Conic Finsler Metrics

Updated 5 July 2026

Conic Finsler Metrics are Finsler structures defined on open conic subsets of the tangent bundle, where admissible directions lie within convex fiberwise cones and norms are positively homogeneous.
They extend the classical framework by incorporating pseudo-Finsler and Lorentzian variants, using tools like the Euler–Lagrange equations and Chern connection to analyze geodesics, curvatures, and separation properties.
Flexible construction methods, such as (F₀,β)-metrics, yield families like Randers, Kropina, and Matsumoto metrics, with applications spanning spacetime models, sub-Finsler, and hyperkähler structures.

Searching arXiv for recent and foundational papers on conic Finsler metrics and closely related conic/sub-conic frameworks. Conic Finsler metrics are Finslerian structures whose natural domain is not the whole tangent bundle $TM$ but an open conic subset $A\subset TM$ , so only directions lying in the fiberwise cones $A_p=A\cap T_pM$ are admissible. In the formulation of Javaloyes–Sánchez, a continuous map $F:A\to[0,\infty)$ is a conic Finsler metric when each fiber $F_p$ is a Minkowski conic norm; equivalently, $F$ is positively homogeneous of degree $1$ and its fundamental tensor is positive-definite on $A\setminus\{0\}$ . If the fundamental tensor is only required to be nondegenerate, one obtains a conic pseudo-Finsler metric. This framework encompasses classical examples such as Randers-, Kropina-, and Matsumoto-type metrics, as well as Lorentzian and spacetime variants, singular cone-surface models, and sub-conic distance structures that are shown to be sub-Finsler (Javaloyes et al., 2011).

1. Foundational definitions and conic domains

Let $M$ be a smooth manifold and $\pi:TM\to M$ its tangent bundle. An open conic domain $A\subset TM$ 0 means that each fiber $A\subset TM$ 1 is open in $A\subset TM$ 2 and invariant under positive scaling. On such a domain, the basic tensorial datum is the fiberwise Hessian

$A\subset TM$ 3

defined on $A\subset TM$ 4. The standard global notion is recovered when $A\subset TM$ 5; positivity of $A\subset TM$ 6 yields a Finsler metric, while nondegeneracy without positivity yields a pseudo-Finsler metric. The conic case differs in that the admissible directions are restricted a priori, and $A\subset TM$ 7 unless $A\subset TM$ 8 in the Minkowski model (Javaloyes et al., 2011).

Class	Domain	Tensor condition
Standard Finsler metric	$A\subset TM$ 9	$A_p=A\cap T_pM$ 0 positive-definite
Pseudo-Finsler metric	$A_p=A\cap T_pM$ 1	$A_p=A\cap T_pM$ 2 nondegenerate
Conic Finsler metric	Open conic $A_p=A\cap T_pM$ 3	$A_p=A\cap T_pM$ 4 positive-definite on $A_p=A\cap T_pM$ 5
Conic pseudo-Finsler metric	Open conic $A_p=A\cap T_pM$ 6	$A_p=A\cap T_pM$ 7 nondegenerate on $A_p=A\cap T_pM$ 8

A useful fiberwise characterization is given by the gauge of a unit ball. If $A_p=A\cap T_pM$ 9 is a closed, star-shaped subset meeting every ray in $F:A\to[0,\infty)$ 0, with smooth boundary $F:A\to[0,\infty)$ 1 on which the position vector is everywhere transverse, then

$F:A\to[0,\infty)$ 2

is a Minkowski conic pseudo-norm with unit ball $F:A\to[0,\infty)$ 3. Moreover, $F:A\to[0,\infty)$ 4 is a conic Minkowski norm iff $F:A\to[0,\infty)$ 5 is strongly convex; if $F:A\to[0,\infty)$ 6 is convex, then $F:A\to[0,\infty)$ 7 is a strict conic norm exactly when $F:A\to[0,\infty)$ 8 is strictly convex (Javaloyes et al., 2011).

Several structural subtleties are specific to the conic setting. The fiberwise domains $F:A\to[0,\infty)$ 9 need not all be diffeomorphic; one often assumes each $F_p$ 0 connected or convex. One may also ask that $F_p$ 1 be maximal, meaning that it is not the restriction of a larger domain where $F_p$ 2 extends smoothly. A further issue is the existence of a Riemannian lower bound $F_p$ 3 with $F_p$ 4, which becomes decisive for topology and distance properties (Javaloyes et al., 2011).

2. Length, separation, and geodesics

An admissible curve is piecewise smooth and satisfies $F_p$ 5 for all $F_p$ 6. Its $F_p$ 7-length is

$F_p$ 8

and the associated Finslerian separation is

$F_p$ 9

In the pseudo-Finsler case with $F$ 0 and $F$ 1, one shows that $F$ 2 is a generalized distance: it is generally non-symmetric, but satisfies positivity, the triangle inequality, and mutual continuity of forward and backward balls. The forward and backward open balls form bases of the manifold topology. In the conic case, fewer properties survive: the forward balls are open, but need not form a basis unless $F$ 3 is lower bounded, and $F$ 4 may even vanish identically or fail continuity (Javaloyes et al., 2011).

The geodesic theory is formulated either through Euler–Lagrange equations or through the Chern connection. For a conic pseudo-Finsler Lagrangian $F$ 5, the geodesic equation can be written as

$F$ 6

where the spray coefficients are

$F$ 7

Equivalently,

$F$ 8

and $F$ 9 is constant along the geodesic (Lu, 2023).

When $1$0 is nondegenerate on $1$1, one defines the Chern connection and the exponential map. In the conic Finsler case, $1$2 is defined in a neighborhood of $1$3, Gauss Lemma still holds, and a conic geodesic ball

$1$4

is strictly convex in the sense that the radial geodesic segment from $1$5 to any $1$6 in the ball is the unique length-minimizer among admissible curves lying entirely in that ball. A plausible implication is that local minimization theory remains robust even when global metric behavior is degraded by the conic restriction (Javaloyes et al., 2011).

3. Construction methods and canonical families

A broad construction principle starts from conic Finsler metrics $1$7 on a common conic domain $1$8, together with one-forms $1$9, and a smooth positive $A\setminus\{0\}$ 0-homogeneous function

$A\setminus\{0\}$ 1

where $A\setminus\{0\}$ 2 is conic. One then defines

$A\setminus\{0\}$ 3

The general fundamental-tensor theorem shows that the new metric is positive-semidefinite on $A\setminus\{0\}$ 4 if all $A\setminus\{0\}$ 5 and $A\setminus\{0\}$ 6 is positive-semidefinite, and positive-definite if in addition $A\setminus\{0\}$ 7 (Javaloyes et al., 2011).

Several standard families are recovered as corollaries. The sum $A\setminus\{0\}$ 8 is conic Finsler. For $A\setminus\{0\}$ 9,

$M$ 0

is conic Finsler on the obvious domain. Randers-type metrics $M$ 1 are conic Finsler on $M$ 2, Kropina-type metrics

$M$ 3

are conic Finsler on $M$ 4, and Matsumoto-type metrics

$M$ 5

are conic Finsler where $M$ 6 (Javaloyes et al., 2011).

A particularly important canonical form is the $M$ 7-metric

$M$ 8

For this class one computes explicitly that $M$ 9 is positive-definite on those $\pi:TM\to M$ 0 for which

$\pi:TM\to M$ 1

The familiar specializations are: Randers, $\pi:TM\to M$ 2; Kropina, $\pi:TM\to M$ 3; and Matsumoto, $\pi:TM\to M$ 4 (Javaloyes et al., 2011).

These constructions clarify a recurring misconception: classical singular examples such as Kropina and Matsumoto do not simply fail to be Finsler; rather, they typically become well-defined conic Finsler metrics after restricting to the maximal conic domain where smoothness and strong convexity hold. In this sense, the conic category is not a defect of the theory but part of its natural completion (Javaloyes et al., 2011).

4. Pseudo-Finsler, spacetime, and anisotropic conformal variants

A major Lorentzian branch of the subject concerns conic Finsler spacetimes built from $\pi:TM\to M$ 5-metrics. Let $\pi:TM\to M$ 6 be Lorentzian of signature $\pi:TM\to M$ 7, let $\pi:TM\to M$ 8, and define

$\pi:TM\to M$ 9

The domain of $A\subset TM$ 00 is a conic subbundle $A\subset TM$ 01, and a Finsler spacetime requires a smaller conic subbundle $A\subset TM$ 02 with convex fibers on which $A\subset TM$ 03, the fundamental tensor has Lorentzian signature $A\subset TM$ 04, and $A\subset TM$ 05 at the boundary. The main theorem identifies necessary and sufficient conditions for the Lorentzian-signature requirement: $A\subset TM$ 06 and, writing $A\subset TM$ 07,

$A\subset TM$ 08

The null boundary is determined by

$A\subset TM$ 09

so the Finsler light-cone splits into the usual metric light-cone and an additional cone determined by zeros of $A\subset TM$ 10 (Voicu et al., 2023).

This framework yields explicit families and parameter ranges. Pure Lorentz metrics $A\subset TM$ 11 are always allowed. Randers, Bogoslovsky–Kropina, Kundt, and exponential metrics are obtained by particular choices of $A\subset TM$ 12, with admissible parameter regions derived by imposing the two scalar inequalities above. At the level of symmetry, any Killing field of $A\subset TM$ 13 that also preserves $A\subset TM$ 14 is automatically a Killing field of the full $A\subset TM$ 15, while extra isometries occur only for special $A\subset TM$ 16 satisfying a specific ODE (Voicu et al., 2023).

A different pseudo-Finsler direction studies anisotropic conformal change on conic pseudo-Finsler surfaces: $A\subset TM$ 17 where $A\subset TM$ 18 is $A\subset TM$ 19-homogeneous in $A\subset TM$ 20. Here the main issue is preservation of nondegeneracy. In dimension $A\subset TM$ 21, using the modified Berwald frame, one obtains

$A\subset TM$ 22

hence

$A\subset TM$ 23

A key invariant criterion is that the geodesic spray is preserved if and only if

$A\subset TM$ 24

equivalently $A\subset TM$ 25. In that case the Barthel and Berwald connections remain unchanged. The same formalism also gives sufficient conditions for projective flatness and dual flatness of $A\subset TM$ 26 (Elgendi et al., 2024).

5. Curvature, hypersurfaces, and variational theory

The curvature theory of conic Finsler manifolds is expressed through the Chern or Cartan connection and the associated flag curvature

$A\subset TM$ 27

On this basis, He–Huang–Dong introduce conic hypersurfaces and isoparametric functions. A nonconstant $A\subset TM$ 28-function $A\subset TM$ 29 is called (du-)isoparametric when

$A\subset TM$ 30

and its level sets form an isoparametric family. In particular, each level set has constant dun-mean curvature, and in constant flag curvature the condition is equivalent to constancy of the principal curvatures (He et al., 2021).

In a conic Minkowski space, the basic isoparametric hypersurfaces are conic hyperplanes, conic hyperspheres, and conic cylinders. Hyperplanes have all principal curvatures zero; hyperspheres have constant principal curvature $A\subset TM$ 31; cylinders have two distinct constant principal curvatures $A\subset TM$ 32 and $A\subset TM$ 33. The classification theorem states that, in a conic Minkowski space endowed with any volume form of zero $A\subset TM$ 34-curvature, the only isoparametric hypersurfaces with one constant principal curvature are conic hyperplanes and conic hyperspheres, and those with two distinct constant principal curvatures are exactly the conic cylinders. The same work also exhibits local helicoids in a special conic $A\subset TM$ 35-space. For Kropina spaces of constant flag curvature arising from Zermelo data $A\subset TM$ 36, every isoparametric hypersurface is precisely an isoparametric hypersurface in the Riemannian space $A\subset TM$ 37, with the same number of distinct principal curvatures and the same multiplicities (He et al., 2021).

The Landsberg–Berwald problem has a distinct conic form in homogeneous geometry. On the unique connected non-Abelian $A\subset TM$ 38-dimensional real Lie group, realized as the $A\subset TM$ 39 group, left-invariant conic Finsler metrics with nowhere vanishing spray can be classified under constant curvature, Landsberg, and Berwald conditions. The main rigidity statement is that every left-invariant conic Landsberg metric on this group must be Berwald. This leads to the homogeneous conic Landsberg conjecture, and the $A\subset TM$ 40-dimensional homogeneous case is proved: if $A\subset TM$ 41 is a homogeneous conic Finsler surface and $A\subset TM$ 42 is Landsberg, then $A\subset TM$ 43 is Berwald (Xu, 2022).

Variational theory is governed by the Morse index theorem. For a $A\subset TM$ 44 conic pseudo-Finsler manifold, a nonconstant geodesic $A\subset TM$ 45 connecting two submanifolds $A\subset TM$ 46 and perpendicular to both endpoints admits the index form

$A\subset TM$ 47

If $A\subset TM$ 48 is positive-definite along the geodesic, then the Morse index equals the sum of the multiplicities of the $A\subset TM$ 49-focal points along $A\subset TM$ 50; with two variable endpoints one obtains the refined formula

$A\subset TM$ 51

Among the consequences is that a minimizing geodesic has no conjugate points in the interior, and that the conic exponential map is a local $A\subset TM$ 52-diffeomorphism on each ray until the first focal instant (Lu, 2023).

6. Singular and non-holonomic extensions

Conic Finsler ideas also appear in singular flat-surface geometry. On a $A\subset TM$ 53-translation surface $A\subset TM$ 54, one fixes a norm $A\subset TM$ 55 on $A\subset TM$ 56 whose unit ball is convex, star-shaped, and invariant under the rotation $A\subset TM$ 57. The translation atlas then induces a compatible Finsler norm

$A\subset TM$ 58

on $A\subset TM$ 59 away from the cone points. Near a cone point of angle $A\subset TM$ 60, the same formula holds sectorwise on the Euclidean cone cover, and the metric remains continuous across the gluing. The resulting distance agrees with the usual length metric, minimizing geodesics exist in each homotopy class, and a canonical piecewise-straight minimal segment exists between any two points, meeting cone points with exterior angles $A\subset TM$ 61 on both sides. In local charts, geodesics are straight, while at cone points a minimizing geodesic breaks so that both interior angles satisfy $A\subset TM$ 62. The same theory constructs a Liouville current

$A\subset TM$ 63

encoding the lengths of closed curves (Pozzetti et al., 2 Apr 2026).

A more non-holonomic extension is given by sub-conic metrics. Let $A\subset TM$ 64 be a smooth Riemannian manifold and let $A\subset TM$ 65 be a distribution of cones, meaning that each fiber $A\subset TM$ 66 is a cone and that $A\subset TM$ 67 is a union of graphs of smooth vector fields. The associated sub-conic distance is

$A\subset TM$ 68

where the infimum is taken over $A\subset TM$ 69-admissible curves. Writing $A\subset TM$ 70 and defining

$A\subset TM$ 71

one obtains a sub-Finsler structure whose closed unit ball is

$A\subset TM$ 72

The main theorem states that if $A\subset TM$ 73 satisfies Chow’s condition, then the sub-conic distance $A\subset TM$ 74 coincides with the corresponding sub-Finsler distance. In particular, every sub-conic metric is sub-Finsler (Korshunov, 2024).

The hyperkähler application is the sub-twistor metric on the period domain

$A\subset TM$ 75

Each positive $A\subset TM$ 76-plane $A\subset TM$ 77 defines a twistor sphere

$A\subset TM$ 78

and the union of tangent planes $A\subset TM$ 79 forms a conical distribution $A\subset TM$ 80. At a point $A\subset TM$ 81,

$A\subset TM$ 82

By transitivity of $A\subset TM$ 83 and Chow’s theorem, the induced sub-twistor metric is genuinely Finsler on $A\subset TM$ 84, and it is then pulled back to the birational Teichmüller space of a hyperkähler manifold to complete Verbitsky’s proof of the Global Torelli theorem (Korshunov, 2024).