On Finsler Surfaces with Isotropic Main Scalar

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper under review seems interesting, but I would like to ask the authors few questions:

1. What conclusion do you draw from your results?

2. What is "H" in abstract?

3. Is the flag curvature (equation (2.5)) same as the sectional curvature?

4. Can we talk about the relationships between intrinsic and extrinsic invariants for a Finsler surface?

Comments on the Quality of English Language

English language seems good to me.

Author Response

Deer Referee

Thank you very much for your comments. I provides some answers for your questions as follows:

1- I tried to give an extension of well-known Berwald's theorem. 

2- The H-curvature is the horizontal derivation of E-curvature along Finslerian geodesic.

3- Yes. But in general the flag curvature is a function of x, y and P. Indeed, K=K(x, y, P). Thus the flag curvature is extension of sectional curvature.

4- In this paper, we did not consider the relationships between intrinsic and extrinsic invariants for a Finsler surface.

Tayebi

Reviewer 2 Report

Comments and Suggestions for Authors

Two-dimensional Finsler spaces have many particularly special properties which are not characteristic for the higher-dimensional Finsler spaces. In particular for two-dimensional Finsler spaces there is defined notion of main scalar function I. In case when the main scalar is isotropic the expressions for corresponding Finsler metric are well known. In particular, the metric is Berwald if and only if the main scalar is constant.

 The mean of Berwald curvature is the trace of Berwald curvature B. By taking the horizontal derivation of the mean of Berwald curvature along Finslerian geodesics one obtains the H curvature.

 The authors deal with positively curved Finsler surface with isotropic main scalar I and show that H = 0 if and only if I = 0. It follows that F has isotropic scalar flag curvature if and only if it is Riemannian.

 The paper is mostly mathematically clearly written with the motivation and context of the research represented in the Introduction. The subject of Finsler surfaces is of interest to a number of geometers and this paper will be of interest to them.

 There are small corrections to be made, some mathematical and some considering English language, which are listed below the report.  Since the author of this report is also not a native English speaker, an English language checking is recommended.

 Therefore, I recommend this paper to be considered for publication in Mathematics, after some minor corrections.

 Line 20: with ->of

 Line 22:dose->does

 Lines 30-32: the sentences in which authors try to introduce main scalar I should be refrased to be more precise mathematically and correct in English

In particular sentence in line 30: “It is natural to ask that which parameter characterize many geometrical properties of two-dimensional metric”, since firstly in mathematics “many geometrical properties” is not precisely defined, nor generally in mathematics such notion needs to exist.

 Formula (1.4): epsilon should be under square root at the right hand side

 Line 46: epsilon= -1 for 0<s<1

 Line 47: that->to

 Line 334: dose->does
I would suggest instead of that something in the line of “Finsler surfaces have many geometric properties different from higher dimensional Finsler spaces. In particular for two-dimensional Finsler metrics there exists a notion of main scalar, denoted by,…, defined by, …which characterizes the following properties… ”

Comments on the Quality of English Language

There are small corrections to be made, some considering English language, which are listed below the report.  Since the author of this report is also not a native English speaker, an English language checking is recommended.

Author Response

Dear Referee

Thank you very much for your comments. I will revise the sentence and the mathematical notions.

Best Regards

Tayebi