Advanced Manifold–Metric Pairs

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

 In the paper under review, the authors describe a mathematical formalism for generalized metric spaces and manifolds. They build several types of D-dimensional manifolds and their corresponding metric pairs using functionals. Furthermore, the different types of manifolds they combine are built not only on generalized spacetime spaces of extra dimensions, but also on the concepts of information, probability, and entropy. In particular, they construct this formalism using simple concepts of mathematical physics, field theory, topology, algebra, probabilities, and statistics.
 The subject of this work is very novel and interesting. The manuscript is well written, well motivated and of some interest for researchers working in geometry and physics. The examples are very helpful for analyzing of the concepts. Many of results are theoretically correct and are supported by well-constructed examples. Also, the study is acceptable in terms of language and expression.  I will recommend the manuscript for possible publication in the journal with some recommendations.

Recommendations

  1- The main problem of this generalization metric is that the metric may be indefinite. For example in Eq 21, the amount of $ds^3(V,V)$ can be positive, negative and 0. 
 
  2- From one perspective, it is as if the author has defined a tensor on a manifold. What is the necessity and fundamental similarity that he has considered this to be a metric? 
  
  3- The article should be reviewed to correct some typographical and grammatical errors. For example, somewhere "Dimension" should be written as "dimension".
 
  4- Some formulas are out of the article layout. Please rewrite them. Lines: 312, 338, 538,... .
  
  5. I don't understand the need for section 3.7? Please explain further the need for this section to be added.

  6. The references should be rechecked and  rearranged as Journal style.  In addition, some articles that have not yet been peer-reviewed or published have been used, which should be reduced. The author has referenced many of his own articles, so one should try to cite only those that are necessary.

Comments for author File: Comments.pdf

Author Response

We thank the reviewer for this comment. We provide the reply attached.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper deals with a construction of advanced manifold pairs and found their several applications.  the applications part is interesting. However, I have thoroughly scrutinized the paper and I have some observations/comments given as follows:

1. The abstract of the paper should be rewritten by focusing the investigated results and the methodology used.
2.  The first three sentences of the introduction should be deleted, because they will  mislead the prospective readers, e.g. manifold is a generalization of a space, but later the author used in several times in the text manifold as a space. Mathematically, a space is nonempty set equipped with a structure or structures (that may be algebraic, geometric etc.)
3. The author have written advanced generalized tensors, what does it mean? Also the author has written Sophisticated mathematical and theoretical physics structures...What does it mean?
4. The author has written in the introduction, extra dimension, spectrum dimensions, manifold dimension, probabilistic set of dimension etc. The author should clearly explain these concepts along with their interrelations in the introduction.
5. In section 2, the author has mentioned (1,3) manifold, what does it mean?
6. The author should clearly write the statement of Theorem2.1.
7. In D-dimensional manifold, the author has consider a metric. Is it always possible that a manifold is metrizable? The conditions of metrizability should be written clearly.
8. In equation (1), the  author has taken the metric as mapping from C*C to R+, please mention here about C.
9. Before equation (5), the  author has has written ... as defined 1 and 2, there must be brackets in 1 and 2 and the same would be corrected in other places.
10. In example of FLRW spacetime, ( section3.1.1), please compare the manifold pair structure with wrapped product structure.
11. The metric considered in the paper, should be compare with the Hessian structure.

Comments on the Quality of English Language

The presentation of the paper is poor. The English should be improved throughout the text, especially the introduction and abstract and the theorem.

Author Response

We thank the reviewer for this comment. We provide our reply attached to this document.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Review on the paper “Advanced manifold-metric pairs” by Pierros Ntelis.

 

This paper is concerned with a new abstract mathematical formalism for metric spaces and manifolds pairs that can help in the development of geometry and topology. In doing so, the authors construct several types of $D$-dimensional advanced manifold-metric spaces pairs using special functionals. An attempt is made to unify different types of manifolds using not only concepts and constructions in spacetime, but also mathematical logic and its foundations, mathematical physics, field theory, topology, algebra, probability, and statistics.

In my opinion, the article needs to be seriously reworked. The statements and their proofs need to be formatted in a way that is easy for readers. The article also needs to be structured in a new way.

It is written in bad mathematical language, it looks more like a literary work with formulas inserted. There are no formulated theorems and statements, there is no structure of proofs. It is necessary to rewrite the whole work again, making it more mathematical.

 

Comments.

Line 5. Change the word “generalised” by “generalized”.

It is important to reorganize (to rewrite) the paper by formulating Theorems and Propositions (see Section 2.1.2). Also, it is necessary to clearly indicate where the proof of the statements begins and ends.

Line 312. It is necessary to rewrite the line. Same with line 338.

It is necessary to rewrite the formula (90).

Author Response

We thank the reviewer for this comment. We provide our reply attached to this document.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors addressed all my comments and suggestions. The revised version is much better than original one. Hence it may be accepted for publication.

Reviewer 3 Report

Comments and Suggestions for Authors

According to my opinion, this version is fine. The paper can be published in the present form.