{"id":"e756c0bf-7064-405b-b94e-13f369fdc561","arxiv_id":"2411.14988","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Eta-Ricci solitons on Kenmotsu 3-manifolds are shown to force local hyperbolic geometry under several curvature conditions, but the paper's existence example misstates the soliton parameters.","lead":"This paper studies eta-Ricci solitons, a generalization of Ricci solitons, on three-dimensional Kenmotsu manifolds and derives conditions under which such manifolds become locally hyperbolic. The results are technical classification statements in contact geometry; they are unlikely to change practice beyond a narrow subfield, and the paper contains internal errors in its main example.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorems 3.1 and 4.1 are proved only for ξ-potential but stated for arbitrary V; the example also has inconsistent soliton constants.","rationale":"The reader's weakest assumption correctly identifies the gap between the proofs, which use the Reeb field ξ as the potential, and the statements of Theorems 3.1 and 4.1, which claim an arbitrary vector field V. This is the most load-bearing concern because it affects the two central classification results; without an argument forcing V=ξ, those theorems are unsupported. The example error is also real, but the underlying manifold does satisfy the η-Ricci soliton equation with λ=μ=1, so the existence claim is salvageable with corrected constants. Because the theorem statements as written overreach and the example's numerical claim is false, the reject verdict stands unchanged.","tokens_in":8231,"tokens_out":10109,"duration_ms":90428,"concrete_test":"Re-derive Proposition 2.1 starting from (1.3) with a general vector field V instead of ξ. Concretely, compute L_Vg for V not equal to ξ and verify whether comparison with (2.12) yields the form S=-(λ+1)g-(μ-1)ηη. If (2.16) depends on the substitution V=ξ, then Theorems 3.1 and 4.1 are not established for arbitrary V.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2.1 assumes an η-Ricci soliton of the form (g,ξ,λ,μ) and uses L_ξg=2(g-ηη) to derive the key identity S=-(λ+1)g-(μ-1)ηη (eq. 2.16). Sections 3 and 4 differentiate (2.16) to conclude μ=1 and λ=1, so Theorems 3.1 and 4.1 are conditional on the potential being ξ. Yet both theorems are stated for an arbitrary field V. Unless the authors prove that any η-Ricci soliton on a Kenmotsu 3-manifold with Codazzi-type or cyclic-parallel Ricci tensor must have V=ξ—which is not shown—the classification does not follow for general V. Separately, the final example computes S=-2g and L_ξg=2(g-ηη); substituting these into (1.3) forces 2λ-2=0 and 2μ-2=0, i.e. λ=μ=1, contradicting the claimed λ=-1, μ=3 and the authors' own formula (2.16). The manifold therefore does admit a proper η-Ricci soliton, but with λ=μ=1, so the numerical claim in Section 7 is false as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8411,"tokens_out":5291,"duration_ms":47983,"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":[{"comment":"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":"Theorems 3.1 and 4.1 (also Theorem 5.1)"},{"comment":"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":"Section 7, final claim (p. 14)"},{"comment":"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.","section":"Section 6, paragraph before Theorem 6.1"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Proposition 2.1"},{"comment":"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.","section":"Equation (3.3)"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main theorems as stated cannot be accepted because they claim a result for arbitrary potential vector fields while the proofs only support the ξ-potential case. This is a gap that can be repaired by restating the theorems with the V=ξ hypothesis, but that would substantially reduce the claimed scope. The example error is a straightforward miscalculation and should be corrected. If the authors are unwilling to restrict the theorems, the paper would need to be rejected; however, given that the computations are mostly sound under the ξ-potential assumption, I think a major revision is the appropriate route."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nThe 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.\n\nSecond, 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.\n\nThird, 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.\n\nI 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.\n\nWho'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.","headline":"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.","tokens_in":8988,"tokens_out":9615,"would_cite":false,"duration_ms":77050,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C15","53C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["η-Ricci soliton","Kenmotsu 3-manifolds","Codazzi type of Ricci tensor","cyclic parallel Ricci tensor","φ-Ricci symmetric","R.R=Q(S,R)","hyperbolic space"],"falsifier":"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.","tokens_in":7951,"feed_emoji":"📐","tokens_out":21070,"duration_ms":193348,"temperature":0.7,"pith_summary":"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.","feed_headline":"Under four curvature conditions, Kenmotsu 3-manifolds turn hyperbolic","feed_subtitle":"A soliton equation plus any one of four Ricci conditions pins λ=μ=1 and S=-2g, leaving H(-1).","key_machinery":"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 $H(-1)$.","core_discovery":"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 $\\lambda=-1$, $\\mu=3$.","pith_inferences":["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."],"forward_implications":["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 $\\mu=3$."],"supporting_citations":[{"why":"Introduces the $\\eta$-Ricci soliton equation (1.3) that the paper studies.","marker":"[8]"},{"why":"Defines Kenmotsu manifolds and supplies the structural identities (2.4)-(2.10).","marker":"[21]"},{"why":"Gives the 3-dimensional Kenmotsu curvature and Ricci formulas (2.11) and (2.12) used in every comparison.","marker":"[9]"},{"why":"Provides the definitions of Codazzi-type and cyclic-parallel Ricci tensor that drive Sections 3 and 4.","marker":"[17]"},{"why":"States the condition $R.R=Q(S,R)$ and the fact that every 3-dimensional Riemannian manifold satisfies it, which underlies Section 6.","marker":"[12]"}],"fun_headline_variants":["Four Ricci conditions force Kenmotsu 3-manifolds to H(-1)","η-Ricci solitons on Kenmotsu 3-manifolds: four paths to H(-1)","Rigidity: η-Ricci solitons on Kenmotsu 3-manifolds imply H(-1)","Kenmotsu 3-manifolds: four curvature conditions force H(-1)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Four Ricci conditions force Kenmotsu 3-manifolds to H(-1)","η-Ricci solitons on Kenmotsu 3-manifolds: four paths to H(-1)","Rigidity: η-Ricci solitons on Kenmotsu 3-manifolds imply H(-1)","Kenmotsu 3-manifolds: four curvature conditions force H(-1)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000802,"raw_usage":{"total_tokens":3522,"prompt_tokens":937,"completion_tokens":2585,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":553,"completion_tokens_details":{"reasoning_tokens":2479}},"tokens_in":553,"tokens_out":2585,"duration_ms":17311,"temperature":1.0,"reasoning_tokens":2479,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:42:08.531924+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the $\\eta$-Ricci soliton equation (1.3) that the paper studies."},{"cited_title":"Kenmotsu, A class of almost contact Riemannian manifolds , Tohoku Math","cited_arxiv_id":null,"evidence_quote":"Defines Kenmotsu manifolds and supplies the structural identities (2.4)-(2.10)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the 3-dimensional Kenmotsu curvature and Ricci formulas (2.11) and (2.12) used in every comparison."},{"cited_title":"Gray, Einstein-like manifolds which are not Einstein , Geom","cited_arxiv_id":null,"evidence_quote":"Provides the definitions of Codazzi-type and cyclic-parallel Ricci tensor that drive Sections 3 and 4."},{"cited_title":"Deszcz, On conformally flat Riemannian manifolds satisfying certai n curvature conditions, Tensor (N.S.) 49 (1990), 134-145","cited_arxiv_id":null,"evidence_quote":"States the condition $R.R=Q(S,R)$ and the fact that every 3-dimensional Riemannian manifold satisfies it, which underlies Section 6."}],"review_version":1}