Conical Finsler Metric

• Conical Finsler metrics are Finsler structures defined on open conic domains of the tangent bundle, ensuring smoothness, 1-homogeneity, and strong convexity on specified directions.
• They also appear in pseudo-Finsler and Lorentz-Finsler geometries where cone structures encode causal or timelike directions and establish anisotropically conformal classes.
• Additional formulations include quotient constructions on weighted projective spaces, polyhedral models, and constant-curvature indicatrices, expanding Finsler theory into diverse geometric applications.

Searching arXiv for relevant papers on conical Finsler metrics and adjacent frameworks. Conical Finsler metrics are Finsler structures whose essential geometry is organized by a cone, either because the metric is defined only on an open conic domain in each tangent space, because it descends from a quotient of a cone by a scaling action, or because its singularities are locally modeled by cone-like spaces. In the research literature, the expression covers several mathematically distinct constructions. One line treats conic Finsler manifolds as Finsler metrics defined on proper open cones in the tangent bundle rather than on all of $TM\setminus\{0\}$ (Youssef et al., 2018, Javaloyes et al., 2011). Another line studies Lorentz-Finsler and pseudo-Finsler structures on cone domains, where the cone encodes causal directions and the metric is defined only on timelike or admissible vectors (Javaloyes et al., 2018, Torromé et al., 2013). A different usage appears in quotient geometry, where weighted projective spaces inherit a scale-invariant Finsler structure from an ambient cone modulo weighted radial dilations, yielding what is explicitly described as a “conical Finsler metric” on the quotient (Shaska, 7 May 2025). There are also singular flat and polyhedral settings—piecewise flat Finsler surfaces, $1/n$-translation surfaces, and triangular or polyhedral norms—where curvature is concentrated at cone points and the metric is locally Minkowski away from singularities (Xu et al., 2016, Pozzetti et al., 2 Apr 2026, Reid, 2023). Finally, the older “Finsleroid” literature uses “conical” in a more shape-theoretic sense for metrics whose indicatrices are constant-curvature, cone-like deformations of Riemannian spheres (Asanov, 2010, Asanov, 2009).

1. Conic domain formulations

A primary modern meaning of conical Finsler metric is domain-theoretic: the metric is defined not on the whole slit tangent bundle, but only on a proper open cone in each tangent space. In this sense, a conic Finsler manifold is a pair $(A,F)$ where $A\subset TM$ is an open conic subset, meaning that for every $v\in A$ and every $\mu>0$, one has $\mu v\in A$, and $F$ satisfies the usual Finsler conditions on $A$ rather than on all of $TM\setminus\{0\}$ (Youssef et al., 2018). The same distinction appears in the systematic treatment of generalized Finsler metrics, where a conic open domain $1/n$0 is required to be fiberwise open and positively homogeneous, and a conic Finsler metric is defined by positive 1-homogeneity together with positive-definite fundamental tensor on $1/n$1 (Javaloyes et al., 2011).

This formulation is motivated by examples in which strong convexity fails or smoothness breaks down outside a preferred cone of directions. The generalized theory shows that many metrics obtained from homogeneous combinations of Finsler metrics and one-forms are genuinely Finsler only on a proper conic subdomain determined by explicit inequalities for the fundamental tensor (Javaloyes et al., 2011). This includes classical $1/n$2-type metrics such as Kropina and Matsumoto metrics, whose natural domains exclude directions where the defining denominator vanishes or where the Hessian loses positive definiteness (Javaloyes et al., 2011).

The conic-domain viewpoint is therefore not an auxiliary technicality but part of the definition. It isolates the maximal region where the Finsler structure is smooth, 1-homogeneous, and strongly convex. This suggests a broad interpretation: a conical Finsler metric is often best understood as a directionally restricted Finsler geometry rather than a global norm on all tangent directions. A plausible implication is that many non-regular examples traditionally treated as singular Finsler metrics are more naturally classified as regular conic Finsler metrics on their true domains.

2. Cone structures, pseudo-Finsler geometry, and causality

A second major framework places conical Finsler metrics inside pseudo-Finsler and Lorentz-Finsler geometry. In the pseudo-Finsler setting, one considers a triple $1/n$3 where $1/n$4 is open, each fiber $1/n$5 is a nonempty open cone, $1/n$6 for $1/n$7, and the fiberwise Hessian of $1/n$8 is nondegenerate of constant index (Torromé et al., 2013). Here “conical” refers directly to the cone-valued domain of admissible directions, often interpreted as timelike cones in Lorentz-Finsler applications (Torromé et al., 2013).

This cone-based viewpoint is systematized in the theory of smooth strong cone structures $1/n$9, where each fiber $(A,F)$0 is a strong cone and the interior $(A,F)$1 defines timelike directions (Javaloyes et al., 2018). A Lorentz-Finsler metric $(A,F)$2 is then defined on a conic domain $(A,F)$3, is 2-homogeneous, extends smoothly as $(A,F)$4 to the boundary cone, and has Lorentzian fundamental tensor on the interior and boundary (Javaloyes et al., 2018). One of the central results is that cone structures are in bijection with anisotropically conformal classes of Lorentz-Finsler metrics: two such metrics determine the same cone structure if and only if they differ by a positive anisotropic conformal factor (Javaloyes et al., 2018).

The cone-triple formalism $(A,F)$5, with $(A,F)$6 and $(A,F)$7 a Finsler metric on $(A,F)$8, gives an explicit way to build these structures. The associated continuous Lorentz-Finsler metric has the form

$(A,F)$9

with the light cone determined by $A\subset TM$0 (Javaloyes et al., 2018). Although this initial $A\subset TM$1 may fail to be smooth along the distinguished timelike direction, the same work proves that every cone structure can be smoothed to a genuine Lorentz-Finsler metric without changing the underlying cone (Javaloyes et al., 2018).

This setting also supports a Lie-theoretic symmetry theory. For conical pseudo-Finsler metrics, the differential of an isometry preserves the associated Sasaki-type pseudo-Riemannian metric on the cone domain, and the isometry group is a finite-dimensional Lie group (Torromé et al., 2013). That result places conical pseudo-Finsler geometry on a structural footing parallel to pseudo-Riemannian geometry.

3. Quotients of cones and weighted projective constructions

A more specific and geometrically concrete notion of conical Finsler metric arises on weighted projective spaces. Weighted projective space $A\subset TM$2 is defined as the quotient of $A\subset TM$3 by the weighted scaling action

$A\subset TM$4

with $A\subset TM$5 (Shaska, 7 May 2025). In the complex case, the quotient is typically an orbifold, with quotient singularities and local cone behavior near points with nontrivial stabilizer (Shaska, 7 May 2025).

The construction in “Finsler Metric Clustering in Weighted Projective Spaces” is explicitly presented as a concrete example of what one might call a conical Finsler metric (Shaska, 7 May 2025). The ambient weighted norm

$A\subset TM$6

is used to normalize the radial scaling, and a Finsler norm $A\subset TM$7 is defined on the quotient so that it is 1-homogeneous in $A\subset TM$8, invariant under the weighted scaling of $A\subset TM$9, and non-degenerate on the quotient tangent space (Shaska, 7 May 2025). The induced path metric

$v\in A$0

is then a genuine metric on $v\in A$1 (Shaska, 7 May 2025).

The paper emphasizes several conical features of this geometry: the underlying space is a quotient of the cone $v\in A$2, the Finsler norm is scale-invariant in the base point, representatives are normalized by cutting the cone with a weighted unit sphere, and orbifold points correspond to cone-angle singularities (Shaska, 7 May 2025). This is arguably the most literal realization of a conical Finsler metric in the current literature: a Finsler structure descended from an ambient cone by homogeneous scaling.

A rational analogue $v\in A$3 is also defined on rational points of weighted projective space using weighted gcd normalization, preserving the same quotient logic over $v\in A$4 (Shaska, 7 May 2025). This suggests that the conical paradigm extends not only across complex-analytic and differential-geometric settings but also into arithmetic geometry.

4. Singular cones, polyhedral models, and flat conical backgrounds

Another major family of conical Finsler metrics appears in singular flat settings, where the base space itself has cone points. A piecewise flat Finsler surface is a triangulated surface whose triangles are Minkowski triangles, glued compatibly along edges (Xu et al., 2016). Each triangle is flat in the Minkowski sense; all curvature and non-smoothness are concentrated at vertices. At a vertex $v\in A$5, the tangent cone $v\in A$6 is the union of Minkowski cones $v\in A$7 glued along rays, giving a direct Finsler analogue of a Euclidean cone (Xu et al., 2016).

In this framework, geodesics are straight in each triangle and satisfy the edge-crossing equation

$v\in A$8

a Snell-type law across edges (Xu et al., 2016). At vertices, geodesic extendability is controlled by a directional curvature $v\in A$9, defined using angular measures on the indicatrix (Xu et al., 2016). The sign of this curvature determines whether a geodesic can pass through the cone apex, whether the extension is unique, or whether no extension exists (Xu et al., 2016). In Landsberg type piecewise flat surfaces, the directional dependence disappears and vertex curvature becomes isotropic, yielding a combinatorial Gauss–Bonnet formula

$\mu>0$0

for compact surfaces without boundary (Xu et al., 2016).

A closely related but smoother flat-cone setting appears in $\mu>0$1-translation surfaces. These are surfaces modeled on $\mu>0$2-structures with cone points of angles $\mu>0$3 and Euclidean geometry away from the singular set (Pozzetti et al., 2 Apr 2026). Compatible Finsler metrics are defined by pulling back a $\mu>0$4-invariant norm on $\mu>0$5 to the regular tangent bundle (Pozzetti et al., 2 Apr 2026). A central result is that CAT(0) geodesics of the underlying flat cone metric are also geodesics for every compatible Finsler metric, and they minimize Finsler length in each homotopy class (Pozzetti et al., 2 Apr 2026). In this setting, the conical background fixes the geodesic geometry, while the Finsler norm changes only the length functional.

The same work constructs Liouville currents for these compatible Finsler metrics by decomposing them into multi-foliation currents associated with $\mu>0$6-metrics, i.e. norms built from $\mu>0$7-invariant webs of directions (Pozzetti et al., 2 Apr 2026). This indicates that conical flat structures can support a linearized current-theoretic description parallel to the classical theory of measured foliations and geodesic currents.

Polyhedral norms also occur in convex projective geometry. For cubic differentials on surfaces, a triangular asymmetric Finsler metric

$\mu>0$8

has triangular unit balls and is explicitly described as polyhedral and conical/piecewise linear in tangent directions (Reid, 2023). In degenerations of convex projective structures, the Danciger–Stecker domain-shape metric converges to this triangular Finsler metric as the cubic differential becomes large (Reid, 2023). Here the “conical” character is not from a tangent-domain restriction but from the polyhedral, sectorial structure of the norm.

5. Finsleroid and constant-curvature indicatrix models

A historically influential but terminologically distinct use of “conical” concerns indicatrix geometry rather than domain restriction. In Asanov’s work, a Finsleroid is a Finsler space whose indicatrix is a constant-curvature hypersurface (Asanov, 2009). The later paper “Finsler connection preserving angle in dimensions $\mu>0$9” formulates this class via conformal automorphisms between a Finsler space $\mu v\in A$0 and a Riemannian space $\mu v\in A$1 (Asanov, 2010). If the automorphism is positively homogeneous of degree $\mu v\in A$2, then the indicatrix has constant curvature

$\mu v\in A$3

and the Riemannian and Finsler norms are related by

$\mu v\in A$4

(Asanov, 2010).

The paper explicitly characterizes these geometries as “cone-like” deformations of the Riemannian sphere, especially when $\mu v\in A$5, and treats them as Finsleroid or conical metrics (Asanov, 2010). The Finslerian two-vector angle is a scaled Riemannian angle,

$\mu v\in A$6

and this permits the construction of an explicit angle-preserving, metrical, nonlinear Finsler connection (Asanov, 2010). Under indicatrix homogeneity, the connection is the “export” of the Levi-Civita connection of the Riemannian metric through the conformal automorphism (Asanov, 2010).

The $\mu v\in A$7-space and Finsleroid-type metrics provide explicit examples with an axial 1-form $\mu v\in A$8, a charge parameter $\mu v\in A$9, and constant-curvature indicatrix determined by

$F$0

(Asanov, 2010). This line of work treats “conical” as referring to a special cone-like profile of the unit ball or indicatrix rather than to a cone domain in the tangent bundle. A plausible implication is that the phrase “conical Finsler metric” has historically been used in two different senses—domain-theoretic and indicatrix-shape-theoretic—and both usages remain active in current literature.

6. Homogeneity, rigidity, and specialized applications

Conical Finsler metrics also appear in homogeneous and symmetry-constrained settings. On the 2-dimensional non-Abelian Lie group $F$1, left invariant conic Finsler metrics are determined by conic Minkowski norms on the Lie algebra and written in polar form $F$2 on an angular cone near a preferred direction (Xu, 2022). Within this class, left invariant conic Landsberg metrics with nowhere vanishing spray are classified, as are those of constant curvature and those of Berwald type (Xu, 2022). A central result is that every left invariant conic Landsberg metric on this $F$3 is Berwald (Xu, 2022). The same paper proposes a homogeneous conic Landsberg conjecture and proves the 2-dimensional case: every homogeneous conic Landsberg surface is Berwald (Xu, 2022).

The relation between conic metrics and structural rigidity also appears in the study of semi-concurrent vector fields. There, a conic Finsler manifold is again defined as a Finsler structure on an open conic subset $F$4, and explicit non-Riemannian examples admitting semi-concurrent vector fields are constructed only in the conic category, not in the regular global one (Youssef et al., 2018). The paper formulates the conjecture that there is no regular non-Riemannian Finsler metric admitting a semi-concurrent vector field, so that any such metric is either Riemannian or conic Finslerian (Youssef et al., 2018). This positions conic Finsler geometry as the natural habitat for certain algebraically constrained non-Riemannian examples.

The metrical approach to Finsler geometry also suggests a way to formalize conical Finsler metrics through metric–connection compatibility on an open conic subset $F$5 (Minguzzi, 2021). Although that work is written on the slit tangent bundle, it explicitly notes that its constructions are local and compatible with conic restrictions, so that Barthel, Chern, and Cartan connections can be defined on conic domains under the same compatibility axioms (Minguzzi, 2021). This suggests that much of standard Finsler connection theory extends verbatim to the conic setting once the domain is treated correctly.

6. Conceptual synthesis and terminological cautions

The literature supports at least four precise meanings of conical Finsler metric.

First, there is the strict conic-domain meaning: a Finsler metric defined on an open conic subset $F$6, usually because smoothness or strong convexity fails outside $F$7 (Javaloyes et al., 2011, Youssef et al., 2018, Xu, 2022). This is now standard in generalized Finsler and Lorentz-Finsler geometry.

Second, there is the cone-structure meaning: a Lorentz-Finsler or pseudo-Finsler metric on a cone domain whose boundary encodes causal or admissible directions, with geometry organized by a smooth family of strong cones $F$8 (Javaloyes et al., 2018, Torromé et al., 2013). This is the dominant meaning in spacetime applications.

Third, there is the quotient-of-a-cone meaning: a Finsler metric induced on a quotient space obtained by collapsing a radial scaling direction in an ambient cone, as in weighted projective spaces (Shaska, 7 May 2025). This meaning is especially geometric and global.

Fourth, there is the indicatrix-shape meaning, where “conical” refers to a cone-like deformation of the unit sphere or a Finsleroid indicatrix of constant curvature (Asanov, 2010, Asanov, 2009). This usage is older and conceptually different from the domain-based one.

A common misconception is that all conical Finsler metrics are singular or merely incomplete versions of ordinary Finsler metrics. The literature does not support that simplification. In many cases, the conic domain is the natural domain of regularity, and the metric is perfectly smooth and strongly convex there (Javaloyes et al., 2011, Javaloyes et al., 2018). Another misconception is that “conical” always refers to cone singularities in the base space. Piecewise flat Finsler surfaces and $F$9-translation surfaces do fit that picture (Xu et al., 2016, Pozzetti et al., 2 Apr 2026), but conic-domain metrics on smooth manifolds do not require singular base geometry.

Taken together, these strands suggest that “conical Finsler metric” is best treated as a family resemblance term rather than a single canonical definition. The unifying motif is positive homogeneity organized by a cone: a cone of admissible tangent directions, a cone quotient in the base geometry, a cone singularity in the local model, or a cone-like indicatrix in each tangent space. The precise meaning in any given context is determined by which of these structures is primary.