Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please, find attached the report

Comments for author File: Comments.pdf

Author Response

REVIWER 1

We thank the referee for the careful review and constructive comments.

Comment 1:

While the manuscript is clearly written and contains some interesting elements, it lacks the level of novelty required for a research paper. The authors are encouraged to better highlight the original aspects of their work and provide some more results. The calculations are correct, as far as this Reviewer could check.

Response 1:

In prior work, catenoidal, helicoidal, and more general rotational hypersurfaces were treated in separate SCI-indexed papers. Here, we study these themes jointly through the lens of parallel hypersurfaces and derive original results for both the hypersurfaces and their parallels. In particular, beyond results on the parallel of the helicoidal hypersurface, we also obtain significant new results for the helicoid itself.

Comment 2:

The Authors build on the previous result (Cf. Theorem 1 p. 3) published in Görgülü, A.  On the curvates of the parallel hypersurfaces.” Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 41 (1992).

The obtained results (Theorems 2, 3, 4) are direct consequence of the cited Theorem, and so they are not sufficiently new. The paper continues with a rich collection of examples which, while illustrative, do not compensate for the limited level of originality required for a research article.

Response 2:

Building on Theorem 1, which states the principal curvature relations for parallel hypersurfaces in E^n, we have specialized these results to E^4 and consolidated the results for the principal curvatures (former Corollary 1), the Gaussian curvature (former Theorem 2), and the mean curvature (Corollary 3) into a single statement as the new Corollary 1. Former Corollary 2 remains unchanged. Former Corollary 3 has been removed. Former Theorem 4 is now stated, without proof, as the new Corollary 3.

Comment 3:

The paper is clear and sufficiently well written. Bobliography should be modified as follows:

– p.1 line +16: either remove citations number [1], [2], [3], [4], [5], [7] and [8] or provide more detailed evidence of their importance.

Response 3:

The specified references have been removed from the manuscript, and new references more closely related to the topic have been added.

Comment 4:

The graphs in the figures are the ones of projections of a 4−dimensional objects into 3− dimenisonal space, by fixing some parameter. However, it is not clear what this means or what the choice of the parameter entails. It would be helpful if the authors could clarify this point. As a suggestion, they might also consider showing several plots for different parameter values, which could make the presentation more illuminating, but still the authors should explain this aspect in more detail.

Response 4:

In all figures, the 4D parametrizations are projected to E³ by explicit linear coordinate combinations, not by suppressing a coordinate. Concretely, we use maps that sum a pair of coordinates to form one plot coordinate—either placing ?₃+?₄ into the third plot coordinate (keeping the first two unchanged), or placing ?₁+?₂ into the first plot coordinate (with the remaining coordinates assigned accordingly). This procedure is described in the manuscript (lines 138–143), where we explain the coordinate-summing projection and the meaning of “fixing a parameter” (e.g., ?=?/6) as selecting a specific 3D slice before projection.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

mathematics-3946229-review report

Provide a point:

1. This paper investigates parallel hypersurfaces in four-dimensional Euclidean space (E4), deriving expressions for their Gaussian and mean curvatures in terms of the base hypersurface's curvature functions. It identifies conditions for these hypersurfaces to be flat or minimal and applies the theory to various hypersurfaces, including rotational, hyperspherical, catenoidal, and helicoidal hypersurfaces, with detailed curvature computations and visualizations.
2. On page 2, line 52, there are many theorems in this paper, it is recommended to add a section on the structural arrangement of the paper.
3. On page 2, line 63, suggest the authors to provide definitions for inner product $ \langle, \rangle $ and norm $ \| \| $.  
4. On page 3, line 78, suggest the authors to provide the definition for distance $ d( , ) $.
5. On page 3, line 87, suggest authors to separate “, ” from the equation by a space, as it may be mistaken for $ K’ $.
6. On page 4, line 94, correct “following result”to “following results”.
7. On page 6, line 131, does Example 1 have any specific physical background or practical significance?
8. On page 8, line 157, add “By (8), we obtain the following theorem:”.
9. On page 14, line 253, the formula is too small to be clear, it is recommended to enlarge it.
10. On page 15, line 260, suggest the authors to supplement the proof of Theorem 15.
11. Can the main conclusions of this paper be extended to n-dimensional Euclidean space?

Conclusion:

I suggest accepting this manuscript after minor revisions.

Author Response

REVIEWER 2

We thank the referee for the careful review and constructive comments.

Comment 1:

This paper investigates parallel hypersurfaces in four-dimensional Euclidean space (E4), deriving expressions for their Gaussian and mean curvatures in terms of the base hypersurface's curvature functions. It identifies conditions for these hypersurfaces to be flat or minimal and applies the theory to various hypersurfaces, including rotational, hyperspherical, catenoidal, and helicoidal hypersurfaces, with detailed curvature computations and visualizations.

Response 1:

We thank the refree.

Comment 2:

On page 2, line 52, there are many theorems in this paper, it is recommended to add a section on the structural arrangement of the paper.

Response 2:

In the revised manuscript, we added a paragraph at the end of the Introduction to guide the reader through the structure of the article. It briefly outlines Section 2 (preliminaries), Section 3 (general framework for parallel hypersurfaces in E^4 together with Theorem 1 and Corollary 1-3), and its applications in Subsections 3.1–3.2 (two rotational parameterizations, hypersphere, catenoid, and helicoid), followed by Section 4 (conclusions).

Comment 3:

On page 2, line 63, suggest the authors to provide definitions for inner product and norm .

Response 3:

We have inserted, between lines 62–66 in the revised version, a detailed explanation of the inner product at the beginning of the Introduction.

Comment 4:

On page 3, line 78, suggest the authors to provide the definition for distance d( , ).

Response 4:

In the revised version of the paper, Definition 1 in Section 3 has been removed. The definition of the parallel hypersurface given in Definition 2 (Definition 1 in the revised version) is sufficient for the paper and has been used in all computations.

Comment 5:

On page 3, line 87, suggest authors to separate “, ” from the equation by a space, as it may be mistaken for $ K’$.

Response 5:

On page 3, line 87, Theorem 2 appeared in the earlier version of the paper. During our revisions, we removed Theorem 2 and incorporated its content under Corollary 1. The issue mentioned by the reviewer has been corrected in the new Corollary 1.

Comment 6:

On page 4, line 94, correct “following result”to “following results”.

Response 6:

In the revised version of the paper, Corollary 1, Theorem 2, and Theorem 3 have been combined under a single Corollary 1. Corollary 3 has been removed, and Theorem 4 has been introduced as a new result. During this revision, the sentence mentioned by the reviewer was deleted, and at the beginning of the new Corollary 1, we added the sentence: “As a direct consequence of Theorem 1, the following results for the four-dimensional Euclidean space are readily obtained.”

Comment 7:

On page 6, line 131, does Example 1 have any specific physical background or practical significance?

Response 7:

No, Example 1 does not have a direct physical or practical meaning.
It serves a purely theoretical purpose—to show how the derived formulas can be applied to a specific analytic case and to illustrate the computational process with a simple, smooth example. The chosen functions only make the algebra manageable and allow for an easy visualization of a rotational hypersurface and its parallel in four-dimensional Euclidean space.

Comment 8:

On page 8, line 157, add “By (8), we obtain the following theorem:”.

Response 8:

The addition suggested by the reviewer has been made; the sentence “By (8), we obtain the following theorem:” has been inserted on page 8, line 157 (page 7, line 154 in the revised paper).

Comment 9:

On page 14, line 253, the formula is too small to be clear, it is recommended to enlarge it.

Response 9:

We enlarge the formula. (Page 12, line 241 in the revised version)

Comment 10:

On page 15, line 260, suggest the authors to supplement the proof of Theorem 15.

Response 10:

The proof of Theorem 15 (Theorem 12 in the revised version) is added.

Comment 11:

Can the main conclusions of this paper be extended to n-dimensional Euclidean space?

Response 11:

The results in E⁴ were obtained by augmenting the base-surface parameterizations with the surface’s unit normal vector field. Accordingly, extending these results to n-dimensional Euclidean space requires parameterizations of the base surfaces in Eⁿ. Once such parameterizations are available, the study can be generalized to n-dimensions.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Authors addressed the pointed out issues.

I suggest to accept the paper in the present form.