New classes of projectively related Finsler metrics of constant flag curvature

Georgeta Cretu

REVIEW 3 major objections 4 minor 17 references

New classes of projectively related Finsler metrics of constant flag curvature

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper defines a Weyl-type curvature tensor whose vanishing characterizes constant flag curvature for Finsler metrics in dimension at least three, then uses it to construct three new projectively flat families.

desk verdict New explicit Finsler families are worth attention, but the projective-invariance claim that motivates them is unproved because a Hamel factor does not kill the dJdhP term. read the letter →

arxiv 1908.05305 v2 pith:DV7IHLOD submitted 2019-08-14 math.DG

Georgeta Cretu This is my paper · ORCID

classification math.DG MSC 53C6053B40

keywords projectivelyflatFinslermetricsWeyl-typecurvaturetensorconstantflagHamelfunctionRanderssquarerelatedconformalchangesof

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to give a tensorial characterization of Finsler metrics with constant flag curvature, the Finsler analogue of constant sectional curvature. It defines a Weyl-type curvature tensor $W_1$ of type $(1,2)$ and proves that, on manifolds of dimension at least three, a Finsler metric has constant flag curvature exactly when $W_1$ vanishes. It also shows that $W_1$ is invariant under projective deformations only when the projective factor is a Hamel function, and uses that invariance to build three new families of projectively flat Finsler metrics of constant flag curvature. Two of the new families have zero flag curvature and one has negative flag curvature; the constructions recover the Funk, generalized Funk, and Berwald metrics as particular cases.

What carries the argument

The central object is the Weyl-type curvature tensor $W_1=R-\frac{1}{2(n-1)}d_J(\operatorname{Tr}\Phi)\wedge J$, a vector-valued semi-basic 2-form built from the curvature tensor of the nonlinear connection and the Jacobi endomorphism of a spray; $\operatorname{Tr}\Phi$ is the Ricci scalar and $d_J$ is the vertical exterior derivative. Its design makes it vanish exactly when the curvature tensor has the constant-flag-curvature form $R=\frac{1}{2(n-1)}d_J(\operatorname{Tr}\Phi)\wedge J$. The matching mechanism is the deformation formula for $W_1$ under projective changes, together with the notion of Hamel function: a 1-homogeneous function $P$ satisfying $\delta_S P=0$, i.e. a first integral of the spray, which is the condition that kills the projective-deformation term and yields invariance.

What would settle it

Compute the right side of formula (3.17) for a concrete Finsler spray whose projective factor is a Hamel function but has $d_Jd_hP\neq 0$; if the term $d_Jd_hP\otimes C$ survives, the claimed invariance fails. A direct substitution check that $W_1=0$ for each of the families (5.16), (5.32), and (5.48) would confirm the constant-flag-curvature conclusion computationally.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.1: for $\dim M \ge 3$, a Finsler metric has constant flag curvature if and only if the Weyl-type tensor $W_1 = R - \frac{1}{2(n-1)} d_J(\operatorname{Tr}\Phi)\wedge J$ vanishes, where $R$ is the curvature tensor of the nonlinear connection and $\Phi$ is the Jacobi endomorphism. The proof goes by showing that vanishing of $W_1$ forces the Ricci scalar to be a function of position only, and a Finslerian Schur lemma upgrades that to constancy. In dimension two, the same conclusion needs the extra condition $d_h\alpha=0$. The paper further claims that under a projective deformation $\bar S = S - 2PC$, the tensor changes by $\bar W_1 = W_1 + \frac12\delta_S P\wedge J + d_J d_h P\otimes C$, so $W_1$ is invariant precisely when the projective factor $P$ is a Hamel function; Proposition 3.5 then states that a Hamel-factor projective deformation of a constant-flag-curvature metric is again of constant flag curvature. On this basis the paper presents the explicit projectively flat families (5.16), (5.32), and (5.48).

Load-bearing premise

The invariance claim for $W_1$ under Hamel projective factors assumes that the extra term $d_Jd_hP\otimes C$ in the deformation formula vanishes or cancels, and the paper does not establish that vanishing.

Editorial extensions

If this is right

In dimension at least three, constant flag curvature is equivalent to the single tensorial equation $W_1=0$, giving a concrete test that does not require solving for geodesics.
A projective deformation with Hamel projective factor turns any constant-flag-curvature Finsler metric into another such metric; in dimension two the additional condition $d_h\alpha=0$ is needed.
The family (5.16) is a projectively flat Randers-type metric whose flag curvature is the same negative constant $-\nu^2$ as the seed metric.
The families (5.32) and (5.48) are projectively flat Finsler metrics of zero flag curvature, obtained respectively from a square-type deformation and a conformal-type deformation.
The Funk, generalized Funk, and Berwald metrics appear as special cases, showing that the new families include and extend known projectively flat constant-flag-curvature examples.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same deformation scheme should work starting from any projectively flat Randers metric whose projective factor is proportional to the metric, not just the particular seed (4.17); the computations only use that proportionality and the closure of the 1-form.
If the unexamined term $d_Jd_hP\otimes C$ is nonzero for some Hamel function, then the true invariant is a restricted version of $W_1$, and the families here would still be covered because their projective factors satisfy the stronger condition.
The pair $(W_1, d_h\alpha)$ in dimension two suggests that a fully projectively invariant characterization of constant flag curvature in low dimension may require two tensors rather than one.
One could test whether the zero-curvature families (5.32) and (5.48) have a known Riemannian or Randers specialization beyond the Funk and Berwald cases, which would connect them to existing classifications.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the new explicit families are the real contribution, but the general invariance theorem that supposedly generates them has a hole. The paper defines a (1,2) Weyl-type tensor W1 and proves (modulo a gap in a lemma) that its vanishing characterizes constant flag curvature in dimension ≥3, with a separate 2D condition. The three families in Section 5 — (5.16), (5.32), (5.48) — are new, specialize to Funk and Berwald metrics in the right limits, and the flag-curvature computations for them look direct and consistent. I did not redo every line, but the structure is credible.

The soft spot is the projective invariance of W1. Lemma 3.3 gives (3.17): under S̄=S−2PC, W̄1−W1 = 1/2 δSP ∧ J + dJdhP⊗C. The paper claims that when P is a Hamel function (δSP=0) the tensor is invariant. That does not follow: the second term remains. The stress-test example is concrete and correct: on R^2 take S0=y^i∂/∂x^i and P=x^1y^1+x^2y^2. Then δS0P=0, so P is Hamel for S0, but dJdh0P = dy^1∧dx^1+dy^2∧dx^2 ≠ 0. Thus W̄1−W1≠0. The proof of Proposition 3.5 for dim≥3 relies on exactly this missing implication, so the proposition is not established as stated. There is also equation (3.6), dhF^2=0, asserted without proof; it may be a standard property of the canonical spray, but the paper doesn't say why.

None of this necessarily kills the examples, because their constant flag curvature is checked directly, not by the invariance argument. But it does mean the paper's advertised mechanism—Hamel projective factors preserve constant flag curvature—is unproved, and the W1 tensor is not invariant under all Hamel factors. The authors need either to add a hypothesis like dJdhP=0, or find a new argument that uses Finsler metrizability of both sprays.

Bottom line: this deserves a serious referee, primarily because of the explicit families and the attempt at a clean criterion. It should be a major revision, not a desk reject. I'd want the invariance gap resolved before citing the families in my own work.

Referee Report

3 major / 4 minor

Summary. The paper defines a (1,2)-type Weyl tensor W1 = R - (1/(2(n-1))) dJ(TrΦ) ∧ J and claims that a Finsler metric has constant flag curvature if and only if W1 vanishes in dimension at least three, with an extra condition in dimension two. It studies projective deformations S → S - 2PC, derives the transformation formula (3.17), and asserts that W1 is projectively invariant under Hamel projective factors. Section 4 characterizes projectively flat Randers metrics whose projective factor is proportional to the metric. Section 5 constructs three families of projectively flat Finsler metrics, (5.16), (5.32), and (5.48), of constant flag curvature, with the flag curvature verified by direct computation and special cases recovering the Funk, generalized Funk, and Berwald metrics.

Significance. If the characterization is correct, the new W1 tensor gives a clean spray-theoretic criterion for constant flag curvature, and the explicit families in Section 5 are useful, checkable examples of projectively related Finsler metrics. The direct flag-curvature computations and the recovery of known projectively flat metrics are genuine strengths. However, the central invariance mechanism involving Hamel projective factors is flawed as stated, and the main theoretical claims in the abstract and in Proposition 3.5 depend on that mechanism. The explicit family computations may survive, but the paper's general framing needs revision.

major comments (3)

[§3, Lemma 3.3 and Proposition 3.5] Formula (3.17) contains the term dJdhP ⊗ C, but the text immediately concludes that W1 is invariant whenever δSP = 0. This conclusion is not justified: δSP = 0 does not imply dJdhP = 0. For the flat spray S0 = y^i ∂/∂x^i on R^2 and P = x^1 y^1 + x^2 y^2, one has δS0P = 0, yet dh0P = y^i dx^i and dJdh0P = dy^1 ∧ dx^1 + dy^2 ∧ dx^2 ≠ 0, so formula (3.17) gives a nonzero change of W1 even though W1 = 0 for the flat spray. Since Proposition 3.5 relies exactly on the implication from δSP = 0 to invariance for dim ≥ 3, and also feeds into the two-dimensional case through Theorem 3.2, the proposition is not established as stated. The proposition needs an additional hypothesis such as dJdhP = 0, or a new proof. The explicit families in Section 5 are not invalidated because their constant flag curvature is checked directly, but the sentence after (5.32) that invokes Proposition 3.5 should be replaced by the direct verification.
[§3, proof of Theorem 3.1] In the converse direction of Theorem 3.1, equation (3.6), namely dhF^2 = 0, is asserted as a consequence of (3.5) without proof; this requires the standard identity S(F^2) = 0 for a Finsler geodesic spray, which is not stated. Moreover, the passage from (3.8) to (3.9) and the claim that isotropy converts (3.9) into (3.10) use identities for the action of curvature-type operators on F^2 that are not explained. The notation d_R F^2 in (3.7) is also undefined. These steps are probably repairable, but as written the 'if' direction of the main characterization is incomplete and should be rewritten with the missing identities supplied.
[§3, proof of Theorem 3.2] The converse of Theorem 3.2 contains the unexplained assertion that vanishing of W1 implies dJκ = 0, without first defining κ or proving that the proportionality factor is fiber-independent. This is a load-bearing step because Proposition 3.5 uses Theorem 3.2 for the two-dimensional case. The proof should be completed by showing explicitly how W1 = 0 forces the scalar curvature to be fiber-independent before the extra condition dhα = 0 makes it constant.

minor comments (4)

[§5.1, equation (5.17)] In the final displayed equality of (5.17), the expression should be -ν^2 F^2, not -ν^2 F, to match the preceding terms and the conclusion κ = -ν^2.
[§5, positivity conditions] The paper does not specify the open domains or parameter conditions under which the metrics (5.16), (5.32), and (5.48) are positive and strongly convex. Please state the conditions on ν, η, e, f, and v that ensure these are genuine Finsler metrics.
[Throughout] There are typographical errors: 'Hammel' should be 'Hamel', 'onlt' should be 'only', and the running title contains 'CUR V ATURE'. In the proof of Proposition 3.5, 'Lemma 3.17' should be 'Lemma 3.3'.
[§3, equation (3.7)] The notation d_R F^2 is used without definition. Please define how the curvature tensor R acts on a function of the tangent bundle, or replace the notation with an explicit formula.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: W1 is a derived obstruction, not a fitted input; the Section 5 families are checked by direct curvature computation.

full rationale

The derivation chain does not reduce to its inputs. W1 is defined in (3.2) as the difference between R and (1/(2(n-1)))dJ(TrΦ)∧J; the forward half of Theorem 3.1 follows from the constant-flag-curvature form (2.20), but the converse is a substantive argument through metrizability, isotropy, and Schur's lemma, so the equivalence is not merely definitional. The Section 5 constructions impose explicit projective-factor ansätze (e.g., P=νb, P=2cF) and then verify constant flag curvature by direct formulas (5.17), (5.31), (5.49); the Hamel condition is a sufficient mechanism used to select examples, not a restatement of the conclusion. The paper's main self-citation is [3] (Bucataru–Cretu), used for W0 and for a Hamel-equivalence identity; W1 and the new families are derived here, and the curvature checks are self-contained, so this citation is not load-bearing. A genuine mathematical gap exists at Lemma 3.3 / Proposition 3.5: from δSP=0 the term dJdhP⊗C in (3.17) does not vanish in general, and the text drops it without proof; that is a correctness risk, not circularity. No fitted parameter is renamed as a prediction, and no uniqueness claim is imported from the authors' prior work. Hence score 1.

Assumptions & free parameters 5 free parameters · 4 assumptions · 1 invented entities

The paper's conclusions rest on standard Finsler-geometry background (sprays, Jacobi endomorphism, Berwald connection, Schur lemma, Hamel characterization) and on two imported transformation formulas from Bucataru-Muzsnay. The new input is the tensor W1 and the ansatz solutions for the deformation functions. No new physical entities or empirically fitted constants are introduced; the listed parameters are constants in the example families. The main unstated load-bearing assumption is dJdhP = 0 in the invariance claim.

free parameters (5)

c = real constant, κ = -c^2
Proportionality constant in the initial projectively flat Randers metric (4.17); determines the negative flag curvature.
ν = real constant, related to c by ν = -c in the first family
Deformation parameter in (5.15) and (5.16); the new metric has flag curvature -ν^2.
η = positive real constant
Scaling factor in the square metric family (5.32); does not affect flag curvature.
e = constant vector in R^n
Parameter in the 1-form b (5.15) for the Randers deformation; must be chosen so that the metric is positive.
f = real constant
Constant in (5.15) ensuring positivity of the metric.

assumptions (4)

standard math Finslerian Schur lemma (Matsumoto)
Used in Theorem 3.1 to conclude the scalar flag curvature κ is constant once dJκ=0.
standard math Hamel's characterization of projectively flat metrics
Used in Section 4 to characterize projectively flat Randers metrics.
standard math Curvature and Jacobi endomorphism transformation formulas under projective change (3.18)-(3.19), from Bucataru-Muzsnay
Imported from cited literature [5]; basis of Lemma 3.3.
domain assumption dhF^2 = 0 for the geodesic spray of a Finsler metric
Asserted in (3.6) without proof; standard in Finsler geometry but unstated.

invented entities (1)

Weyl-type curvature tensor W1
purpose: Characterize constant flag curvature and track projective invariance
Defined as W1 = R - 1/(2(n-1)) dJ(TrΦ) ∧ J. It is a derived mathematical object from existing curvature quantities, not a postulated physical entity.

how reviews work

0 comments

Cite this review

Pith. "Pith review of New classes of projectively related Finsler metrics of constant flag curvature." pith.science (2026). https://pith.science/paper/DV7IHLOD

@misc{pith2026190805305,
  author       = {Pith},
  title        = {Pith review of: New classes of projectively related Finsler metrics of constant flag curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DV7IHLOD}},
  note         = {Machine review of arXiv:1908.05305}
}

abstract

We define a Weyl-type curvature tensor of $(1,2)$-type to provide a characterization for Finsler metrics of constant flag curvature. This Weyl-type curvature tensor is projective invariant only to projective factors that are Hamel functions. Based on this aspect we construct new families of projectively related Finsler metrics that have constant flag curvature.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

and Matsumoto M.: The theory of sprays and Finsler spaces with applications in physics and biology , FTPH 58, Kluwer Academic Publishers, 1993

Antonelli P.L., Ingarden R.S. and Matsumoto M.: The theory of sprays and Finsler spaces with applications in physics and biology , FTPH 58, Kluwer Academic Publishers, 1993

work page 1993
[2]

and Shen Z.,: Finsler metrics of constant positive curvature on the Lie gr oup S3, J

Bao D. and Shen Z.,: Finsler metrics of constant positive curvature on the Lie gr oup S3, J. London Math. Soc. (2) 66(2002), 453–467

work page 2002
[3]

Bucataru I., Cret ¸u G.: A characterisation for Finsler metrics of constant curvatu re and a Finslerian version of Beltrami theorem, Journal of Geometric Analysis, DOI: 10.1007/s12220-019- 00158-7, arXiv:1808.05001v2, 2019

work page Pith review arXiv 2019
[4]

Applications to dynamical sys tems, Romanian Academy, 2007

Bucataru I., Miron R.: Finsler-Lagrange geometry. Applications to dynamical sys tems, Romanian Academy, 2007

work page 2007
[5]

Bucataru I., Muzsnay Z.: Projective and Finsler metrizability: parametrization ri gidity of geodesics , Int. J. Math., 23 no. 6 (2012), 1250099, 15 pages. NEW CLASSES OF PROJECTIVELY RELATED FINSLER METRICS OF CONS TANT FLAG CUR V ATURE 19

work page 2012
[6]

Cui N., Shen Y.B.: Projective change between two classes of (α, b )-metrics, Diﬀerential Geometry and its Applications 27 (2009) 566-573

work page 2009
[7]

Grifone J.: Structure presque tangente et connections I , Ann. Inst. Fourier, 22 (1972), 287–334

work page 1972
[8]

Grifone J., Muzsnay Z.: Variational Principles For Second-Order Diﬀerential Equa tions, W orld Scientiﬁc, 2000

work page 2000

Show all 17 references

[9]

S.: Conformal geometry of generalized metric spaces , Proc.nat.Acad

Knebelman M. S.: Conformal geometry of generalized metric spaces , Proc.nat.Acad. Sci.USA, 15(1929):33–41 and 376–379

1929
[10]

Matsumoto M.: Foundations of Finsler geometry and special Finsler spaces , Kaiseisha Press, 1986

1986
[11]

Mikes J.,: Geodesic mappings of aﬃne-connected and Riemannian spaces , J. Math. Sci. 78, 3 (1996)

1996
[12]

Mo X.: A global classiﬁcation result for Randers metrics of scalar curvature on closed manifolds , Nonlinear Analysis, 69(2008), 2996–3004

2008
[13]

and Yang C.: Some constructions of projectively ﬂat Finsler metrics , Scientia Sinica

Mo X., Shen Z. and Yang C.: Some constructions of projectively ﬂat Finsler metrics , Scientia Sinica. 49(2006), 703–714

2006
[14]

of the American Mathematical Society, Volume 355, Number 4, 1713-1728, 2002

Shen Z.: Projectively ﬂat Finsler metrics of constant ﬂag curvature , Trans. of the American Mathematical Society, Volume 355, Number 4, 1713-1728, 2002

2002
[15]

and Yu C.: On Einstein Square Metrics , Publicationes mathematicae, 85(3) September 2012, DOI: 10.5486/PMD.2014.6015

Shen Z. and Yu C.: On Einstein Square Metrics , Publicationes mathematicae, 85(3) September 2012, DOI: 10.5486/PMD.2014.6015

2012
[16]

Shibata C.: On invariant tensors of β -changes of Finsler metrics , J . Math. Kyoto Univ. (JMKYAZ),24-1 (1984), 163–188

1984
[17]

F aculty of Mathematics, Alexandru Ioan Cuza University, Ias ¸i, Romania E-mail address : cretuggeorgeta@gmail.com

Szilasi J., Lovas R., Kert´ esz D.: Connections, sprays and Finsler structures , W orld Scientiﬁc, 2014. F aculty of Mathematics, Alexandru Ioan Cuza University, Ias ¸i, Romania E-mail address : cretuggeorgeta@gmail.com

2014

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

and Matsumoto M.: The theory of sprays and Finsler spaces with applications in physics and biology , FTPH 58, Kluwer Academic Publishers, 1993

Antonelli P.L., Ingarden R.S. and Matsumoto M.: The theory of sprays and Finsler spaces with applications in physics and biology , FTPH 58, Kluwer Academic Publishers, 1993

work page 1993

[2] [2]

and Shen Z.,: Finsler metrics of constant positive curvature on the Lie gr oup S3, J

Bao D. and Shen Z.,: Finsler metrics of constant positive curvature on the Lie gr oup S3, J. London Math. Soc. (2) 66(2002), 453–467

work page 2002

[3] [3]

Bucataru I., Cret ¸u G.: A characterisation for Finsler metrics of constant curvatu re and a Finslerian version of Beltrami theorem, Journal of Geometric Analysis, DOI: 10.1007/s12220-019- 00158-7, arXiv:1808.05001v2, 2019

work page Pith review arXiv 2019

[4] [4]

Applications to dynamical sys tems, Romanian Academy, 2007

Bucataru I., Miron R.: Finsler-Lagrange geometry. Applications to dynamical sys tems, Romanian Academy, 2007

work page 2007

[5] [5]

Bucataru I., Muzsnay Z.: Projective and Finsler metrizability: parametrization ri gidity of geodesics , Int. J. Math., 23 no. 6 (2012), 1250099, 15 pages. NEW CLASSES OF PROJECTIVELY RELATED FINSLER METRICS OF CONS TANT FLAG CUR V ATURE 19

work page 2012

[6] [6]

Cui N., Shen Y.B.: Projective change between two classes of (α, b )-metrics, Diﬀerential Geometry and its Applications 27 (2009) 566-573

work page 2009

[7] [7]

Grifone J.: Structure presque tangente et connections I , Ann. Inst. Fourier, 22 (1972), 287–334

work page 1972

[8] [8]

Grifone J., Muzsnay Z.: Variational Principles For Second-Order Diﬀerential Equa tions, W orld Scientiﬁc, 2000

work page 2000