On Spherically Symmetric Sprays

Benling Li; Yajing Gui

arxiv: 2604.11196 · v2 · submitted 2026-04-13 · 🧮 math.DG

On Spherically Symmetric Sprays

Yajing Gui , Benling Li This is my paper

Pith reviewed 2026-05-10 15:38 UTC · model grok-4.3

classification 🧮 math.DG

keywords spherically symmetric spraysprojectively flatisotropic curvaturespray geometryFinsler geometryHilbert's fourth problemzero curvature

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The pith

Projectively flat spherically symmetric sprays with isotropic curvature are fully classified, including explicit zero-curvature cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper first shows that sprays invariant under orthogonal group actions have geodesic coefficients reducible to a form with two functions α and β depending on radial distance and a directional variable. It then classifies the projectively flat ones among them that have isotropic curvature, giving explicit expressions especially when the curvature is zero. This connects directly to questions about metrics whose geodesics are straight lines. Weakly isotropic curvature cases are further characterized by systems of partial differential equations. Such results organize the interplay between symmetry and curvature properties in this geometric setting.

Core claim

For projectively flat spherically symmetric sprays we derive a complete classification of those with isotropic curvature, and in particular we obtain the explicit form of those with zero curvature. Furthermore, we characterize sprays of weakly isotropic curvature in this class by a system of partial differential equations.

What carries the argument

The canonical expression for the geodesic coefficients of spherically symmetric sprays, given by G^i = |y| α(r,s) y^i + |y|^2 β(r,s) x^i with r = |x|^2 and s = ⟨x,y⟩/|y|, which captures the invariance.

If this is right

Geodesic coefficients simplify to depend on only two scalar functions under spherical symmetry.
Projectively flat examples with isotropic curvature receive explicit descriptions.
Zero-curvature projectively flat sprays in this class have specific closed-form expressions.
Weakly isotropic curvature sprays satisfy a determined system of PDEs.
This framework unifies aspects of symmetry and curvature analysis in spray geometry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These classifications could be used to check metrizability for symmetric Finsler structures.
Similar reductions might apply under other group symmetries in spray or Finsler geometry.
Concrete examples from the classification can be tested for additional properties like completeness or convexity.
Links to Hilbert's fourth problem suggest potential new examples of metrics with straight geodesics.

Load-bearing premise

Invariance under the orthogonal group allows the geodesic coefficients to be expressed exactly using only the two functions α and β of r and s, without extra terms.

What would settle it

Discovery of a projectively flat spherically symmetric spray with isotropic curvature whose geodesic coefficients cannot be written in the classified forms would refute the completeness of the classification.

read the original abstract

This paper studies spherically symmetric sprays, i.e., sprays that are invariant under orthogonal transformations. We first establish a canonical form for such sprays, showing that their geodesic coefficients can be expressed as \(G^i = |y|\alpha(r,s) y^i + |y|^2\beta(r,s) x^i\), where \(r = |x|^2\) and \(s = \langle x,y\rangle/|y|\). For projectively flat spherically symmetric sprays -- which are directly related to Hilbert's fourth problem on characterizing metrics whose geodesics are straight lines -- we derive a complete classification of those with isotropic curvature, and in particular, we obtain the explicit form of those with zero curvature. Furthermore, we characterize sprays of weakly isotropic curvature in this class by a system of partial differential equations. These results may provide a unified framework for understanding symmetry and curvature in spray geometry and could offer new insights into the metrizability problem in Finsler geometry.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a canonical form for O(n)-invariant sprays and classifies the projectively flat isotropic-curvature cases, but the reduction to exactly two functions is the step that needs checking.

read the letter

The punchline is that the paper supplies a canonical form for spherically symmetric sprays and a classification of the projectively flat isotropic-curvature cases within that class, with explicit zero-curvature examples. It does a solid job setting up the symmetry reduction. Starting from invariance under the orthogonal group, it arrives at geodesic coefficients depending on just two scalar functions α and β of the radial and angular invariants r and s. That reduction is what makes the later substitutions into the projective flatness and curvature isotropy conditions feasible, leading to the claimed complete classification and the PDE system for the weakly isotropic subcase. The explicit forms for zero curvature are concrete and could be verified directly. The main soft spot is the foundation of that canonical form itself. The stress-test correctly identifies it as the critical step: the claim that O(n) invariance forces exactly G^i = |y| α(r,s) y^i + |y|^2 β(r,s) x^i with no additional terms. If the paper's derivation of this expression from the invariance conditions is complete and correct, then the classification follows. But if there are missed invariant terms or constraints between α and β, the list of solutions would be incomplete. Since the abstract states the form first and then the classification, the paper must contain that derivation, but it is worth a referee walking through the invariance calculation line by line. No other major issues stand out from the description. The work connects to Hilbert's fourth problem in a natural way without exaggeration. This paper is aimed at researchers in Finsler geometry and spray geometry who already work with curvature tensors and projective flatness. Someone outside that niche would find it dense, but within the subfield it offers a useful organizing framework. I would send it to peer review. The results are specific and the topic is connected to an open question, so referees can assess the technical details.

Referee Report

2 major / 2 minor

Summary. The manuscript studies sprays on R^n that are invariant under the orthogonal group O(n). It first derives a canonical form for the geodesic coefficients of such spherically symmetric sprays, G^i = |y| α(r,s) y^i + |y|^2 β(r,s) x^i where r = |x|^2 and s = <x,y>/|y|. For the subclass of projectively flat sprays it then gives a complete classification of those with isotropic curvature, supplies explicit expressions for the zero-curvature members, and characterizes the weakly isotropic curvature case by a system of PDEs. The work is motivated by connections to Hilbert's fourth problem and the metrizability problem in Finsler geometry.

Significance. If the reduction to the two-function canonical form is rigorously established and the subsequent substitutions into the projective-flatness and curvature-isotropy conditions are carried out without omission, the explicit classifications and the PDE characterization constitute a concrete advance in the study of symmetric sprays. The results supply a unified framework that could be used to test metrizability conjectures or to generate new examples of projectively flat Finsler metrics with prescribed curvature.

major comments (2)

[canonical-form section (prior to the projective-flatness analysis)] The section deriving the canonical form: the assertion that O(n)-invariance forces the geodesic spray to reduce exactly to the two scalar functions α(r,s) and β(r,s) with no additional invariant terms must be verified by exhibiting the full set of O(n)-invariant scalar functions on the tangent bundle and showing that all higher-order or independent invariants are either absent or algebraically dependent on r and s. This reduction is load-bearing for every subsequent classification.
[classification theorem] The classification theorem for projectively flat isotropic-curvature sprays: once the canonical form is substituted into the projective-flatness condition, the resulting PDE system on α and β must be solved completely; the manuscript should state explicitly whether the solution set is exhausted by the listed families or whether additional branches exist when the curvature-isotropy parameter is allowed to vary.

minor comments (2)

[preliminaries] Notation: the symbols r and s are introduced with clear definitions, but the manuscript should confirm that these are the only independent O(n)-invariants used throughout the curvature calculations.
[weakly-isotropic section] The PDE system for weakly isotropic curvature is presented as a characterization; a brief remark on the regularity assumptions (e.g., smoothness class of α and β) under which the system is derived would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

Referee: [canonical-form section (prior to the projective-flatness analysis)] The section deriving the canonical form: the assertion that O(n)-invariance forces the geodesic spray to reduce exactly to the two scalar functions α(r,s) and β(r,s) with no additional invariant terms must be verified by exhibiting the full set of O(n)-invariant scalar functions on the tangent bundle and showing that all higher-order or independent invariants are either absent or algebraically dependent on r and s. This reduction is load-bearing for every subsequent classification.

Authors: We agree that an explicit enumeration of the O(n)-invariant scalars strengthens the derivation. The manuscript derives the canonical form by imposing O(n)-invariance on the spray coefficients G^i, which must transform as a vector field. In the revision we will insert a dedicated paragraph (or short subsection) that first identifies the two fundamental O(n)-invariant scalars r = |x|^2 and s = ⟨x,y⟩/|y| on the tangent bundle, then shows via the orthogonal decomposition of T_x R^n into radial and tangential parts that any O(n)-invariant vector field on T R^n is necessarily a linear combination of y^i and x^i whose coefficients depend only on r and s. Higher-order contractions or tensorial invariants reduce algebraically to functions of these two scalars by the representation theory of O(n) and direct differentiation under the group action. This confirms the absence of independent additional terms and makes the reduction fully rigorous. revision: partial
Referee: [classification theorem] The classification theorem for projectively flat isotropic-curvature sprays: once the canonical form is substituted into the projective-flatness condition, the resulting PDE system on α and β must be solved completely; the manuscript should state explicitly whether the solution set is exhausted by the listed families or whether additional branches exist when the curvature-isotropy parameter is allowed to vary.

Authors: After substituting the canonical form into the projective-flatness and isotropic-curvature conditions we obtain an over-determined system of PDEs in α and β. This system is reduced by differentiation and algebraic elimination to a set of ordinary differential equations (or algebraic relations) whose complete integration yields precisely the families listed in the theorem, including the explicit zero-curvature solutions. In the revised manuscript we will append a short paragraph at the conclusion of the classification section that states: “The solution set is exhausted by the families enumerated above; no additional branches arise for any value of the curvature-isotropy parameter.” This makes the completeness explicit without altering the listed solutions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper first derives the canonical form G^i = |y| α(r,s) y^i + |y|^2 β(r,s) x^i directly from the definition of spherical symmetry (invariance under the orthogonal group O(n)), expressing the geodesic coefficients in terms of the two independent invariants r = |x|^2 and s = <x,y>/|y|. This is a standard symmetry reduction in spray geometry and does not presuppose the later classification results. The classification of projectively flat cases with isotropic curvature, the explicit zero-curvature forms, and the PDE system for weakly isotropic curvature are then obtained by substituting the derived form into the projective-flatness and curvature-isotropy equations and solving the resulting system of PDEs. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the chain is self-contained from the symmetry assumptions and the standard definitions of sprays and curvature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard definition of a spray on the tangent bundle and the domain assumption that orthogonal invariance forces the stated two-function dependence; no free parameters or new entities are introduced in the abstract.

axioms (2)

standard math A spray is a vector field on the slit tangent bundle satisfying the standard homogeneity conditions.
Invoked implicitly when writing the geodesic coefficients G^i.
domain assumption Invariance under the orthogonal group reduces the spray to the form involving only α(r,s) and β(r,s).
This is the key step used to obtain the canonical expression.

pith-pipeline@v0.9.0 · 5456 in / 1326 out tokens · 44574 ms · 2026-05-10T15:38:46.207800+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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[1] [1]

Asma and M

M. Asma and M. Zoltan,The holonomy of spherically symmetric projective Finsler metrics of constant curvature, J. Geom. Anal.34(2024), no. 8, Paper No. 257, 15 pp

work page 2024

[2] [2]

Buc˘ ataru and Z

I. Buc˘ ataru and Z. Muzsnay,Finsler metrizable isotropic sprays and Hilbert’s fourth problem, J. Aust. Math. Soc. 97(2014), no. 1, 27–47

work page 2014

[3] [3]

Berwald,Untersuchung der Kr¨ ummung allgemeiner metrischer R¨ aume auf Grund des in ihnen herrschenden Parallelismus, Math

L. Berwald,Untersuchung der Kr¨ ummung allgemeiner metrischer R¨ aume auf Grund des in ihnen herrschenden Parallelismus, Math. Z.26(1927), no. 1, 176

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Berwald, ¨Uber dien-dimensionalen Geometrien konstanter Kr¨ ummung, in denen die Geraden die k¨ urzesten sind, Math

L. Berwald, ¨Uber dien-dimensionalen Geometrien konstanter Kr¨ ummung, in denen die Geraden die k¨ urzesten sind, Math. Z.30(1929), 449–469

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S. G. Elgendi,On Riemann curvature of spherically symmetric metrics, Math. Phys. Anal. Geom.27(2024), no. 3, Paper No. 14, 19 pp. ON SPHERICALLY SYMMETRIC SPRAYS 13

work page 2024

[6] [6]

L. B. Huang and X. H. Mo,On spherically symmetric Finsler metrics of scalar curvature, J. Geom. Phys.62 (2012), no. 11, 2279–2287

work page 2012

[7] [7]

Ilmavirta and K

J. Ilmavirta and K. M¨ onkk¨ onen,The geodesic ray transform on spherically symmetric reversible Finsler manifolds, J. Geom. Anal.33(2023), no. 4, Paper No. 137, 27 pp

work page 2023

[8] [8]

Khani-Moghaddam, M

T. Khani-Moghaddam, M. Rafie-Rad and A. Tayebi,On general spherically symmetric Finsler metrics, Facta Univ. Ser. Math. Inform.40(2025), no. 5, 1065–1082

work page 2025

[9] [9]

Li,On the classification of projectively flat Finsler metrics with constant flag curvature, Adv

B. Li,On the classification of projectively flat Finsler metrics with constant flag curvature, Adv. Math.257 (2014), 266–284

work page 2014

[10] [10]

Li,Finsler metrics of weakly isotropic flag curvature, Comm

B. Li,Finsler metrics of weakly isotropic flag curvature, Comm. Anal. Geom.28(2020), no. 1, 113–139

work page 2020

[11] [11]

Li and Z

B. Li and Z. Shen,Sprays of isotropic curvature, Internat. J. Math.29(2018), no. 1, 1850003, 12 pp

work page 2018

[12] [12]

Y. Li, X. H. Mo and Y. Yu,Inverse problem of sprays with scalar curvature, Internat. J. Math.30(2019), no. 9, 1950041, 15 pp

work page 2019

[13] [13]

Lou and B

Y. Lou and B. Li,Sprays of weakly isotropic curvature, Differential Geom. Appl.82(2022), Paper No. 101896, 12 pp

work page 2022

[14] [14]

P. W. Michor,Topics in Differential Geometry, Graduate Studies in Mathematics 93, American Mathematical Society, Providence, RI, 2008

work page 2008

[15] [15]

G. W. Schwarz,Smooth functions invariant under the action of a compact Lie group, Topology14(1975), no. 1, 63–68

work page 1975

[16] [16]

Sol´ orzano and B

N. Sol´ orzano and B. Leandro,On spherically symmetric Finsler metric with scalar and constant flag curvature, Differential Geom. Appl.79(2021), Paper No. 101828, 17 pp

work page 2021

[17] [17]

Tayebi and B

A. Tayebi and B. Najafi,Some constructions of projectively related spherically symmetric Finsler metrics, Facta Univ. Ser. Math. Inform.40(2025), no. 2, 411–419

work page 2025

[18] [18]

Yang,Some classes of sprays in projective spray geometry, Differential Geom

G. Yang,Some classes of sprays in projective spray geometry, Differential Geom. Appl.29(2011), no. 4, 606–614

work page 2011

[19] [19]

Yang,On sprays of scalar curvature and metrizability, J

G. Yang,On sprays of scalar curvature and metrizability, J. Geom. Anal.33(2023), no. 4, Paper No. 120, 25 pp

work page 2023

[20] [20]

Zhou,Spherically symmetric Finsler metrics inR n, Publ

L. Zhou,Spherically symmetric Finsler metrics inR n, Publ. Math. Debrecen80(2012), no. 1-2, 67–77

work page 2012

[21] [21]

Zhu and H

H. Zhu and H. Zhang,Projective Ricci flat spherically symmetric Finsler metrics, Internat. J. Math.29(2018), no. 11, 1850078, 13 pp. School of Mathematics and Statist Ningbo University, 315211 Ningbo, China Email address:gyj mathematics@163.com School of Mathematics and Statist Ningbo University, 315211 Ningbo, China Email address:libenling@nbu.edu.cn

work page 2018