Weighted Projective Ricci Curvature in Finsler Geometry

A. Tayebi; B. Najafi; T. Tabatabaeifar

REVIEW 4 major objections 3 minor 25 references

Weighted Projective Ricci Curvature in Finsler Geometry

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

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Pith's one-line read This paper introduces weighted projective Ricci curvature and proves that projectively flat Finsler metrics with isotropic weighted projective Ricci and isotropic S-curvature are Randers or Kropina metrics.

desk verdict The new weighted projective Ricci curvature is a reasonable extension, and the Randers/Kropina computations are useful, but the main classification theorem is false: any non-Randers/Kropina locally Minkowski norm satisfies the hypotheses. read the letter →

arxiv 1908.09465 v1 pith:WAR4YETI submitted 2019-08-26 math.DG

T. Tabatabaeifar , B. Najafi , A. Tayebi This is my paper · ORCID

classification math.DG MSC 53B4053C60

keywords weightedprojectiveRiccicurvatureFinslergeometryRandersmetricKropinaS-curvatureprojectivelyflat(αβ)-metricinvariance

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

This paper introduces a new projective invariant, the weighted projective Ricci curvature: for a Finsler metric $F$ and a fixed reference metric $F_0$, it is $\mathrm{Ric}+(n-1)(\bar{S}^2+\bar{S}_{|k}y^k)$, where $\bar{S}$ packages the $S$-curvature of $F$ with the logarithm of the volume ratio. The paper characterizes exactly when a Randers metric $F=\alpha+\beta$ or a Kropina metric $F=\alpha^2/\beta$ is weighted projective Ricci flat with respect to its Riemannian part $\alpha$. Its main theorem says that a projectively flat Finsler metric with isotropic weighted projective Ricci curvature and isotropic $S$-curvature is forced to be one of these two $(\alpha,\beta)$-metric types. A sympathetic reader would care because this is a rigidity statement: a mild curvature condition, together with projective flatness, pins the metric down to the two simplest non-Riemannian families.

What carries the argument

The load-bearing object is the weighted projective Ricci quantity $$\mathrm{WPRic}_0=\mathrm{Ric}+(n-1)\left(\bar{S}^2+\bar{S}_{|k}y^k\right),\qquad \bar{S}=\frac{S+d\ln\Sigma}{n+1},$$ where $\Sigma$ is the ratio of the fixed reference volume form to the Busemann-Hausdorff volume form of $F$. Because this combination is invariant under projective changes of metric with a fixed volume form, it pairs naturally with projective flatness. Flatness writes the spray as $G^i=Py^i$, which turns the Ricci trace into $(n-1)(P^2-P_0)$; feeding the two isotropy conditions into that identity produces the quadratic equation whose two solution branches are exactly the Randers and Kropina metric forms.

What would settle it

Compute $F_{|s}y^s$ directly for a projectively flat Finsler metric with $G^i=Py^i$: the missing identity is equivalent to $F_{x^s}y^s=2PF$. If one exhibits a projectively flat metric with isotropic $S$-curvature, such as a non-Randers projectively flat $(\alpha,\beta)$-metric, for which this equality fails, then equation (6.5) is missing the term $cF_{|s}y^s$ and the paper's argument does not force the metric to be Randers or Kropina.

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Extended reading notes

Core claim

The central claim is Theorem 1.4: if a projectively flat Finsler metric has $\mathrm{WPRic}_0=(n-1)\sigma F^2$ and $S=(n+1)cF$ for scalar functions $\sigma(x)$ and $c(x)$, then $F$ is either a Randers metric or a Kropina metric. The proof reduces the two curvature assumptions, together with the projectively flat spray form $G^i=Py^i$ and the resulting Ricci formula $\mathrm{Ric}=(n-1)(P^2-P_0)$, to a quadratic equation in $F$ whose coefficients depend only on $x$. When the quadratic coefficient $\sigma-c^2$ vanishes, the solution is a quotient of the form $\alpha^2/\beta$, a Kropina metric; otherwise the positive solution has the form $\sqrt{\text{quadratic}}+\text{linear}$, a Randers metric. The paper also proves separate characterizations for Randers and Kropina metrics to be weighted projective Ricci flat with respect to their Riemannian part.

Load-bearing premise

The proof's load-bearing step is an unproved cancellation: when the two isotropy conditions are inserted into the projectively flat Ricci formula, the paper drops a term involving the horizontal derivative of $F$ along $y$, effectively assuming $F_{|s}y^s=0$; the paper gives no reason this derivative must vanish, and if it does not, the Randers/Kropina conclusion does not follow from the stated equations.

Editorial extensions

If this is right

Any projectively flat Finsler metric satisfying the two isotropy conditions is C-reducible, in the sense of belonging to the Randers or Kropina classes, so the rigidity constrains the whole metric function rather than only its geodesic spray.
For a complete Finsler manifold, the inequality $\mathrm{WPRic}_0\ge\mathrm{Ric}$ or $\mathrm{WPRic}_0\le\mathrm{Ric}$ holds if and only if the $S$-curvature vanishes, and then the $S$-curvature is an exact one-form; the Funk metric on the ball shows the completeness assumption cannot be relaxed to positive completeness.
A Randers metric $F=\alpha+\beta$ is weighted projective Ricci flat with respect to $\alpha$ exactly when its Riemannian part has $\mathrm{Ric}=t^m{}_m\alpha^2+2t_{00}$ and the divergence $s^m{}_{0;m}=0$.
A Kropina metric $F=\alpha^2/\beta$ is weighted projective Ricci flat with respect to $\alpha$ exactly when equations (1.4) and (1.5) hold, and the proof ties this to the conformal-form condition $r_{00}=\sigma\alpha^2$ and hence to vanishing $S$-curvature.
Because weighted projective Ricci curvature is projectively invariant with respect to a fixed volume form, it supplies a curvature quantity that is unchanged under projective equivalence, which is what makes the flatness classification meaningful.

Reading between the lines

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

If the identity $F_{|s}y^s=0$ used in the proof should fail for some projectively flat metric, the quadratic equation would become higher-degree and the decisive Randers/Kropina dichotomy could admit other metric families; this would leave the classification open rather than false.
The same construction may be tested with other projectively invariant quantities, for instance replacing the $S$-curvature term by the mean Landsberg curvature, to see whether a comparable two-family rigidity appears.
The Funk metric example suggests that on non-complete domains the sign comparison between weighted projective Ricci and Ricci curvature is governed by boundary terms rather than by vanishing $S$-curvature, so the weighted quantity may be useful for probing completeness.
A direct check of $F_{x^s}y^s=2PF$ on known projectively flat families with isotropic $S$-curvature would settle the missing identity by calculation, without needing the full classification.

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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

The paper's useful contribution is the definition of WPRic0 as a projective invariant extending Shen's PRic, plus the explicit necessary-and-sufficient conditions for Randers and Kropina metrics to be weighted projective Ricci flat. Those two computational theorems are the kind of thing specialists would want on record, if the algebra checks out. The proof of Theorem 1.1, however, is mis-stated: the argument concludes S = -d ln Sigma, not S = 0, so the theorem should say the weighted S-curvature vanishes. And the Randers section drops a factor (n+1) between (4.4) and (4.5). These look like typos, but they erode confidence in the computations. The real problem is Theorem 1.4. It is false as stated. Take any non-Randers, non-Kropina Minkowski norm on R^n, for instance F = ((y1)^4+(y2)^4)^{1/4} + epsilon sqrt((y1)^2+(y2)^2), epsilon small. This is a regular Finsler metric, independent of x, hence projectively flat with S=0 and Ric=0. With F0=F, WPRic0=0, so it has isotropic weighted projective Ricci and isotropic S-curvature (both with zero function). It is neither Randers nor Kropina. The stress-test's L^4 norm alone is not strongly convex, but the perturbed version fixes that, so the counterexample is genuine. The proof fails in exactly the degenerate case: equation (6.5) becomes 0=0, and the two-case split disappears. Even in the nondegenerate case, the step from a quadratic equation for F to 'Randers' is a non-sequitur: a square root of an arbitrary homogeneous degree-two function is not a Riemannian metric unless it is a quadratic form, and that is never shown. The passage from (6.4) to (6.5) also discards c F|_s y^s without justification. So the main theorem is not just missing a proof; it is wrong. Who gets value from this? Finsler specialists working on projective invariants, and only if they treat Theorems 1.2 and 1.3 as computational claims to be verified. The citation pattern is unremarkable; self-citations are inline. I would not rely on the paper for a theorem. Recommendation: desk reject, or accept only after a major rewrite that removes or sharply restricts Theorem 1.4 and fixes the algebraic slips.

Referee Report

4 major / 3 minor

Summary. This paper defines a weighted projective Ricci curvature WPRic0 = Ric + (n-1)(Sbar^2 + Sbar|_k y^k) with Sbar = (S + d ln Sigma)/(n+1) for a Finsler metric F relative to a fixed Finsler metric F0 and fixed volume forms. It then states four results: a comparison theorem between WPRic0 and Ric under completeness (Theorem 1.1), necessary and sufficient conditions for a Randers metric to be weighted projective Ricci flat (Theorem 1.2), analogous conditions for Kropina metrics (Theorem 1.3), and a classification of projectively flat Finsler metrics with isotropic weighted projective Ricci and isotropic S-curvature as Randers or Kropina metrics (Theorem 1.4). The proofs use standard formulas for sprays, S-curvature, and Ricci curvature, and several examples are included.

Significance. If the results were correct, the paper would provide a useful extension of Shen's projective Ricci curvature and a strong rigidity statement for projectively flat Finsler metrics. The formulas in Theorems 1.2 and 1.3 are concrete and the examples are instructive. However, the central classification Theorem 1.4 is false as stated, and the proof of Theorem 1.1 is internally inconsistent. These are not minor presentation issues; they invalidate the main claims of the paper.

major comments (4)

[Theorem 1.4, Section 6] Theorem 1.4 is false as stated. On R^2, let F(y) = ((y^1)^4 + (y^2)^4)^{1/4}, a strongly convex Minkowski norm. Since F is independent of x, its spray coefficients vanish, so F is projectively flat, Ric = 0, and S = 0. Taking F0 = F gives Sigma = 1 and eta = 0, hence Sbar = 0 and WPRic0 = Ric = 0. Thus F has isotropic S-curvature S = (n+1)cF with c = 0 and isotropic weighted projective Ricci WPRic0 = (n-1)sigma F^2 with sigma = 0. But F is neither Randers nor Kropina: a Randers metric F = alpha + beta has unit sphere satisfying a quadratic equation after squaring alpha = 1 - beta, and a Kropina metric F = alpha^2/beta has unit sphere satisfying the quadratic equation alpha^2 = beta, whereas the L^4 norm has the quartic unit sphere (y^1)^4 + (y^2)^4 = 1. In this example equation (6.5) degenerates to 0 = 0, so the case split in the proof does not produce the claimed conclusion.
[Section 6, Eqs. (6.4)-(6.5)] Independently of the counterexample, the algebraic passage from (6.4) to (6.5) is not justified as written. Expanding the term (cF + eta)|_s y^s requires an identity for F|_s y^s, which the paper neither states nor proves. More concretely, the sign of c0 in (6.5) is inconsistent with the expansion: substituting (cF+eta)^2 + (cF+eta)|_s y^s into (6.4) gives a term -(2c eta + c0)F, not -(2c eta - c0)F, when the equation is rearranged as in (6.5). Even after correcting this sign, the proof gives no argument that every solution of the scalar equation (6.5) must have the Randers or Kropina form, and the degenerate 0 = 0 case is not addressed.
[Section 3, Theorem 1.1] Theorem 1.1 and its proof do not agree. The comparison argument proves that phi(0) = 0 for phi = (S + d ln Sigma)/(n+1), i.e. S = -d ln Sigma. This is not the same as S = 0 when F0 is arbitrary. If S = -d ln Sigma, then Sbar = 0 and hence WPRic0 = Ric, so both inequalities WPRic0 >= Ric and WPRic0 <= Ric hold even though S need not vanish. Thus the 'if and only if S = 0' statement in Theorem 1.1 is false unless one additionally assumes d ln Sigma = 0, for example F0 = F with the same volume form. The corollary for PRic requires this additional hypothesis.
[Section 4, Eqs. (4.4)-(4.5)] The proof of Theorem 1.2 contains an unexplained factor change. Equation (4.4) states S = (n+1)[e00/(2F) - (s0 + rho0)], but equation (4.5) states S = (r00 - 2 alpha s0)/(2F), dropping the factor (n+1). The subsequent computation of S|_m y^m and the final formula (4.7) use the unnormalized expression, so the derivation of Theorem 1.2 does not follow from (4.4) as written. If (4.5) is intended to introduce a normalized S-curvature, this redefinition must be stated explicitly.

minor comments (3)

[Section 6, first paragraph] The sentence 'For simplicity, let us ut eta = ...' contains a typo; it should read 'let us put eta = ...'.
[Theorem 1.2, statement] The statement says 'for some scalar function c = c(x)' but the displayed conditions (i) and (ii) do not contain c; this leftover parameter should be removed or explained.
[Section 2, Example 2.1] The symbol beta is used both for a general 1-form in the Randers/Kropina discussion and for the normalized form (1/(n+1)) d ln Sigma in Example 2.1; this reuse is confusing and should be disambiguated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the classification claims are derived from definitions and standard formulas, not assumed as inputs.

full rationale

The paper introduces the weighted projective Ricci curvature as a new definition, WPRic0 := Ric + (n-1){S^2 + S|_k y^k} with S = (S + d ln(Sigma))/(n+1), and then derives consequences from this definition together with cited tensor formulas. Theorem 1.2 and Theorem 1.3 insert the known geodesic coefficients, Ricci-curvature formulas, and S-curvature formulas for Randers and Kropina metrics, respectively, and the resulting conditions (i)-(ii) and (1.4)-(1.5) are not assumed in the definitions of weighted projective Ricci flatness. Theorem 1.4 uses the projectively flat form G^i = P y^i, computes Ric = (n-1)(P^2 - P0), and rearranges the isotropic condition into a quadratic equation in F; calling the two algebraic cases Kropina and Randers follows from the form of the solution, not from an assumed classification. The self-citations in the paper appear in the introduction and examples and are not the load-bearing justification for Theorems 1.1-1.4. The main mathematical weakness is the step from (6.4) to (6.5), where the term c F|_s y^s is dropped when expanding (cF+eta)|_s y^s; this is an unproved identity and a correctness gap, not a circularity, because it does not make the conclusion equivalent to the hypotheses by construction. On the circularity criterion defined for this analysis, the derivation is self-contained and no step reduces to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper's results rest on a new curvature definition and on cited tensor formulas. No numerical parameters are fitted; the main risk is not hidden degrees of freedom but unverified tensor algebra and notation slips.

assumptions (4)

standard math Standard formulas for S-curvature and Ricci curvature of Randers metrics (Eq. 4.3-4.4) and Kropina metrics (Eq. 5.2-5.3) from [14] and [24].
The computations for Theorems 1.2 and 1.3 rely on these cited tensor formulas as inputs; the paper does not re-derive them.
standard math Projectively flat Finsler metrics have geodesic coefficients of the form G^i = P(x,y) y^i.
Used in Section 6 to derive Ric=(n-1)(P^2-P0) and the quadratic classification.
standard math Berwald metrics have vanishing S-curvature, as cited from [19].
Used in Examples 2.1 and 2.2 to simplify WPRic0.
domain assumption For Randers metrics, the Busemann-Hausdorff volume ratio satisfies ln(Σ)=(n+1)ln ρ, stated in Section 4 before Theorem 1.2.
This volume formula enters the weighted definition and the Randers computation; it is quoted without derivation.

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Cite this review

Pith. "Pith review of Weighted Projective Ricci Curvature in Finsler Geometry." pith.science (2026). https://pith.science/paper/WAR4YETI

@misc{pith2026190809465,
  author       = {Pith},
  title        = {Pith review of: Weighted Projective Ricci Curvature in Finsler Geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WAR4YETI}},
  note         = {Machine review of arXiv:1908.09465}
}

read the original abstract

In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

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Pith tools

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

[1] [1]

Akbar-Zadeh, Champ de vecteurs projectifs sur le ﬁbre unitaire , J

H. Akbar-Zadeh, Champ de vecteurs projectifs sur le ﬁbre unitaire , J. Math. Pures Appl. 65(1986), 47-79

work page 1986

[2] [2]

Asanov, Finsler Geometry, Relativity and Gauge Theories , D

G.S. Asanov, Finsler Geometry, Relativity and Gauge Theories , D. Reidel Publish- ing Company, Dordrecht, Holland, 1985

work page 1985

[3] [3]

Bacso and M

S. Bacso and M. Matsumoto, On Finsler spaces of Douglas type. A generalization of the notion of Berwald space , Publ. Math. Debrecen, 51 (1997), 385-406

work page 1997

[4] [4]

Bao and Z

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

work page 2002

[5] [5]

Cheng and Z

X. Cheng and Z. Shen, Randers metrics of scalar ﬂag curvature , J. Australian. Math. Soc, 87(2009), 359-370

work page 2009

[6] [6]

Cheng, Y

X. Cheng, Y. Shen and X. Ma, On a class of projective Ricci ﬂat Finsler metrics , Publ. Math. Debrecen, 7528 (2017), 1-12. Weighted Projective Ricci Curvature in Finsler Geometry 18

work page 2017

[7] [7]

Ingarden, Geometry of thermodynamics, Diﬀ

R.S. Ingarden, Geometry of thermodynamics, Diﬀ. Geom. Methods in Theor. Phys., XV Intern. Conf. Clausthal 1986, World Scientiﬁc, Singapore, 198 7

work page 1986

[8] [8]

Kropina, On projective two-dimensional Finsler spaces with a specia l metric , Trudy Sem

V.K. Kropina, On projective two-dimensional Finsler spaces with a specia l metric , Trudy Sem. Vektor. Tenzor. Anal. 11(1961), 277-292

work page 1961