---
title: Conic Finsler Metrics
url: https://www.emergentmind.com/topics/conic-finsler-metrics
type: topic
---

# Conic Finsler Metrics

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 [1111.5066].

## 1. Foundational definitions and conic domains

Let \(M\) be a smooth manifold and \(\pi:TM\to M\) its tangent bundle. An open conic domain \(A\subset TM\) means that each fiber \(A_p\) is open in \(T_pM\) and invariant under positive scaling. On such a domain, the basic tensorial datum is the fiberwise Hessian
\[
g_{ij}(v)\;=\;\tfrac12\frac{\partial^2}{\partial v^i\partial v^j}\bigl(F(v)^2\bigr),
\]
defined on \(A\setminus\{0\}\). The standard global notion is recovered when \(A=TM\); positivity of \(g\) 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 \(0\notin A\) unless \(A=V\) in the Minkowski model [1111.5066].

| Class | Domain | Tensor condition |
|---|---|---|
| Standard Finsler metric | \(A=TM\) | \(g\) positive-definite |
| Pseudo-Finsler metric | \(A=TM\) | \(g\) nondegenerate |
| Conic Finsler metric | Open conic \(A\subset TM\) | \(g\) positive-definite on \(A\setminus\{0\}\) |
| Conic pseudo-Finsler metric | Open conic \(A\subset TM\) | \(g\) nondegenerate on \(A\setminus\{0\}\) |

A useful fiberwise characterization is given by the gauge of a unit ball. If \(B\subset A\) is a closed, star-shaped subset meeting every ray in \(A\), with smooth boundary \(S=\partial B\subset A\) on which the position vector is everywhere transverse, then
\[
F(v)=\inf\{\lambda>0:\;v/\lambda\in B\}
\]
is a Minkowski conic pseudo-norm with unit ball \(B\). Moreover, \(F\) is a conic Minkowski norm iff \(S\) is strongly convex; if \(A\) is convex, then \(F\) is a strict conic norm exactly when \(B\) is strictly convex [1111.5066].

Several structural subtleties are specific to the conic setting. The fiberwise domains \(A_p\) need not all be diffeomorphic; one often assumes each \(A_p\) connected or convex. One may also ask that \((A,F)\) be maximal, meaning that it is not the restriction of a larger domain where \(F\) extends smoothly. A further issue is the existence of a Riemannian lower bound \(g_0\) with \(F(v)\ge \sqrt{g_0(v,v)}\), which becomes decisive for topology and distance properties [1111.5066].

## 2. Length, separation, and geodesics

An admissible curve is piecewise smooth and satisfies \(\dot\gamma(t)\in A\) for all \(t\). Its \(F\)-length is
\[
\ell_F(\gamma)\;=\;\int_a^b F\bigl(\dot\gamma(t)\bigr)\,dt,
\]
and the associated Finslerian separation is
\[
d_F(p,q)\;=\;\inf\{\ell_F(\gamma):\gamma\text{ admissible from }p\to q\}.
\]
In the pseudo-Finsler case with \(A=TM\) and \(F\ge \sqrt{g_0}\), one shows that \(d_F\) 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 \(F\) is lower bounded, and \(d_F\) may even vanish identically or fail continuity [1111.5066].

The geodesic theory is formulated either through Euler–Lagrange equations or through the Chern connection. For a conic pseudo-Finsler Lagrangian \(L=F^2\), the geodesic equation can be written as
\[
\ddot x^i + 2\,G^i\bigl(x,\dot x\bigr)=0,
\]
where the spray coefficients are
\[
4\,G^i(x,y)
\;=\;g^{i\ell}(x,y)\Bigl(\,
\frac{\partial^2L}{\partial x^k\partial y^\ell}\,y^k
-\frac{\partial L}{\partial x^\ell}\Bigr).
\]
Equivalently,
\[
\nabla_{\dot y}\dot y(t)\;=\;0,
\]
and \(L(\dot y)\) is constant along the geodesic [2302.00611].

When \(g\) is nondegenerate on \(A\setminus\{0\}\), one defines the Chern connection and the exponential map. In the conic Finsler case, \(\exp_p\) is defined in a neighborhood of \(0\in A_p\), Gauss Lemma still holds, and a conic geodesic ball
\[
\exp_p\!\bigl(\{v\in A_p:F(v)<r\}\bigr)
\]
is strictly convex in the sense that the radial geodesic segment from \(p\) to any \(q\) 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 [1111.5066].

## 3. Construction methods and canonical families

A broad construction principle starts from conic Finsler metrics \(F_1,\dots,F_n\) on a common conic domain \(A\subset TM\), together with one-forms \(B_{n+1},\dots,B_{n+m}\), and a smooth positive \(2\)-homogeneous function
\[
L\;:\;B\times M\;\longrightarrow\;\mathbb R,
\]
where \(B\subset\mathbb R^{n+m}\) is conic. One then defines
\[
F(v)\;=\;\sqrt{\,L\bigl(F_1(v),\dots,F_n(v),\,B_{n+1}(v),\dots,B_{n+m}(v);\;p\bigr)\,}.
\]
The general fundamental-tensor theorem shows that the new metric is positive-semidefinite on \(A\) if all \(\{L\}_{x_k}\ge 0\) and \(\mathrm{Hess}_x(L)\) is positive-semidefinite, and positive-definite if in addition \(\sum_{k=1}^n \{L\}_{x_k}>0\) [1111.5066].

Several standard families are recovered as corollaries. The sum \(F=F_1+\dots+F_n\) is conic Finsler. For \(q>0\),
\[
F(v)
=\Bigl(\sum_{k=1}^nF_k(v)^q\;+\;\sum_{\alpha=n+1}^{n+m}B_\alpha(v)^q\Bigr)^{1/q}
\]
is conic Finsler on the obvious domain. Randers-type metrics \(F=F_0+B\) are conic Finsler on \(\{F_0+B>0\}\), Kropina-type metrics
\[
F=F_0^{\,q+1}/B^q
\]
are conic Finsler on \(\{B>0\}\), and Matsumoto-type metrics
\[
F=F_0^{\,q+1}/|F_0-B|^q
\]
are conic Finsler where \((F_0-(q+1)B)(F_0-B)>0\) [1111.5066].

A particularly important canonical form is the \((F_0,\beta)\)-metric
\[
F(v)=F_0(v)\,\varphi\!\bigl(\tfrac{\beta(v)}{F_0(v)}\bigr).
\]
For this class one computes explicitly that \(g\) is positive-definite on those \(v\) for which
\[
\varphi(s)-s\,\varphi'(s)>0
\quad\text{and}\quad
\varphi(s)\,\varphi''(s)\ge 0.
\]
The familiar specializations are: Randers, \(\varphi(s)=1+s\); Kropina, \(\varphi(s)=s^{-q}\); and Matsumoto, \(\varphi(s)=|1-s|^{-q}\) [1111.5066].

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 [1111.5066].

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

A major Lorentzian branch of the subject concerns conic Finsler spacetimes built from \((\alpha,\beta)\)-metrics. Let \(a=a_{ij}(x)\,dx^i\otimes dx^j\) be Lorentzian of signature \((+,-,-,-)\), let \(b=b_i(x)\,dx^i\), and define
\[
A(x,y)=a_{ij}(x)\,y^i\,y^j,\qquad B(x,y)=b_i(x)\,y^i,\qquad
L(x,y)=A(x,y)\,\Psi(B^2/A).
\]
The domain of \(L\) is a conic subbundle \(\mathcal A\subset TM\setminus\{0\}\), and a Finsler spacetime requires a smaller conic subbundle \(\mathcal T\subset\mathcal A\) with convex fibers on which \(L>0\), the fundamental tensor has Lorentzian signature \((+,-,-,-)\), and \(L\to 0\) at the boundary. The main theorem identifies necessary and sufficient conditions for the Lorentzian-signature requirement:
\[
\Psi(s)-s\,\Psi'(s)>0,
\]
and, writing \(\sigma(s):=(\Psi-s\Psi')^2\Psi\),
\[
(s-\langle b,b\rangle)\,\frac{d}{ds}\ln\sigma(s)>-1.
\]
The null boundary is determined by
\[
L=0 \iff A=0 \text{ or } \Psi(s)=0,
\]
so the Finsler light-cone splits into the usual metric light-cone and an additional cone determined by zeros of \(\Psi\) [2302.09937].

This framework yields explicit families and parameter ranges. Pure Lorentz metrics \(\Psi\equiv \mathrm{const}>0\) are always allowed. Randers, Bogoslovsky–Kropina, Kundt, and exponential metrics are obtained by particular choices of \(\Psi\), with admissible parameter regions derived by imposing the two scalar inequalities above. At the level of symmetry, any Killing field of \(a\) that also preserves \(b\) is automatically a Killing field of the full \(L\), while extra isometries occur only for special \(\Psi\) satisfying a specific ODE [2302.09937].

A different pseudo-Finsler direction studies anisotropic conformal change on conic pseudo-Finsler surfaces:
\[
\bar F(x,y)=e^{\phi(x,y)}F(x,y),
\]
where \(\phi\) is \(0\)-homogeneous in \(y\). Here the main issue is preservation of nondegeneracy. In dimension \(2\), using the modified Berwald frame, one obtains
\[
\det(\bar g_{ij})
=
\varepsilon\,e^{4\phi}\bigl[\sigma-(\phi_{;2})^2+\varepsilon\bigr]\det(g_{ij}),
\]
hence
\[
\bar F\text{ is pseudo-Finsler}
\quad\Longleftrightarrow\quad
\sigma-(\phi_{;2})^2+\varepsilon\neq 0
\quad\text{everywhere on }\mathcal A.
\]
A key invariant criterion is that the geodesic spray is preserved if and only if
\[
\delta_i\phi=0,
\]
equivalently \(d_h\phi=0\). In that case the Barthel and Berwald connections remain unchanged. The same formalism also gives sufficient conditions for projective flatness and dual flatness of \(\bar F\) [2404.15659].

## 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
\[
K(\Pi,y)\;=\; \frac{u^k y^\ell\,R^i{}_{k\ell j}(x,y)\,u^j\,y_i}
{\bigl[g_{pq}(x,y)\,y^p y^q\;\cdot\;g_{rs}(x,y)\,u^r u^s \,-\,(g_{pq}(x,y)\,y^p u^q)^2\bigr]}.
\]
On this basis, He–Huang–Dong introduce conic hypersurfaces and isoparametric functions. A nonconstant \(C^2\)-function \(f\) is called (du-)isoparametric when
\[
F(\nabla f)=a(f),\qquad \Delta f=b(f),
\]
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 [2106.05056].

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 \(1/r\); cylinders have two distinct constant principal curvatures \(0\) and \(1/r\). The classification theorem states that, in a conic Minkowski space endowed with any volume form of zero \(S\)-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 \((a,B)\)-space. For Kropina spaces of constant flag curvature arising from Zermelo data \((h,W)\), every isoparametric hypersurface is precisely an isoparametric hypersurface in the Riemannian space \((M,h)\), with the same number of distinct principal curvatures and the same multiplicities [2106.05056].

The Landsberg–Berwald problem has a distinct conic form in homogeneous geometry. On the unique connected non-Abelian \(2\)-dimensional real Lie group, realized as the \(ax+b\) 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 \(2\)-dimensional homogeneous case is proved: if \((G/H,F)\) is a homogeneous conic Finsler surface and \(F\) is Landsberg, then \(F\) is Berwald [2212.06549].

Variational theory is governed by the Morse index theorem. For a \(C^6\) conic pseudo-Finsler manifold, a nonconstant geodesic \(\gamma\) connecting two submanifolds \(P,Q\) and perpendicular to both endpoints admits the index form
\[
I(V,W)
=\int_0^T\Bigl\{
g_{\dot\gamma}\!\bigl(D_tV,D_tW\bigr)
-
g_{\dot\gamma}\!\bigl(R_{\dot\gamma}(V,\dot\gamma)\dot\gamma,W\bigr)
\Bigr\}\,dt
+\text{boundary terms}.
\]
If \(g_{\dot\gamma(t)}\) is positive-definite along the geodesic, then the Morse index equals the sum of the multiplicities of the \(P\)-focal points along \(\gamma\); with two variable endpoints one obtains the refined formula
\[
\Ind\,I
=\sum_{0<t<T}p_P(t)+\Ind(A_\gamma).
\]
Among the consequences is that a minimizing geodesic has no conjugate points in the interior, and that the conic exponential map is a local \(C^3\)-diffeomorphism on each ray until the first focal instant [2302.00611].

## 6. Singular and non-holonomic extensions

Conic Finsler ideas also appear in singular flat-surface geometry. On a \(1/n\)-translation surface \(\Sigma\), one fixes a norm \(\|\cdot\|\) on \(\mathbb R^2\) whose unit ball is convex, star-shaped, and invariant under the rotation \(R_{2\pi/n}\). The translation atlas then induces a compatible Finsler norm
\[
F_x(v)=\|(\mathrm d\phi_i)_x(v)\|
\]
on \(T_x\Sigma\) away from the cone points. Near a cone point of angle \(\theta(p)=2\pi k/n\), 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 \(\ge \pi\) on both sides. In local charts, geodesics are straight, while at cone points a minimizing geodesic breaks so that both interior angles satisfy \(\alpha_\pm\ge \pi\). The same theory constructs a Liouville current
\[
\mu_F^{\rm Liouville}=\int_{S^1}\mu_\theta\,d\nu(\theta)
\]
encoding the lengths of closed curves [2604.02243].

A more non-holonomic extension is given by sub-conic metrics. Let \((M,g)\) be a smooth Riemannian manifold and let \(D\subset TM\) be a distribution of cones, meaning that each fiber \(D_p\) is a cone and that \(D\) is a union of graphs of smooth vector fields. The associated sub-conic distance is
\[
d_D(x,y)
=
\inf_{\gamma}
\int_0^1 \sqrt{g(\dot\gamma(s),\dot\gamma(s))}\,ds,
\]
where the infimum is taken over \(D\)-admissible curves. Writing \(S_p=\langle D_p\rangle\) and defining
\[
\nu_{g,D}(v)
=\inf\Bigl\{\sum_{i=1}^k\lambda_i\;\Big|\;
v=\sum_{i=1}^k \lambda_iX_i,\;
X_i\in D_p,\; g(X_i,X_i)=1,\; \lambda_i>0\Bigr\},
\]
one obtains a sub-Finsler structure whose closed unit ball is
\[
B_{sF}(p)=\overline{\Conv\bigl(D_p\cap B_g(p)\bigr)}.
\]
The main theorem states that if \(D\) satisfies Chow’s condition, then the sub-conic distance \(d_D\) coincides with the corresponding sub-Finsler distance. In particular, every sub-conic metric is sub-Finsler [2410.18255].

The hyperkähler application is the sub-twistor metric on the period domain
\[
P=\{\,W\subset V\mid \dim W=2,\;q|_W>0,\;\text{$W$ oriented}\,\}.
\]
Each positive \(3\)-plane \(T\subset V\) defines a twistor sphere
\[
Tw_T=\{\,W\in P\mid W\subset T\}\cong S^2,
\]
and the union of tangent planes \(T_W(Tw_T)\) forms a conical distribution \(D^{\mathrm{tw}}\subset TP\). At a point \(W\in P\cong \Hom(W,W^\perp)\),
\[
D^{\mathrm{tw}}_W
=
\{\,A\in\Hom(W,W^\perp)\mid \mathrm{rank}\,A\le 1,\;\mathrm{Im}\,A>0\}\cup\{0\}.
\]
By transitivity of \(SO(3,b-3)\) and Chow’s theorem, the induced sub-twistor metric is genuinely Finsler on \(P\), 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 [2410.18255].

Source: https://www.emergentmind.com/topics/conic-finsler-metrics