connection satisfies

$L$7

A major conceptual simplification in the recent literature is the identification of anisotropic connections with vertically trivial linear Finsler connections: if $L$8 are anisotropic Christoffel symbols, then

$L$9

and conversely

$F(x,v)$0

For pseudo-Finsler metrics there is a unique torsion-free, metric-compatible anisotropic connection, the Levi-Civita–Chern anisotropic connection, which is exactly the Chern connection viewed intrinsically on the base manifold (Javaloyes et al., 2021).

Geodesics are defined by the parallel transport of their own tangent field,

$F(x,v)$1

or equivalently as critical points of the energy

$F(x,v)$2

The first and second variation formulas lead to the index form and Jacobi equation,

$F(x,v)$3

with the usual notions of conjugate and focal points adapted to admissible conic domains. The coordinate-free treatment through the Chern connection as a family of affine connections is one of the main advances of the variational theory in pseudo-Finsler geometry (Javaloyes et al., 2014).

Special local coordinates can be built by applying normal-coordinate constructions to the pullback of a horizontally torsionless connection. This avoids the differentiability problems of ordinary Finsler normal and Fermi coordinates, which in general are not sufficiently differentiable, and yields Taylor expansions of the metric, connection, and Finsler Lagrangian in terms of curvature, Cartan torsion, and Landsberg tensor. In Lorentz-Finsler applications, these coordinates are used to formulate a Finslerian version of the equivalence principle (Minguzzi, 2016).

The isometry theory is likewise more rigid than might be expected from the direction dependence. By passing to the pseudo-Riemannian Sasaki metric $F(x,v)$4 on the slit tangent bundle, the isometry group of a pseudo-Finsler structure embeds as a closed subgroup of a pseudo-Riemannian isometry group. The resulting theorem is that $F(x,v)$5, endowed with the $F(x,v)$6-topology, is a Lie transformation group; every pseudo-Finsler isometry is $F(x,v)$7 and is determined by its second jet at any point (Torromé et al., 2013).

4. Lorentzian spacetime structures and pseudo-Finsler gravity

Pseudo-Finsler metrics acquire additional structure in spacetime applications. For Lorentzian $F(x,v)$8-metrics,

$F(x,v)$9

with $1$0 for a Lorentzian metric $1$1 and $1$2, the supplied literature gives a complete criterion for when such a metric defines a Finsler spacetime in the sense of Hohmann–Pfeifer–Voicu. The criterion requires a connected conic timelike subbundle $1$3 on which

$1$4

together with

$1$5

and the boundary condition $1$6 on $1$7. This simultaneously encodes Lorentzian signature of the fundamental tensor and the existence of a future-pointing timelike cone (Voicu et al., 2023).

Within this spacetime framework, explicit subclasses can be characterized sharply. Randers-type deformations define Finsler spacetimes iff

$1$8

Bogoslovsky–Kropina metrics, Kundt metrics, and exponential metrics each satisfy precise signature and cone conditions, and in several cases the timelike cones either coincide with those of the underlying Lorentzian metric or are obtained by intersecting those cones with the half-space $1$9 (Voicu et al., 2023).

Pseudo-Finsler geometry also supports Palatini-type gravitational formalisms. In the Einstein–Hilbert–Palatini framework for pseudo-Finsler metrics $F(x,Kv)=K\,F(x,v),$0, the independent affine variable is not a linear connection on $F(x,Kv)=K\,F(x,v),$1 but a homogeneous nonlinear connection $F(x,Kv)=K\,F(x,v),$2 on a conic domain $F(x,Kv)=K\,F(x,v),$3. Replacing scalar curvature by the Finslerian Ricci scalar yields coupled affine and metric equations for $F(x,Kv)=K\,F(x,v),$4. A central conclusion is that the mean Landsberg tensor $F(x,Kv)=K\,F(x,v),$5 is the obstruction to recovering the classical Palatini picture: if $F(x,Kv)=K\,F(x,v),$6, the metric nonlinear connection solves the affine equation and the classical Palatini conclusions are recovered; if $F(x,Kv)=K\,F(x,v),$7, the metric and affine structures necessarily decouple (Javaloyes et al., 2021).

5. Principal classes and explicit constructions

The supplied literature contains several structurally important families of pseudo-Finsler metrics and pseudo-Finslerian constructions.

Class	Defining expression	Result in the supplied papers
Randers-type spacetime	$F(x,Kv)=K\,F(x,v),$8	Finsler spacetime iff $F(x,Kv)=K\,F(x,v),$9 (Voicu et al., 2023)
Bogoslovsky–Kropina spacetime	$\int F\!\left(x(t),\frac{dx}{dt}\right)\,dt$0	Exact Lorentzian cone classification by sign of $\int F\!\left(x(t),\frac{dx}{dt}\right)\,dt$1 and range of $\int F\!\left(x(t),\frac{dx}{dt}\right)\,dt$2 (Voicu et al., 2023)
Zermelo translation	$\int F\!\left(x(t),\frac{dx}{dt}\right)\,dt$3	Translations of semi-Riemannian metrics are exactly pseudo-Randers-Kropina metrics (Javaloyes et al., 2014)
Pseudo-Finsleroid metrics	$\int F\!\left(x(t),\frac{dx}{dt}\right)\,dt$4 or separated variants	Constant indicatrix curvature $\int F\!\left(x(t),\frac{dx}{dt}\right)\,dt$5 with axial anisotropy (Asanov, 2015, Asanov, 2017)
Kropina metrics in gravity	$\int F\!\left(x(t),\frac{dx}{dt}\right)\,dt$6	Einstein iff $\int F\!\left(x(t),\frac{dx}{dt}\right)\,dt$7 is Einstein and $\int F\!\left(x(t),\frac{dx}{dt}\right)\,dt$8 is Killing (Heefer et al., 5 Jun 2026)

Zermelo navigation provides one of the clearest geometric constructions. Javaloyes and Vitório define the translation of a pseudo-Finsler metric by shifting each indicatrix by a vector field $\int F\!\left(x(t),\frac{dx}{dt}\right)\,dt$9. In the pseudo-Finsler setting, this leads to two distinct types, straight and reverse translations, depending on the orientation of the translated indicatrix. When the starting metric is semi-Riemannian, the translated metrics are precisely pseudo-Randers and pseudo-Kropina metrics, and vanishing Matsumoto tensor is equivalent to being such a translation. Under a homothetic wind field, the translated flag curvature differs from the original by the non-positive constant $$g_{ab}(x,v)=\frac{1}{2}\,\frac{\partial^2}{\partial v^a\partial v^b}\big[F^2(x,v)\big].$$0, and the geodesic flow is obtained by composing original geodesics with the flow of $$g_{ab}(x,v)=\frac{1}{2}\,\frac{\partial^2}{\partial v^a\partial v^b}\big[F^2(x,v)\big].$$1 (Javaloyes et al., 2014).

A different strand of explicit model building is the pseudo-Finsleroid program. In one-axis form, the metric function $$g_{ab}(x,v)=\frac{1}{2}\,\frac{\partial^2}{\partial v^a\partial v^b}\big[F^2(x,v)\big].$$2 describes a relativistic geometry with a preferred spacelike direction and Lorentzian signature $$g_{ab}(x,v)=\frac{1}{2}\,\frac{\partial^2}{\partial v^a\partial v^b}\big[F^2(x,v)\big].$$3, while preserving the constant negative curvature $$g_{ab}(x,v)=\frac{1}{2}\,\frac{\partial^2}{\partial v^a\partial v^b}\big[F^2(x,v)\big].$$4 of the indicatrix. The three-dimensional section orthogonal to the time direction inherits constant positive curvature $$g_{ab}(x,v)=\frac{1}{2}\,\frac{\partial^2}{\partial v^a\partial v^b}\big[F^2(x,v)\big].$$5, so the four-dimensional construction acts as a relativistic lift of the earlier three-dimensional Finsleroid geometry (Asanov, 2015). The two-axes pseudo-Finsleroid class extends this to a geometry with a vertical distinguished direction and a horizontal distinguished direction, together with an angle-separated ansatz and an explicit angle-regular solution whose indicatrix again has constant curvature $$g_{ab}(x,v)=\frac{1}{2}\,\frac{\partial^2}{\partial v^a\partial v^b}\big[F^2(x,v)\big].$$6 (Asanov, 2017).

6. Transformations, metrizability, projective geometry, and current physical directions

Pseudo-Finsler metrics admit transformation theories that are much richer than their Riemannian analogues. For conic pseudo-Finsler surfaces, an anisotropic conformal change

$$g_{ab}(x,v)=\frac{1}{2}\,\frac{\partial^2}{\partial v^a\partial v^b}\big[F^2(x,v)\big].$$7

does not automatically preserve the pseudo-Finsler condition. The supplied paper gives a necessary and sufficient nondegeneracy criterion in terms of the modified Berwald frame and the derivatives of $$g_{ab}(x,v)=\frac{1}{2}\,\frac{\partial^2}{\partial v^a\partial v^b}\big[F^2(x,v)\big].$$8, and shows that the geodesic spray is preserved exactly when

$$g_{ab}(x,v)=\frac{1}{2}\,\frac{\partial^2}{\partial v^a\partial v^b}\big[F^2(x,v)\big].$$9

This is strictly more general than the isotropic conformal case, where spray invariance reduces to homothety. The same work derives explicit transformed inverse metrics, Cartan tensor, main scalar, Barthel connection, and Berwald connection, and supplies sufficient conditions for projective flatness and dual flatness (Elgendi et al., 2024).

Projective geometry yields another family of invariants. Using the Schwarzian derivative and either the Funk metric or the Poincaré metric on $F$00, the papers on projectively invariant pseudo-distance construct an intrinsic pseudo-distance $F$01 from chains of projective geodesic segments. The governing equation for the projective parameter is

$F$02

When the Ricci tensor is parallel, this equation has explicit trigonometric, exponential, or fractional-linear solutions depending on the sign of the curvature term. Complete Finsler spaces with positive semi-definite Ricci tensor have trivial projective pseudo-distance, while spaces with negative-definite parallel Ricci tensor have a genuine, and in the complete case complete, projectively invariant distance (Sepasi et al., 2013, Bidabad et al., 2015).

Metrizability questions show that pseudo-Finsler geometry is not merely a reformulation of pseudo-Riemannian geometry. For $F$03-dimensional $F$04-invariant torsion-free affine connections, the supplied classification identifies five Berwald-Finsler classes. Two classes—power-law and exponential—give explicit non-Riemannian Berwald metrics that are not pseudo-Riemann metrizable, while the remaining classes are pseudo-Riemann metrizable under explicit holonomy-dimension and Ricci-symmetry conditions. This establishes that affine geodesics may arise from pseudo-Finsler metrics even when no affinely equivalent pseudo-Riemannian metric exists (Voicu et al., 2024).

Current physical programs use pseudo-Finsler metrics in several distinct ways. One line, in higher-spin theory, expands a homogeneous pseudo-Finsler line element around a pseudo-Riemannian background in symmetric tensors and finds that the linearized Finsler Ricci tensor reproduces the curved-space Fronsdal operator, plus a Stueckelberg-like coupling. The same paper emphasizes, however, that the genuine gauge principle remains problematic and that non-transverse modes are not eliminated in the versions of Finsler dynamics examined there (Tomasiello, 2024). Another line studies Einstein-Kropina metrics in arbitrary signature: these are Einstein precisely when the underlying pseudo-Riemannian metric $F$05 is Einstein and the vector field $F$06 is Killing, but their role in Finsler gravity is unexpectedly rigid. In the Lorentzian and Riemannian cases, Einstein-Kropina solutions of the Pfeifer–Wohlfarth $F$07-vacuum equation are necessarily Berwald and Ricci-flat, and the cosmological constant must vanish (Heefer et al., 5 Jun 2026).

Pseudo-Finsler metrics therefore form a broad geometric category rather than a single model class. The supplied literature presents them as conic, homogeneous, direction-dependent metric structures with nondegenerate but not necessarily positive fundamental tensor; as Lorentzian causal geometries with timelike cones and null boundaries; as variational systems with sprays, anisotropic connections, and Jacobi theory; and as a flexible framework for navigation, metrizability, projective geometry, and generalized gravity. What remains uniform across these developments is the central role of the fiberwise Hessian and the persistent fact that indefinite signature fundamentally changes the analytic and global behavior of Finsler geometry.