… “full of sound and fury, signifying nothing.” And the editor believes it, even if it is self-contradictory, after sitting on the article for half a year.

There are two problems:

- the problem of time; you write a long and dense article, which may be hard to review and the referee, instead of declining to review it, it keeps it until the editor presses him to write a review, then he writes some fast, crappy report, much below the quality of the work required.
- the problem of communication: there is no two way communication with the author. After waiting a considerable amount of time, the author has nothing else to do than to re-submit the article to another journal.

Both problems could be easily solved by open peer-review. See Open peer-review as a service.

The referee can well be anonymous, if he wishes, but a dialogue with the author and, more important, with other participants could only improve the quality of the review (and by way of consequence, the quality of the article).

I reproduce further such a review, with comments. It is about the article “Sub-riemannian geometry from intrinsic viewpoint” arXiv:1206.3093 . You don’t need to read it, maybe excepting the title, abstract and contents pages, which I reproduce here:

Sub-riemannian geometry from intrinsic viewpoint

Marius Buliga

Institute of Mathematics, Romanian Academy

P.O. BOX 1-764, RO 014700

Bucuresti, Romania

Marius.Buliga@imar.ro

This version: 14.06.2012

Abstract

Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Caratheodory distance. One of the most striking features of a regular sub-riemannian space is that it has at any point a metric tangent space with the algebraic structure of a Carnot group, hence a homogeneous Lie group. Siebert characterizes homogeneous Lie groups as locally compact groups admitting a contracting and continuous one-parameter group of automorphisms. Siebert result has not a metric character.

In these notes I show that sub-riemannian geometry may be described by about 12 axioms, without using any a priori given differential structure, but using dilation structures instead.

Dilation structures bring forth the other intrinsic ingredient, namely the dilations, thus blending Gromov metric point of view with Siebert algebraic one.

MSC2000: 51K10, 53C17, 53C23

1 Introduction 2

2 Metric spaces, groupoids, norms 4

2.1 Normed groups and normed groupoids 5

2.2 Gromov-Hausdorff distance 7

2.3 Length in metric spaces 8

2.4 Metric profiles. Metric tangent space 10

2.5 Curvdimension and curvature 12

3 Groups with dilations 13

3.1 Conical groups 14

3.2 Carnot groups 14

3.3 Contractible groups 15

4 Dilation structures 16

4.1 Normed groupoids with dilations 16

4.2 Dilation structures, definition 18

5 Examples of dilation structures 20

5.1 Snowflakes, nonstandard dilations in the plane 20

5.2 Normed groups with dilations 21

5.3 Riemannian manifolds 22

6 Length dilation structures 22

7 Properties of dilation structures 24

7.1 Metric profiles associated with dilation structures 24

7.2 The tangent bundle of a dilation structure 26

7.3 Differentiability with respect to a pair of dilation structures 29

7.4 Equivalent dilation structures 30

7.5 Distribution of a dilation structure 31

8 Supplementary properties of dilation structures 32

8.1 The Radon-Nikodym property 32

8.2 Radon-Nikodym property, representation of length, distributions 33

8.3 Tempered dilation structures 34

9 Dilation structures on sub-riemannian manifolds 37

9.1 Sub-riemannian manifolds 37

9.2 Sub-riemannian dilation structures associated to normal frames 38

10 Coherent projections: a dilation structure looks down on another 41

10.1 Coherent projections 42

10.2 Length functionals associated to coherent projections 44

10.3 Conditions (A) and (B) 45

11 Distributions in sub-riemannian spaces as coherent projections 45

12 An intrinsic description of sub-riemannian geometry 47

12.1 The generalized Chow condition 47

12.2 The candidate tangent space 50

12.3 Coherent projections induce length dilation structures 53

Now the report:

“Referee report for the paper

Sub-riemannian geometry from intrinsic viewpoint

Marius Buliga

for

One of the important theorems in sub-riemannian geometry is a result

credited to Mitchell that says that Gromov-Hausdorff metric tangents

to sub-riemannian manifolds are Carnot groups.

For riemannian manifolds, this result is an exercise, while for

sub-riemannian manifolds it is quite complicate. The only known

strategy is to define special coordinates and using them define some

approximate dilations. With this dilations, the rest of the argument

becomes very easy.

Initially, Buliga isolates the properties required for such dilations

and considers

more general settings (groupoids instead of metric spaces).

However, all the theory is discussed for metric spaces, and the

groupoids leave only confusion to the reader.

His claims are that

1) when this dilations are present, then the tangents are Carnot groups,

[Rmk. The dilations are assumed to satisfy 5 very strong conditions,

e.g., A3 says that the tangent exists – A4 says that the tangent has a

multiplication law.]

2) the only such dilation structures (with other extra assumptios) are

the riemannian manifolds.

He misses to discuss the most important part of the theory:

sub-riemannian manifolds admit such dilations (or, equivalently,

normal frames).

His exposition is not educational and is not a simplification of the

paper by Mitchell (nor of the one by Bellaiche).

The paper is a cut-and-past process from previous papers of the

author. The paper does not seem reorganised at all. It is not

consistent, full of typos, English mistakes and incomplete sentences.

The referee (who is not a spellchecker nor a proofread) thinks that

the author himself could spot plenty of things to fix, just by reading

the paper (below there are some important things that needs to be

fixed).

The paper contains 53 definitions – fifty-three!.

There are 15 Theorems (6 of which are already present in other papers

by the author of by other people. In particular 3 of the theorems are

already present in [4].)

The 27 proofs are not clear, incomplete, or totally obvious.

The author consider thm 8.10 as the main result. However, after

unwrapping the definitions, the statement is: a length space that is

locally bi-lipschitz to a commutative Lie group is locally

bi-lipschitz to a Riemannian manifold. (The proof refers to Cor 8.9,

which I was unable to judge, since it seems that the definition of

“tempered” obviously implies “length” and “locally bi-lipschitz to the

tangent”)

The author confuses the reader with long definitions, which seems very

general, but are only satisfied by sub-riemannian manifolds.

The definitions are so complex that the results are tautologies, after

having understood the assumptions. Indeed, the definitions are as long

as the proofs. Just two examples: thm 7.1 is a consequence of def 4.4,

thm 9.9 is a consequence of def 9.7.

Some objects/notions are not defined or are defined many pages after

they are used.

Small remarks for the author:

def 2.21 is a little o or big O?

page 13 line 2. Which your convention, the curvdim of a come in infinite.

page 13 line -2. an N is missing in the norm

page 16 line 2, what is \nu?

prop 4.2 What do you mean with separable norm?

page 18 there are a couple of “dif” which should be fixed.

in the formula before (15), A should be [0,A]

pag 19 A4. there are uncompleted sentences.

Regarding the line before thm 7.1, I don’t agree that the next theorem

is a generalisation of Mitchell’s, since the core of his thm is the

existence of dilation structures.

Prop 7.2 What is a \Gamma -irq

Prop 8.2 what is a geodesic spray?

Beginning of sec 8.3 This is a which -> This is a

Beginning of sec 9 contains a lot of English mistakes.

Beginning of sec 9.1 “we shall suppose that the dimension of the

distribution is globally constant..” is not needed since the manifold

is connected

thm 9.2 rank -> step

In the second sentence of def 9.4, the existence of the orthonormal

frame is automatic.”

Now, besides some of the typos, the report is simply crap:

- the referee complains that I’m doing it for groupoids, then says that what I am doing applies only to subriemannian spaces.
- before, he says that in fact I’m doing it only for riemannian spaces.
- I never claim that there is a main result in this long article, but somehow the referee mentions one of the theorems as the main result, while I am using it only as an example showing what the theory says in the trivial case, the one of riemannian manifolds.
- the referee says that I don’t treat the sub-riemannian case. Should decide which is true, among the various claims, but take a look at the contents to get an opinion.
- I never claim what the referee thinks are my two claims, both being of course wrong,
- in the claim 1) (of the referee) he does not understand that the problem is not the definition of an operation, but the proof that the operation is a Carnot group one (I pass the whole story that in fact the operation is a conical group one, for regular sub-riemannian manifolds this translates into a Carnot group operation by using Siebert, too subtle for the referee)
- the claim 2) is self-contradictory just by reading only the report.
- 53 definitions (it is a very dense course), 15 theorems and 27 proofs, which are with no argument: “ not clear, incomplete, or totally obvious“
- but he goes on hunting the typos, thanks, that’s essential to show that he did read the article.

There is a part of the text which is especially perverse: The paper is a cut-and-past process from previous papers of the

author.

Mind you, this is a course based on several papers, most of them unpublished! Moreover, every contribution from previous papers is mentioned.

Tell me what to do with these papers: being unpublished, can I use them for a paper submitted to publication? Or else, they can be safely ignored because they are not published? Hmm.

This shows to me that the referee knows what I am doing, but he does not like it.

Fortunately, all the papers, published or not, are available on the arXiv with the submission dates and versions.

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See also previous posts:

- Multiple peer-reviews, a story with a happy-end
- Anonymous peer-review after 15 months
- FoM: denied publication
- The price of publishing with arXiv

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