On the Geometry of the Riemannian Curvature Tensor of Nearly Trans-Sasakian Manifolds
Round 1
Reviewer 1 Report (Previous Reviewer 2)
The author of the paper has sufficiently addressed most of my concerns related to the original submission.
Author Response
The text of the article has been amended in accordance with all the comments of the reviewer.
Reviewer 2 Report (New Reviewer)
This paper presents some the results issued from the fundamental research of the geometry of the Riemannian curvature tensor of nearly trans-Sasakian manifolds. The components of the Riemannian curvature tensor on the space of the associated $G$-structure are counted, and the components of the Ricci tensor are calculated. Some identities are obtained, satisfied by the Riemannian curvature tensors and the Ricci tensor. A number of properties are proved as characterizations for nearly trans-Sasakian manifolds with a closed contact form. The structure of nearly trans-Sasakian manifolds with a closed contact form is obtained. Several classes are singled out in terms of second-order differential-geometric invariants and their local structure is obtained. The $k$-nullity distribution of a nearly trans-Sasakian manifold is studied.
The paper has the following structure:
1. In the Introduction, the author gives the general framework for the current research;
2. The second section emphasizes Preliminary information, necessary for the further development of the paper. Hence, the method and the object of the research are provided;
3. In the third section, the Definition of a nearly trans-Sasakian structure and its structural equations are given;
4. The fourth section is dedicated to the Riemannian curvature tensor of a nearly trans-Sasakian manifold, where the components of the Riemannian curvature tensor and the Ricci tensor of a nearly trans-Sasakian manifold on the space of the associated $G$-structure are calculated, as well as the scalar curvature;
5. In the fifth section, NTS-manifolds of constant curvature are described. It is proved that a nearly trans-Sasakian manifold of constant curvature is either a trans-Sasakian manifold of constant negative curvature or is locally conformal to a closely cosymplectic
manifold of constant curvature. In particular, a nearly trans-Sasakian manifold is zero constant curvature if and only if it is a cosymplectic manifold of constant curvature;
6. The sixth section presents several Curvature identities, such as the contact analogues of the Gray classes of nearly trans-Sasakian manifolds;
7. In the seventh section, the notion of $k$-nullity distribution is introduced;
8. As Conclusions, the author states that the class of NTS-manifolds with non-closed contact form coincides with the class of almost contact metric manifolds homothetic to Sasaki varieties; the class of NTS-manifolds with a closed contact form coincides with the class of almost contact metric manifolds with a closed contact form that are locally conformal to closely cosymplectic manifolds. The article presents some interesting identities, which are satisfied by the Riemannian curvature tensor and the Ricci tensor. It is proved that an NTS-manifold of constant curvature is either a trans-Sasakian manifold of constant negative curvature, or it has a closed contact form, and hence it is locally conformal to a closely cosymplectic manifold of constant curvature;
9. The last section is dedicated to several References, where results from the field of the paper are emphasized.
The results are clearly organized and the topic of the paper adheres to the journal's standards.
In my opinion, the paper deserves publication.
Author Response
The text of the article has been amended in accordance with all the comments of the reviewer.
Reviewer 3 Report (New Reviewer)
The paper under review deals with the study of nearly trans-Sasakian manifolds. This class of manifolds is not of interest in contact geometry. No examples are given.
In the first 3 sections the authors recalls known results.
In Section 4 the Riemannian curvature tensor of a nearly-trans Sasakian manifold is computed. The calculations are standard and straightforward. In the next section nearly-trans Sasakian manifolds of constant curvature are considered. In the last section elementary curvature identities are established.
The present paper contains neither interesting results nor new techniques in the geometry of contact manifolds.
Therefore I will reject it.
The English is good, the Mathematics is not.
Author Response
- The phrase "This class of manifolds is not of interest in contact geometry". This phrase is surprising. The work is devoted to the study of the geometry of one of the classes of almost contact metric manifolds - the class of nearly trans-Sasakian manifolds. The paper does not study the geometry of contact manifolds. The class of nearly trans-Sasakian manifolds intersects with the class of contact manifolds in terms of the class of Sasakian manifolds. It is well known that contact (metric) manifolds are generalizations of Sasakian manifolds.
- The phrase "No examples are given". Corollaries 4, 5, 6 to Theorem 1 give examples of nearly trans-Sasakian manifolds. In the text of the article, after Corollary 6, a corresponding sentence was added.
- The phrase "In the first 3 sections the authors recalled known results". The reviewer speaks in the plural about the author, although the author of the article is one. Yes, there are no new results in the first two sections, these two sections are auxiliary. In the third section, we present new results due to the author of the article. In particular: theorems 2, 3, 5, 6; equalities (83), (85), (87), (89) and consequences from them.
- The phrase "The calculations are standard and straightforward." The author tried to make the calculations understandable and standard. In this the author sees nothing reprehensible, nothing bad. Where, if not in the articles, it is necessary to provide evidence of the statements.
- The phrase "In the last section elementary curvature identities are established". Non-elementary identities can be obtained similarly. But why are they needed if you cannot get information. On the basis of some "elementary" identities, subclasses of nearly trans-Sasakian manifolds are distinguished and a local characterization of the distinguished subclasses is obtained.
- The phrase "The present paper contains neither interesting results nor new techniques in the geometry of contact manifolds". This is the personal opinion of the reviewer and the author cannot discuss it. I will say that the paper studies the geometry of one of the classes of almost contact metric manifolds, non-contact (metric) manifolds. The class of contact manifolds is one of the classes of almost contact metric manifolds.
Author Response File: 
Author Response.pdf
Reviewer 4 Report (New Reviewer)
The author fills a gap in the existing liturature regarding the Riemannian curvature of nearly trans-Sasakian manifolds. Numerous theorem are presented and proved. The paper is well-written and easy to follow. The Introduction does a good job of providing big-picture definitions, laying out the exisiting liturature, showing how the current work fits in, and outlining for the reader the scope and structure of the paper. Second 2 is helpful for preparing the reader with foundational information. The mathematics is presented cleanly in the remaining sections and I did not find any errors. Further there was good connection to the liturature in these sections. The paper ended with a Conclusion that satisfactorily describes what was accomplish in the paper. I am marking the paper as "publish as is" but I have three very minor suggestions that the author could take or leave.
1) The author uses a numbering scheme in which the collective expression is arabic and the sub-expressions are arabic. See for example equation 16 and others. Perhaps the sub-expressions could be Roman (or perhaps letters: a, b, c, ...).
2) Perhaps a few comments could be helpful for the reader in going through equations 52 - 72. Maybe something like "following each of the next 8 equations we will express the consequence of that relation." Or maybe something as simple as explaing the first of the i.e. expressions.
3) There were a few single sentence paragraphs but, for me, that didn't rise to the level of being halting or distracting.
Author Response
The text of the article has been amended in accordance with all the comments of the reviewer.
Round 2
Reviewer 3 Report (New Reviewer)
The author did not improve the manuscript. I maintain the previous report.
My final decision: Reject.
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
This is a good and interesting paper, concerning the Riemann curvature tensor for near Trans-Sasakian manifolds etc. This paper is generally very well written and it seems to me to be mathematically correct, thus this is a paper, which eventually deserves to be published.
However, before it is ready for publication, I have a few questions and comments for the author of this paper:
1. This paper is generally very well organized, except for the beginning of this paper: During the first five pages, there are 3-5 equations without any numbers, so that it is almost impossible for the readers to inquire about this paper, because the readers cannot find a way through this paper. I must therefore insist, that all equations etc, are clearly numbered in the revised version of this paper, and in the whole paper!
2. Dozens of places in this paper, the author has arranged 1), 2), 3), 4), 5) etc. Sometimes it is horizontal oriented, like in equation (27), which is difficult to see, and sometimes it is vertically oriented, like in equation (28), which is easy to see. I strongly suggest, that all these arrangements are vertically oriented!
3. A more serious thing, is that there are hundreds of equations etc, inserted directly into the text, so that it is impossible for the readers to inquire about this paper! But I guess that this is common for many mathematical papers, so what can I do?
Further revision is necessary. Major revision.
Author Response
The author expresses his sincere gratitude to Reviewer № 1 for his helpful remarks. The author took into account all the comments of the reviewer and made the appropriate changes. Formulas are highlighted in a separate line and numbered.
Reviewer 2 Report
The paper "On the Geometry of the Riemannian Curvature Tensor of Nearly Trans-Sasakian Manifolds" by Aligadzhi R. Rustanov presents research on the geometric properties of nearly Trans-Sasakian manifolds, specifically in relation to the Riemannian curvature tensor.
In this research, the author determines the components of the Riemannian curvature tensor and the Ricci tensor on the space of the associated G-structure. The paper also outlines several identities satisfied by these tensors.
A significant part of the research is dedicated to characterizing nearly Trans-Sasakian manifolds with a closed contact form.
Additionally, the author identifies several classes based on second-order differential-geometric invariants.
Lastly, the paper investigates the k-nullity distribution of a nearly Trans-Sasakian manifold.
Comments
Mathematical Notation: I noticed in several instances where mathematical formulas spanned multiple lines, double symbols such as '+', '-', and '=' were used. This usage could potentially cause confusion for readers. I would recommend revising these instances to ensure the formulas are as clear and concise as possible.
Consistency in Terminology: I also noticed some inconsistencies in the way short names for certain structures were introduced and used, for example,
... called nearly trans-Sasakian (in short, NTS-) structures
... an AC-structure (in short, an AC-manifold)
... an almost Hermitian structure (in short, AH-) structure
Author Response
The author expresses his sincere gratitude to Reviewer № 2. The author took into account all the comments of the reviewer. The abbreviated names are removed, the abbreviated names are left only for nearly trans-Sasakian manifolds.
Round 2
Reviewer 1 Report
Clearly unsatisfying revision. This paper is still terrible bad organized and it is impossible for the innocent readers to inquire about this paper. My first opinion was to reject this paper, because as far as I can see, the author of this paper, has not followed the recommendations, which I gave in my first report.
Anyway, I can see that in many mathematical papers, there are dozens of equations inserted directly into the text, or even hundreds of equations inserted directly into the text. In addition, there are many equations, without any numbers, so that it is impossible for the readers to inquire about this paper. Anyway, under pressure, I can accept this paper for the Axioms Journal.
