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arxiv: 1906.11933 · v1 · pith:G7GTRFBMnew · submitted 2019-06-27 · 🧮 math.DG

On Warped Product Gradient Ricci-Harmonic Soliton

Elismar Batista , Levi Adriano , Willian Tokura This is my paper

Pith reviewed 2026-05-25 14:11 UTC · model grok-4.3

classification 🧮 math.DG
keywords warped productgradient Ricci-harmonic solitonsemi-Riemannian manifoldgeodesically completeharmonic mapwarping functiontranslation groupconformal semi-Euclidean space
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The pith

Warped products of semi-Euclidean spaces under codimension-one translation invariance yield infinitely many geodesically complete gradient Ricci-harmonic solitons.

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

The paper studies gradient Ricci-harmonic solitons that take the form of warped product manifolds. It first proves triviality results: the potential function, the warping function, and the harmonic map must be constant when any of them attains a maximum or minimum. To produce nontrivial examples, the authors restrict to the case where both the base and the fiber are conformal to a semi-Euclidean space that is invariant under a codimension-one translation group. This single construction supplies infinitely many geodesically complete examples in the semi-Riemannian setting.

Core claim

By taking the base and fiber to be conformal to semi-Euclidean spaces that remain invariant under the action of a codimension-one translation group, one obtains infinitely many geodesically complete warped-product gradient Ricci-harmonic solitons in the semi-Riemannian category; the same construction is excluded in the Riemannian category by an earlier theorem.

What carries the argument

Warped-product metric whose base and fiber are each conformal to a semi-Euclidean space preserved by a codimension-one translation group; the group action supplies the explicit warping and potential functions that solve the soliton equation.

If this is right

  • The potential function, warping function and harmonic map are constant whenever any of them reaches an extremum.
  • The same translation-invariant conformal data produce infinitely many distinct complete solutions.
  • These solutions exist only in the semi-Riemannian signature and are ruled out in the Riemannian case by a prior global theorem.
  • The construction is fully explicit once the translation group and the conformal factors are fixed.

Where Pith is reading between the lines

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

  • The same symmetry reduction may generate complete examples for other soliton equations that involve harmonic maps.
  • Indefinite-signature completeness becomes accessible through algebraic group actions that are unavailable in the definite case.
  • The method supplies an infinite family of model spaces on which one can test further analytic or curvature properties of Ricci-harmonic solitons.

Load-bearing premise

The base and fiber must be chosen conformal to a semi-Euclidean space that stays invariant under a codimension-one translation group.

What would settle it

An explicit verification that at least one of the constructed metrics fails to satisfy the gradient Ricci-harmonic soliton equation, or that the resulting manifold is not geodesically complete.

read the original abstract

In this paper we study gradient Ricci-Harmonic soliton with structure of warped product manifold. We obtain some triviality results for the potential function, warping function and the harmonic map which reaches maximum or minimum. In order to obtain nontrivial examples of warped product gradient Ricci-harmonic soliton, we consider the base and fiber conformal to a semi-Euclidean space which is invariant under the action of a translation group of co-dimension one. This approach provide infinitely many geodesically complete examples in the semi-Riemannian context, which is not contemplated in the Riemannian case by the Theorem 1.2 in [17].

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.

Referee Report

2 major / 2 minor

Summary. The paper examines gradient Ricci-harmonic solitons on warped product manifolds. It establishes triviality results for the potential function, warping function, and harmonic map when any of these attains a maximum or minimum. Nontrivial examples are constructed by taking the base and fiber to be conformal to semi-Euclidean spaces that are invariant under a codimension-one translation group action; the authors claim this produces infinitely many geodesically complete examples in the semi-Riemannian setting, in contrast to the Riemannian case covered by Theorem 1.2 of [17].

Significance. If the constructions are verified to satisfy the soliton equations, the work would supply new families of geodesically complete examples in the semi-Riemannian category that are unavailable under the corresponding Riemannian rigidity result. The triviality theorems would also contribute to rigidity statements for warped-product gradient Ricci-harmonic solitons.

major comments (2)
  1. [construction section] Construction of examples (the section following the triviality results): the manuscript must explicitly substitute the conformal ansatz (base and fiber metrics conformal to semi-Euclidean space, invariant under the codimension-one translation group) into the warped-product form of the gradient Ricci-harmonic soliton equation and verify that all curvature and Hessian terms cancel, including those arising from the indefinite signature. Without this direct computation, the claim that the resulting metrics are solitons and that the potential function is gradient remains unsubstantiated.
  2. [construction section] Geodesic completeness argument (same construction section): the paper asserts infinitely many geodesically complete examples, but must show that the chosen conformal factors and warping function produce a complete metric on the warped product; the translation invariance alone does not automatically guarantee completeness when the signature is indefinite.
minor comments (2)
  1. [abstract] Abstract, last sentence: 'This approach provide' should read 'This approach provides'.
  2. [preliminaries] Notation for the harmonic map and the potential function should be introduced consistently before the triviality statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive suggestions. We address each major comment below and will incorporate the requested explicit verifications into a revised manuscript.

read point-by-point responses

  1. Referee: [construction section] Construction of examples (the section following the triviality results): the manuscript must explicitly substitute the conformal ansatz (base and fiber metrics conformal to semi-Euclidean space, invariant under the codimension-one translation group) into the warped-product form of the gradient Ricci-harmonic soliton equation and verify that all curvature and Hessian terms cancel, including those arising from the indefinite signature. Without this direct computation, the claim that the resulting metrics are solitons and that the potential function is gradient remains unsubstantiated.

    Authors: We agree that the manuscript would benefit from an explicit substitution of the conformal ansatz into the warped-product soliton equations. In the revised version we will add this direct computation, verifying term-by-term cancellation of all curvature and Hessian contributions while accounting for the indefinite signature. revision: yes

  2. Referee: [construction section] Geodesic completeness argument (same construction section): the paper asserts infinitely many geodesically complete examples, but must show that the chosen conformal factors and warping function produce a complete metric on the warped product; the translation invariance alone does not automatically guarantee completeness when the signature is indefinite.

    Authors: We accept that a self-contained proof of geodesic completeness is needed. The revision will include a detailed argument establishing that the chosen conformal factors and warping function yield a geodesically complete warped-product metric, with explicit attention to issues that may arise from the indefinite signature. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction uses explicit ansatz independent of target equations

full rationale

The paper first derives triviality results for the potential function, warping function, and harmonic map when they attain extrema. It then introduces an explicit ansatz (base and fiber conformal to semi-Euclidean space, invariant under a codimension-one translation group) to generate nontrivial examples. This ansatz is selected to satisfy the warped-product gradient Ricci-harmonic soliton equations and geodesic completeness in the semi-Riemannian setting; it is not obtained by fitting parameters to the soliton PDEs or by reducing to a self-citation. The reference to Theorem 1.2 in [17] is an external contrast with the Riemannian case and does not bear the load of the existence claim. No self-definitional, fitted-input, or uniqueness-imported steps appear.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard domain assumption of warped product structure and an ad hoc construction using conformal semi-Euclidean spaces with translation invariance; no free parameters or new entities are indicated.

axioms (2)
  • domain assumption The manifold admits a warped product structure.
    Central to all results stated in the abstract.
  • ad hoc to paper Base and fiber are conformal to a semi-Euclidean space invariant under a co-dimension one translation group.
    Invoked specifically to obtain the nontrivial complete examples.

pith-pipeline@v0.9.0 · 5624 in / 1245 out tokens · 53789 ms · 2026-05-25T14:11:00.942415+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 1 internal anchor

  1. [1]

    Bang-yen

    C. Bang-yen. Differential geometry of warped product manifolds and subma nifolds. World Scientific, 2017

    work page 2017

  2. [2]

    A. L. Besse. Einstein manifolds. Springer Science & Business Media, 2007

    work page 2007

  3. [3]

    R. L. Bishop and B. ONeill. Manifolds of negative curvature. Transactions of the Amer- ican Mathematical Society, 145 1–49, 1969

    work page 1969

  4. [4]

    M. L. de Sousa and R. Pina. Gradient Ricci solitons with structure of warped product. Results in Mathematics, 71(3-4) 825–840, 2017

    work page 2017

  5. [5]

    Dobarro and E

    F. Dobarro and E. Lami Dozo. Scalar curvature and warped products of Riemannan manifolds. Transactions of the American Mathematical Society, 303(1) 161–168, 1987

    work page 1987

  6. [6]

    Fern´ andez - L´ opes, M

    E. Fern´ andez - L´ opes, M. & Garc´ a-R ´ ıoA remark on compact Ricci solitons. In Math. Ann., page 340: 893. Springer, 2008. D. Gilbarg and N. S. Trudinger . Elliptic partial differential equations of second order. springer, 2015

    work page 2008

  7. [7]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. springer, 2015

    work page 2015

  8. [8]

    P. Grisha. The entropy formula for the Ricci flow and its geometric appli cations. arXiv:math/0211159, 2002

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  9. [9]

    T. Ivey. New examples of complete Ricci solitons. Proc. Amer. Math. Soc., 122(1) 241– 245, 1994

    work page 1994

  10. [10]

    H. X. Guo, R. Philipowski, and A. Thalmaier. On gradient solitons of the Ricci- harmonic flow. Acta Math. Sin. (Engl. Ser.), 31(11) 1798–1804, 2015

    work page 2015

  11. [11]

    Kim and Y

    D.-S. Kim and Y. H. Kim. Compact einstein warped product spaces with nonpositive scalar curvature. Proceedings of the American Mathematical Society, 131(8) 2573–2576, 2003

    work page 2003

  12. [12]

    S. D. Lee, B. H. Kim, and J. H. Choi. Warped product spaces with Ricci conditions. Turkish Journal of Mathematics, 41(6) 1365–1375, 2017

    work page 2017

  13. [13]

    B. Oneill. Semi-Riemannian geometry with applications to relativity . volume 103. Aca- demic press, 1983

    work page 1983

  14. [14]

    Petersen and W

    P. Petersen and W. Wylie. On gradient Ricci solitons with symmetry. Proc. Amer. Math. Soc., 137(6) 2085–2092, 2009

    work page 2085

  15. [15]

    Petersen and W

    P. Petersen and W. Wylie. Rigidity of gradient Ricci solitons. Pacific J. Math., 241(2) 329–345, 2009

    work page 2009

  16. [16]

    Tokura, L

    W. Tokura, L. Adriano, R. Pina, and M. Barboza. On warped product gradient Yamabe. J. Math. Anal. Appl., 473(1) 201–214, 2019

    work page 2019

  17. [17]

    M. Zhu. On the relation between Ricci-Harmonic solitons and Ricci s olitons. J. Math. Anal. Appl., 447(2) 882–889, 2017. ON W ARPED PRODUCT GRADIENT RICCI-HARMONIC SOLITON 13 1 Instituto Federal do Tocantins, Rodovia TO 040 - Km 349 Lotea mento Rio Palmeira, 77300- 000, Dian ´opolis, TO Brazil. E-mail address : elismardb@gmail.com 1 2 Universidade Federal...

    work page 2017