$\eta$-Ricci Solitons on Kenmotsu 3-Manifolds

K. De; U.C. De

REVIEW 3 major objections 4 minor 23 references

$\eta$-Ricci Solitons on Kenmotsu 3-Manifolds

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read On Kenmotsu 3-manifolds, four natural curvature conditions force eta-Ricci solitons to have constants $\lambda=\mu=1$, Ricci tensor $S=-2g$, and local hyperbolic geometry.

desk verdict Routine Kenmotsu 3-manifold soliton paper with two fixable but real errors: theorems stated for arbitrary V but proved only for V=ξ, and an example with wrong soliton constants. read the letter →

arxiv 2411.14988 v1 pith:VYIW654O submitted 2024-11-22 math.DG

K. De , U.C. De This is my paper · ORCID

classification math.DG MSC 53C1553C25

keywords η-RiccisolitonKenmotsu3-manifoldsCodazzitypeofRiccitensorcyclicparallelφ-RiccisymmetricR.R=Q(SR)hyperbolicspace

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 studies $\eta$-Ricci solitons on Kenmotsu 3-manifolds, a class of almost-contact Riemannian spaces modeled on the hyperbolic direction. It claims that four natural curvature conditions on the Ricci tensor, namely Codazzi type, cyclic parallel, $\varphi$-Ricci symmetry, and the curvature identity $R.R=Q(S,R)$, each force the soliton constants to be $\lambda=\mu=1$ and the Ricci tensor to be $S=-2g$; in the first three cases the paper concludes that the manifold is locally isometric to the hyperbolic space $H(-1)$. If correct, this is a uniform rigidity statement: in dimension three these conditions pin down the geometry completely. The paper also constructs an explicit Kenmotsu 3-manifold that it asserts carries a proper $\eta$-Ricci soliton with $\lambda=-1$ and $\mu=3$, showing that nontrivial solitons can occur in this class.

What carries the argument

The key identity is the 3-dimensional Kenmotsu formula $S(X,Y)=\frac{1}{2}[(r+2)g(X,Y)-(r+6)\eta(X)\eta(Y)]$, combined with the Lie-derivative identity $\mathcal{L}_{\xi}g=2[g-\eta\otimes\eta]$ and the soliton equation (1.3). Comparing these gives $S(X,Y)=-(\lambda+1)g(X,Y)-(\mu-1)\eta(X)\eta(Y)$ and hence $\lambda+\mu=2$. Each curvature condition (Codazzi type, cyclic parallel, $\varphi$-Ricci symmetry, or $R.R=Q(S,R)$) is fed into the covariant derivative of that expression; each forces $\mu=1$, then $\lambda=1$, $S=-2g$, $r=-6$. The 3-dimensional curvature expression then turns these data into constant sectional curvature $-1$, which is the local model $H(-1)$.

What would settle it

Substitute the Section 7 example into (1.3) with $V=\xi$, $S=-2g$, $\lambda=-1$, $\mu=3$: the left side is $2(g-\eta\otimes\eta)+2(-2g)+2(-1)g+2(3)\eta\otimes\eta=-4g+4\eta\otimes\eta$, which is not the zero tensor; if this substitution is correct, the example is not a proper $\eta$-Ricci soliton.

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Extended reading notes

Core claim

The central claim is a family of rigidity theorems: on a Kenmotsu 3-manifold, an $\eta$-Ricci soliton whose Ricci tensor is of Codazzi type, cyclic parallel, or $\varphi$-Ricci symmetric must have $\lambda=\mu=1$, Ricci tensor $S=-2g$, scalar curvature $r=-6$, and be locally isometric to the hyperbolic space $H(-1)$; the curvature condition $R.R=Q(S,R)$ likewise forces $\lambda=\mu=1$ and the Einstein condition $S=-2g$. The argument compares the 3-dimensional Kenmotsu Ricci formula with the soliton equation, obtaining $S(X,Y)=-(\lambda+1)g(X,Y)-(\mu-1)\eta(X)\eta(Y)$ and the constraint $\lambda+\mu=2$. Each curvature condition is substituted into the covariant derivative of this expression and forces $\mu=1$; the 3-dimensional curvature identity then converts $S=-2g$ and $r=-6$ into constant sectional curvature $-1$. The final section gives an explicit coordinate model that the paper presents as a Kenmotsu 3-manifold admitting a proper $\eta$-Ricci soliton with $\lambda=-1$, $\mu=3$.

Load-bearing premise

The classification arguments depend on the soliton's potential vector field being the Reeb field $\xi$, because the central comparison formula is derived by computing the Lie derivative along $\xi$; if the potential is truly arbitrary, the theorems as stated are not established.

Editorial extensions

If this is right

Under the Codazzi-type Ricci condition, any proper $\eta$-Ricci soliton on a Kenmotsu 3-manifold must have $\lambda=\mu=1$, $S=-2g$, and be locally isometric to $H(-1)$.
The same rigidity holds when the Ricci tensor is cyclic parallel.
A $\varphi$-Ricci symmetric Kenmotsu 3-manifold admitting a proper $\eta$-Ricci soliton must satisfy $\mu=1$, $\lambda=1$, and be locally isometric to $H(-1)$.
The curvature condition $R.R=Q(S,R)$ forces $\lambda=\mu=1$ and makes the manifold Einstein with $S=-2g$.
The explicit example in Section 7 is asserted to be a Kenmotsu 3-manifold admitting a proper $\eta$-Ricci soliton with $\lambda=-1$ and $\mu=3$.

Reading between the lines

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

An unstated corollary of Theorem 6.1, combined with the paper's own assertion that every 3-dimensional Riemannian manifold satisfies $R.R=Q(S,R)$, is that every $\eta$-Ricci soliton on a Kenmotsu 3-manifold would have $\lambda=\mu=1$ and $S=-2g$; reconciling this with the Section 7 example is a direct consistency check.
The method could be extended to arbitrary potential fields by decomposing $V$ into its Reeb and horizontal components and deriving the analogue of (2.16) without the assumption $V=\xi$.
A nearby testable problem is whether the same rigidity holds for gradient $\eta$-Ricci solitons on Kenmotsu 3-manifolds, where the potential field is the gradient of a smooth function.

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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

Quick take: the paper is a routine application of standard η-Ricci soliton formulas to Kenmotsu 3-manifolds. The classifications are new in the narrow sense that these specific combinations haven't been written down, but they come from direct substitution into known equations. No new ideas.

What's good: Proposition 2.1 is clean. The trace argument in Section 6 checks out under the stated assumption that the potential is ξ. The example, once corrected, would be a legitimate explicit Kenmotsu 3-manifold carrying a proper η-Ricci soliton.

The soft spots are real. First, Theorems 3.1 and 4.1 are stated for an arbitrary potential vector field V, but the proofs rely on (2.15)–(2.16), which use L_ξ g. Nothing in the paper shows that a soliton with Codazzi-type or cyclic-parallel Ricci tensor must have V=ξ. So the theorems as stated are not proved; they hold only for V=ξ. This is a load-bearing gap, though likely fixable by either restricting the statement or adding an argument.

Second, the Section 7 example is internally inconsistent. With S=-2g and L_ξ g=2(g-ηη), equation (1.3) forces λ=μ=1. The paper claims λ=-1, μ=3, which contradicts its own (2.16). The example still works as a proper η-Ricci soliton with λ=μ=1, so the construction is salvageable, but the numbers as written are wrong.

Third, Theorem 5.1 is missing part of its hypothesis: it says 'If M is φ-Ricci symmetric, then μ=1, λ=1...' without stating that an η-Ricci soliton is being considered, so λ and μ are undefined. That's a presentation error, but easily fixed.

I also note that formula (2.12) comes from a paper coauthored by U.C. De; that's ordinary reliance, not a problem, but worth being aware.

Who's this for? Specialists in contact geometry who track classification results on Kenmotsu manifolds. It's a niche contribution. The core computations are mostly correct and the errors are fixable, so I'd send it to a referee rather than desk reject. But it needs major revision before acceptance. If the authors fix the V=ξ gap and the example, it could be a publishable minor result.

Referee Report

3 major / 4 minor

Summary. The paper studies η-Ricci solitons on Kenmotsu 3-manifolds. After deriving a relation λ+μ=2 for solitons with potential vector field ξ (Proposition 2.1, equation (2.16)), the authors claim that under each of four hypotheses—Codazzi-type Ricci tensor, cyclic-parallel Ricci tensor, φ-Ricci symmetry, and the curvature condition R.R=Q(S,R)—the soliton constants must be λ=μ=1 and the manifold is locally isometric to hyperbolic space H(-1), with the last case giving an Einstein manifold. An explicit 3-dimensional example is constructed and claimed to admit a proper η-Ricci soliton with λ=-1, μ=3.

Significance. If the classification theorems were valid as stated, they would contribute to the growing literature on η-Ricci solitons in low-dimensional almost-contact geometry. The paper contains a transparent derivation of equation (2.16) and a mostly correct computational core after that point; in particular, the trace argument in Section 6 is elementary and appears sound under the ξ-potential assumption. The explicit example, once its constants are corrected, does provide a nontrivial Kenmotsu 3-manifold admitting a proper η-Ricci soliton. However, the current version overstates the generality of the main theorems and contains a concrete numerical error in the example, so the significance can only be assessed after substantial revision.

major comments (3)

[Theorems 3.1 and 4.1 (also Theorem 5.1)] Theorems 3.1 and 4.1 are stated for an η-Ricci soliton of the type (g,V,1,1) with an arbitrary potential vector field V, but the proofs use equation (2.16), which is derived in Proposition 2.1 under the explicit assumption that the potential is ξ (see (2.13)-(2.16)). No argument is given that the Codazzi-type condition or the cyclic-parallel condition forces V=ξ. Consequently, the classifications as stated are unsupported for general V. The same issue affects Theorem 5.1, whose proof relies on (5.2) and (5.3), both derived from the ξ-potential formula (2.16). The authors should either prove that the additional curvature hypotheses imply V=ξ, or restrict Theorems 3.1, 4.1, and 5.1 to solitons with potential ξ and adjust the abstract and statements accordingly.
[Section 7, final claim (p. 14)] The numerical values λ=-1 and μ=3 are inconsistent with the defining equation (1.3) and with the authors' own formula (2.16). For the example, the computed Ricci tensor is S=-2g and the Lie derivative is L_ξg=2(g-η⊗η). Substituting into (1.3) gives (2λ-2)g + (2μ-2)η⊗η = 0, forcing λ=μ=1. Thus the example does admit a proper η-Ricci soliton, but with λ=μ=1, not with λ=-1 and μ=3. The statement on page 14 must be corrected.
[Section 6, paragraph before Theorem 6.1] The text states that every 3-dimensional Riemannian manifold satisfies R.R=Q(S,R) identically, citing reference [12]. If so, the hypothesis 'the curvature condition R.R=Q(S,R) holds' in Theorem 6.1 is vacuous in dimension three. The theorem should be reformulated to state directly that any η-Ricci soliton (g,ξ,λ,μ) on a Kenmotsu 3-manifold satisfies λ=μ=1 and is Einstein, and the role of the curvature condition should be clarified. This is not a mathematical error in the proof, but it affects the presentation and the apparent novelty of the result.

minor comments (4)

[Throughout] There are repeated typographical errors: 'Coddazi' should be 'Codazzi' (Sections 3 and abstract), 'admittting' should be 'admitting' (Theorems 3.1 and 4.1), and 'cu rvature' is split in the abstract. The manuscript should be proofread carefully.
[Proposition 2.1] Proposition 2.1 assumes the soliton is proper (μ ≠ 0), but the proof and the relation λ+μ=2 do not actually require properness. The word 'proper' appears to be unnecessary in the proposition statement, and its presence is confusing because the proposition is later used for all η-Ricci solitons.
[Equation (3.3)] The covariant derivative formula (3.3) is written without explicitly stating that it is derived from (2.16) under the ξ-potential assumption. Adding a sentence to that effect would prevent the reader from applying (3.3) outside the intended setting.
[References] Reference [9] by U. C. De and G. Pathak is cited for the 3-dimensional Kenmotsu formulas (2.11) and (2.12). Since one of the present authors is a coauthor of [9], the reliance on (2.12) should be explicitly acknowledged, even if it is a standard formula; this is an ordinary citation practice issue.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proofs are deductive from the defining soliton equation and standard Kenmotsu identities; the paper's weaknesses are mathematical gaps, not circular reductions.

full rationale

I walked the derivation chain. Proposition 2.1 starts from the η-Ricci soliton equation (1.3) with V=ξ and the Kenmotsu Lie-derivative identity (2.15) to obtain (2.16); comparing with the standard 3-dimensional Kenmotsu Ricci formula (2.12) yields λ+μ=2. None of these steps assumes the conclusions λ=μ=1 or S=-2g; they are algebraic consequences of the hypotheses. In Sections 3–6, the additional conditions (Codazzi-type Ricci tensor, cyclic parallel Ricci tensor, φ-Ricci symmetry, and R.R=Q(S,R)) each force μ=1 or λ=1 and then S=-2g, which gives constant curvature -1 via the 3-dimensional curvature formula (3.6); this is ordinary deduction, not a circular reduction. Formula (2.12) is cited from [9], coauthored by U.C. De, but it is a parameter-free standard identity for 3-dimensional Kenmotsu manifolds whose stated assumptions do not include the target results; per the review rules, such an independent cited identity does not create circularity. The real weaknesses of the paper—Theorems 3.1 and 4.1 being stated for arbitrary V while proved only for V=ξ, the missing soliton hypothesis in the statement of Theorem 5.1, and the inconsistent constants in the Section 7 example—are correctness or rigor gaps, not cases where a prediction reduces to its input by construction. No fitted parameter is renamed as a prediction, and no self-citation chain is used to force a conclusion.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted to data and no new entities are introduced. The central claims rest on standard Kenmotsu formulas and on the quoted 3D identity R.R=Q(S,R).

assumptions (3)

domain assumption Formula (2.12): S(X,Y)=1/2[(r+2)g(X,Y)-(r+6)eta(X)eta(Y)] for 3D Kenmotsu manifolds, taken from [9].
This formula is the basis for comparing the Ricci tensor with the soliton equation in Proposition 2.1; it is cited, not proved, from a paper coauthored by one of the present authors.
domain assumption The identity R.R=Q(S,R) holds identically for every 3D Riemannian manifold, as asserted with citation [12].
Section 6 uses this to substitute into equation (6.6); if this identity were false, the derivation of S=-2g would fail.
domain assumption Standard Kenmotsu structure equations (2.4)-(2.10) from [21] and [1],[2] are assumed.
These equations are used throughout to compute Lie derivatives, curvature terms, and the form of the Ricci tensor.

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Cite this review

Pith. "Pith review of $\eta$-Ricci Solitons on Kenmotsu 3-Manifolds." pith.science (2026). https://pith.science/paper/VYIW654O

@misc{pith2026241114988,
  author       = {Pith},
  title        = {Pith review of: $\eta$-Ricci Solitons on Kenmotsu 3-Manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYIW654O}},
  note         = {Machine review of arXiv:2411.14988}
}

abstract

In the present paper we study $\eta$-Ricci solitons on Kenmotsu 3-manifolds. Moreover, we consider $\eta$-Ricci solitons on Kenmotsu 3-manifolds with Codazzi type of Ricci tensor and cyclic parallel Ricci tensor. Beside these, we study $\phi$-Ricci symmetric $\eta$-Ricci soliton on Kenmotsu 3-manifolds. Also Kenmotsu 3-manifolds satisfying the curvature condition $R.R=Q(S,R)$ is considered. Finally, an example is constructed to prove the existence of a proper $\eta$-Ricci soliton on a Kenmotsu 3-manifold.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [12]

Deszcz, On conformally ﬂat Riemannian manifolds satisfying certai n curvature conditions, Tensor (N.S.) 49 (1990), 134-145

R. Deszcz, On conformally ﬂat Riemannian manifolds satisfying certai n curvature conditions, Tensor (N.S.) 49 (1990), 134-145

work page 1990

[2] [1]

D. E. Blair, Lecture notes in Mathematics , Springer-Verlag Berlin, 509,1976. [2] , Riemannian Geometry of contact and symplectic manifolds , Birkh¨ auser, Boston, 2002

work page 1976

[3] [3]

A. M. Blaga, η-Ricci solitons on Lorentzian para-Sasakian manifolds , Filomat 30 (2016), no. 2, 489-496

work page 2016

[4] [4]

, η-Ricci solitons on para-Kenmotsu manifolds , Balkan J. Geom. Appl. 20 (2015), 1-13

work page 2015

[5] [5]

Calin and M

C. Calin and M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f- Kenmotsu manifolds , Bull. Malays. Math. Soc. 33(3) (2010), 361-368

work page 2010

[6] [6]

Chave and G

T. Chave and G. Valent, Quasi-Einstein metrics and their renormalizability prope r- ties, Helv. Phys. Acta. 69 (1996), 344-347

work page 1996

[7] [7]

, On a class of compact and non-compact quasi-Einstein metric s and their renormalizability properties, Nuclear Phys. B. 478 (1996), 758-778

work page 1996

[8] [8]

J. T. Cho and M. Kimura, Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J. 61, no. 2 (2009), 205-212

work page 2009