Construction of a New Hypersurface Family Using the Spherical Product in Minkowski Geometry

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

See the attached report

Comments for author File: Comments.pdf

Author Response

REVIEWER 1

We are grateful to the reviewer for their insightful suggestions and constructive contributions.

Comment 1:

Include at least one worked numerical example demonstrating curvature computation.

Response 1:

Thank you for this helpful suggestion. We have added a worked numerical example (Section 3, pp. 8, line 163 (in the revised version)) that demonstrates how to evaluate our curvature formulas. Also, we give a visualization of the surface.

Comment 2:

Provide full figure captions indicating hypersurface type, parameter values (ci,εi) and software used.

Response 2:

In the revised manuscript, we have updated all figure captions to explicitly state the type of hypersurface, the corresponding parameter values c_i and ε_i and the software used to generate each plot.

Comment 3:

In figures, add coordinate axes or color legends for better geometric visualization.

Response 3:

In the revised version of the manuscript, we have added coordinate axes (with labels).

Comment 4:

In Eq. (3.10), the term W|W|\sqrt{|W|} could be simplified or written more compactly.

Response 4:

We simplified the term.

Comment 5:

Maintain consistent terminology (“hypersurface,” “surface,” “spherical product hypersurface”).

Response 5:

In the revised manuscript, we have carefully checked and unified the terminology. The main object is now consistently referred to as a “spherical product hypersurface” (or simply “hypersurface”), and the word “surface” is only used for genuinely two-dimensional objects where appropriate.

 

Comment 6:

Discuss briefly how the minimal and flat conditions could correspond to vacuum or constant mean curvature hypersurfaces in general relativity, or how the constructions might relate to known spacetime geometries (e.g., Schwarzschild or de Sitter embeddings).

Response 6:

In the revised manuscript, we have added a short remark (line 271-277) explaining how the minimal and flat conditions on our spherical product hypersurfaces relate to vacuum and constant mean curvature hypersurfaces in general relativity, and how suitable choices of the generating curves are connected to standard spacetime geometries such as Minkowski, Schwarzschild, and de Sitter spacetimes.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

Please see the attached file

Comments for author File: Comments.pdf

Author Response

REVIEWER 2

We are grateful to the reviewer for their insightful suggestions and constructive contributions.

Comment 1:

I’m quite concerned about the definition of the vector product in Minkowski space. The authors define x × y × z using a determinant with −e4, but they dont justify this choice or provide any references. This isnt standard notation as far as I’m aware, and since the normal vector (and therefore all curvature calculations) depends on this definition, it really needs proper justification. Without this, it is hard to have confidence in the subsequent results.

Response 1:

The Lorentzian inner product that underlies our definition of the triple vector product

x×y×z (given via the determinant with −e4 ​) is the standard one used in Minkowski space and can be found, for example, in references [7] and [13].

Comment 2:

Also, the distinction between spacelike and timelike hypersurfaces isnt handled very clearly. Corollary 1 gives a condition involving ∥γ′2∥ ≥ ∥γ′1 ∥, but this feels somewhat arbitrary and isnt properly motivated. The authors should provide a more comprehensive treatment of how the causal character of the generating curves affects the resulting hypersurface.

Response 2:

We know from Minkowski geometry that if the unit normal vector of a hypersurface is timelike, then the hypersurface is spacelike, whereas if the unit normal is spacelike, the hypersurface is timelike. In our construction, the condition that the normal vector be spacelike (equivalently, that the hypersurface be timelike) is satisfied precisely when ∥γ’2∥ ≥ ∥γ’1 ∥. We have now clarified and explained this point in more detail in Corollary 1 of the revised manuscript.

 

Comment 3:

The curvature formulas in Equations (3.10) and (3.11) are extremely complicated – honestly, they are almost unreadable in their current form. I understand that these calculations can get messy, but the authors should at least try to simplify them or provide some geometric interpretation. As they stand, it’s difficult to see what these formulas actually tell us about the geometry of these hypersurfaces.

Response 3:

In the revised manuscript, we have simplified Equation (3.10). For Equation (3.11), however, further algebraic simplification appears to be very difficult without losing essential information. We have therefore rewritten it using the compact notation G=G(v,w), which, to the best of our efforts, represents the simplest and most readable form of this expression.

Comment 4:

The proofs are also quite minimal. For Theorem 3.1, the authors just say by using (2.6), (3.6), and (3.9), we obtain the result” this isnt really a proof, it is more of a calculation sketch. For important results like this, readers need to see the actual derivation or at least the key steps.

Response 4:

We would like to point out that the detailed derivation underlying Theorem 3.1 is already presented in the text before the theorem is stated. In fact, in the course of computing the Gaussian curvature, all required derivatives and the coefficients of the fundamental forms are obtained starting from Equation (3.6) and continuing up to Equation (3.9). Thus, the proof is essentially given first in the form of these step–by–step computations, and Theorem 3.1 is then formulated as a direct consequence summarizing these equalities.

Comment 5:

The authorship list seems inconsistent – the fourth author has three  affiliation markers (4,4,4), which doesnt make sense. This needs to be cleaned up.

Response 5:

This has been corrected in the revised manuscript.

Comment 6:

There are several notational problems. In Theorem 3.2, λ4(u) and λ5(v) should probably be λ4(w) and λ5(w) based on their definitions. In the hyperquadrics section, using ci as both a constant and an exponent is really confusing they should use different symbols.

Response 6:

This has been corrected in the revised manuscript.

 

 

Comment 7:

The connection to symmetry, which should be a major theme for this journal, is mentioned briefly but not developed in any depth. The authors should discuss explicitly what symmetries these constructions preserve or create.

Response 7:

In the revised manuscript the following comment added in the conclusion section at third paragraph lines between 265-270: “In addition, spherical product contributes rotational or Lorentzian symmetries, while hyperquadrics obtained through spherical products inherit all symmetries of the associated quadratic form. Consequently, the curvature conditions derived in this paper—such as flatness, minimality, and causal character—are compatible with and, in some cases, dictated by these underlying symmetry groups.”

Comment 8:

Here is what I think the authors should focus on for a revision: First and foremost, they need to properly justify the vector product definition in Minkowski space. This is fundamental to everything that follows.

The curvature formulas need serious work either simplification or geometric interpretation. It would be much more valuable if the authors could  express these curvatures in terms of geometric properties of the generating curves, rather than just presenting these massive expressions.

The proofs need to be expanded, particularly for the main theorems. Readers need to see how these results are actually derived.

The authors should carefully proofread the entire manuscript and fix the numerous typographical and notational errors. The affiliation list needs  correction, and all figures should be included.

Finally, the conclusion overstates the applications. The authors claim relevance to “geometric modeling and mathematical physics” but dont provide any concrete examples or discussion. They should either tone down these claims or provide specific examples of how their results might be applied.

Response 8:

This overall comment (Comment 8) summarizes the issues raised in Comments 1–7. As explained in our detailed responses to those comments, each of the points listed here has been addressed separately, and the corresponding revisions have been carried out in the manuscript.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

The paper is mathematically correct and within the journal’s scope but needs major language and stylistic revision and a more explicit discussion of its geometric contribution beyond prior literature.

Comments for author File: Comments.pdf

Author Response

REVIEWER 3

We are grateful to the reviewer for their insightful suggestions and constructive contributions.

Comment 1:

However, the scientific novelty is modest. The main results-flatness corresponding to one generating curve being straight, and minimality under specific conditions-are direct analogues of known Euclidean results (e.g., Bulca & Arslan, An. St. Univ. Ovidius Constanța, 2012; Büyükkütük & Öztürk, Turk. J. Math., 2024). The transition to Minkowski space changes only the signature of the metric, without introducing new geometric phenomena or invariant formulations. Consequently, the paper extends known constructions rather than developing genuinely new theory.

Response 1:

We thank the reviewer for this remark and for pointing out the relation with the Euclidean results. We agree that our construction is inspired by Euclidean spherical product surfaces and hypersurfaces, but we respectfully disagree that the Minkowskian version “only changes the signature.” In ​ the causal character of tangent and normal directions (spacelike, timelike, lightlike) forces a case-by-case analysis and produces families of spherical product hypersurfaces that have no Euclidean analogue. In particular, our flatness and minimality conditions are not obtained by a formal sign change from the Euclidean formulas; they couple the generating curves through causal constraints and yield timelike, spacelike and lightlike models, together with explicit examples and visualisations that exist only in the Lorentzian setting.

Moreover, there is a well-established literature where the same geometric class is studied separately in Euclidean and Minkowski spaces and the Lorentzian case is treated as a genuine contribution. For instance, Bekkar and Senoussi classify factorable surfaces satisfying Δri=λiri ​ simultaneously in ​ and in Lorentzian ​ ​, obtaining distinct families adapted to each metric. Similarly, Büyükkütük and Öztürk first characterize factorable surfaces in Euclidean 4-space , and then, in a separate paper, classify spacelike factorable surfaces in Minkowski 4-space ​ with different curvature conditions and model examples. Our work on spherical product hypersurfaces in  follows exactly this accepted pattern: the same construction leads to genuinely different Lorentzian geometry, which we believe justifies the novelty of the present manuscript.

Bekkar, M.; Senoussi, B. Factorable surfaces in the three-dimensional Euclidean and Lorentzian spaces satisfying Δri=λiri\Delta r_i = \lambda_i r_iΔri​=λi​ri​. Journal of Geometry 2012, 103(1), 17–29. https://doi.org/10.1007/s00022-012-0117-3 SpringerLink+1

Büyükkütük, S.; Öztürk, G. A Characterization of Factorable Surfaces in Euclidean 4-Space E4E^4E4. Kocaeli Journal of Science and Engineering 2018, 1(1), 15–20. https://doi.org/10.34088/kojose.403665 DergiPark+1

Büyükkütük, S.; Öztürk, G. Spacelike Factorable Surfaces in Four-Dimensional Minkowski Space. Bulletin of Mathematical Analysis and Applications 2017, 9(4), 12–20.

 

Comment 2:

The formulas for the first and second fundamental forms and curvature functions are correct but presented in overly long computational form; a tensorial or invariant approach would improve clarity.

Response 2:

In the manuscript, the coefficients of the first and second fundamental forms are expressed in terms of the mapping G(v,w) and the inner products of its partial derivatives. Moreover, the Gaussian and mean curvature functions are written using the auxiliary functions λi therefore, these formulas are already presented in as compact a form as possible without sacrificing clarity.

Comment 3:

The English language requires significant editing. Common mistakes include missing articles, incorrect verb tenses, and misspellings (“standart embedding”, “hypersurfaces’ projection”, “superhyperbola”)

Response 3:

We have carefully revised the English throughout the manuscript, correcting missing articles, verb tenses, and spelling errors. In particular, the expressions “standart embedding”, “hypersurfaces’ projection”, and “superhyperbola” have been corrected, and similar issues have been systematically fixed in the revised version.

Comment 4:

The figures are illustrative but lack scale or axis labeling; captions could explain geometric meaning rather than software details.

Response 4:

In the revised version, we have added appropriate axis labels and scales to all figures, and we have expanded the figure captions to focus more on the geometric meaning of the depicted hypersurfaces.

Comment 5:

Several references are self-citations or overlap with closely related earlier works; inclusion of broader sources (e.g., O’Neill, Lopez, Magid) is recommended.

Response 5:

In the manuscript, the standard works by López and O’Neill have already been included as References 13 and 15, respectively; see, for example, López, R. “Differential geometry of curves and surfaces in Lorentz–Minkowski space”, Int. Electron. J. Geom. 2014, 7(1), 44–107, and O’Neill, B. “Semi-Riemannian Geometry with Applications to Relativity”, Academic Press, New York, 1983.

Comment 6:

Clear structure and readable organization.

Complete derivations of Gaussian and mean curvatures.

Concrete geometric examples connecting spherical products to hyperquadrics.

Response 6:

We thank the referee for these positive comments.

Comment 7:

Limited originality; mostly computational extension of known Euclidean constructions.

Lack of geometric or physical interpretation.

Language and stylistic issues throughout.

Response 7:

In the revised manuscript, we have added a short remark explaining how the minimal and flat conditions on our spherical product hypersurfaces relate to vacuum and constant mean curvature hypersurfaces in general relativity, and how suitable choices of the generating curves are connected to standard spacetime geometries such as Minkowski, Schwarzschild, and de Sitter spacetimes. In addition, we have carefully reread the entire manuscript and corrected the grammatical and stylistic issues throughout.

Moreover, all of the indicated typographical corrections have been implemented.

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have met all the requirements, and the paper now seems good. I recommend the acceptance in the current form.

Author Response

We would like to thank the reviewer for their valuable time and positive evaluation of our manuscript. We are pleased to learn that the paper meets the requirements and is recommended for acceptance in its current form.

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript is mathematically sound, logically consistent, and written in a style that is appropriate for a research article in geometry. The authors extend the classical notion of spherical products from Euclidean spaces to the Lorentzian setting and successfully derive curvature expressions for these hypersurfaces. The characterization of flat and minimal cases is correct and well-explained. The examples involving hyperquadrics further strengthen the contribution by showing concrete realizations of the general theory.

“standart embedding” → “standard embedding”
•  “is hold” → “holds”
•  “spherical product hypersurface is minimal if…” → consider rephrasing for style

Bibliography
•    Two entries are labelled as “1.”; please correct numbering

Comments on the Quality of English Language

minor grammatical issues in a few sentences

Author Response

Reviewer Comment: “standart embedding” → “standard embedding”

Response: We have corrected all instances of "standart embedding" to "standard embedding" throughout the manuscript.

Reviewer Comment: “is hold” → “holds”

Response: The phrase "is hold" has been corrected to "holds" as suggested.

Reviewer Comment: “spherical product hypersurface is minimal if…” → consider rephrasing for style

Response: We have rephrased this sentence to improve its flow and clarity. It now reads: "If $\gamma$ passes through the origin and is congruent to a straight line, then the spherical product hypersurface is minimal."

Reviewer Comment: Bibliography: Two entries are labelled as “1.”; please correct the numbering

Response: The numbering error in the bibliography has been corrected.