Talk:Fundamental theorem of Riemannian geometry

Which reference should cite a mathematical demonstration? the demonstration itself is THE REFERENCE! Please, be careful with your robot. —Preceding unsigned comment added by 149.132.102.246 (talk) 10:55, 24 September 2010 (UTC)

Koszul

The phrase "first line" might mean "first three terms" and so on. — Preceding unsigned comment added by 92.26.3.31 (talk) 15:17, 10 November 2016 (UTC)

Angle operation

What operation between X and Y is intended by "$\langle Y,Z\rangle$"? Arbitrarily0 ^(talk) 02:34, 24 March 2018 (UTC)

removal of section "Geodesics defined by a metric or a connection"

I've removed the section "Geodesics defined by a metric or a connection". The section was confusingly written, but seems to be describing the following statement: given a metric, there is exactly one torsion-free connection which has the same geodesics as the Levi-Civita connection. That's fine, but is seemingly not related to the fundamental theorem as given on the page. It is a distinct uniqueness theorem for the Levi-Civita connection. The addition of the text seven years ago, saying it was providing a "simple description of the [fundamental] theorem in term of geodesics and torsion", seems to be just incorrect. The Spivak reference also does not discuss it as being related to the fundamental theorem. So I've removed the text. Gumshoe2 (talk) 04:19, 16 January 2022 (UTC)

I think more care ix needed. The statements dismissing Spivak's Vol. 2, "A Comprehensive Introduction to Differential Geometry" are not correct. On Pages 249–253 there's an Addendum I which precisely concerns "Connections with the same geodesics". (Note that the books of Spivak are easy to find as pdf files.) Propositions 14 and 17 give conditions for ${\overline {\nabla }}$ and $\nabla$ to have the same geodesics with the same parametrizations, resp. with possibly different parametrizations. It is equivalent to ${\overline {\nabla }}_{X}X-\nabla _{X}X$ being identically zero or being proportional to $X$. In general, not providing reliable references is not to be recommended; it's not how wikipedia functions in either mathematics or any other domain. Mathsci (talk) 05:59, 16 January 2022 (UTC)

My sentence was "The Spivak reference also does not discuss it as being related to the fundamental theorem." My sentence was not "The Spivak reference also does not discuss it." Gumshoe2 (talk) 07:13, 16 January 2022 (UTC)

Without addressing the above comment, Mathsci has added back in the removed text, along with reverting my more recent edits to a worse explanation of one of the presentations of the proof. I have reverted back. Gumshoe2 (talk) 16:35, 16 January 2022 (UTC)

Unexplained reverts

User:Gumshoe2 is reverting content that is properly sourced. He is trying to remove WP:RS that he does not like. I have reverted that content. The edits should be discussed in a dispassionate way. It's easy to check that the material is classical. Mathsci (talk) 16:43, 16 January 2022 (UTC)

As I explained on this very talk page within the last 24 hours, the material does not belong on this page, and the sources likewise do not assert that the material is related to the topic of the page. Mathsci has reverted without addressing this. (So I have reverted back again.) Gumshoe2 (talk) 16:47, 16 January 2022 (UTC)

After another reversion by mathsci without addressing comments here, I have reverted again. Gumshoe2 (talk) 17:00, 16 January 2022 (UTC)

Removing three WP:RSs seems like vandalism. Please you can explain why you are deleting content in this reckless way. Please seek WP:consensus and/or explain yourself: at the moment, there are no edit summaries giving detailed explanations.

The classical material of Cartan, Weyl and Eisenhart has been summarised from the WP:RSs of Spivak and Nomizu & Sasaki; it's easy to check that out on mathscinet, zbl or arXiv if you are in some of doubt. Mettler & Paternain research in that area and give Spivak as a standard reference. Since Patternain & co regard this as standard material that I have already summarised, I will restore the content: at the moment that particular deletion seems impossible to distinguish from vandalism. The new material was created by me, so, given that it is supported by WP:RSs, the WP:ONUS is now on you to explain yourself. Please do not delete WP:RSs. Mathsci (talk) 17:39, 16 January 2022 (UTC)

I have explained each edit here on the talk page. You are edit warring, as you have not addressed my comment, which is that what is written is not obviously related to the page topic. Gumshoe2 (talk) 17:45, 16 January 2022 (UTC)

Here is a non-specialist opinion (non-specialist opinions are relevant, since, per WP:TECHNICAL, Wikipedia articles must be written for a larger audience that specialists of the subject).

About the section added by Mathsci: The relation between the content of this section and the fundamental theorem is not explained. The lack of such an explanation suggests to readers that such a relation does not exists, and thus that the section does not belong to the article. So, I agree with the revert of this section.

Tone: the reverted edits begin each section with a discussion of the sources. This does belongs to Wikipedia articles, when the sources are compatible, which is the case here. So, this is another reason for agreeing with Gumshoe2. D.Lazard (talk) 18:32, 16 January 2022 (UTC)

The stated result, whilst interesting, has no relation whatsoever to the Fundamental theorem of Riemannian geometry. It doesn't matter if it is cited with reliable sources, and there is no requirement for content to be kept just because it is reliably cited. I see no reason why it should be on this page. It might be of passing interest on the page Geodesic or affine connection although on either page it would be out of place and not particularly notable. In this case I think WP:NOTTEXTBOOK and it just doesn't belong here. Tazerenix (talk) 20:26, 16 January 2022 (UTC)

Kobayashi & Nomizu use the word "Proofs". The second proof, Proof B, is the coordinate-free proof in the text, which has no steps omitted. K & N give a summary, instead of spelling everything out. Their first proof, Proof A, uses the Cartan structural equations. It's non-technical, treated elsewhere but not here.

Yesterday there were two WP:RS in the article; the Spivak reference was removed. Then a lot of tangential references were added. In general, when an WP:RS is not used directly or where there are no in-line citations for specific content, it's standard to add them in a section "Further references" or "Further reading" instead of removing them. I have no strong views on whether the Addendum on geodesics should be added or not: I've not really had a chance to look at relevant papers nor to read the CUP book of Nomizu & Sasaki. Mathsci (talk) 22:09, 16 January 2022 (UTC)

This is barely coherent as a response to the discussion. Anyway, I have reverted your most recent edit, since Milnor's book is specifically cited in the main text. Gumshoe2 (talk) 22:29, 16 January 2022 (UTC)

Since you have now removed the part of the text that you were previously insistent on keeping, I have also removed the relevant references, which are no longer referred to anywhere in the main text. Gumshoe2 (talk) 22:32, 16 January 2022 (UTC)

Please reread In general, when an WP:RS is not used directly or where there are no in-line citations for specific content, it's standard to add them in a section "Further references" or "Further reading" instead of removing them. I have no strong views on whether the Addendum on geodesics should be added or not: I've not really had a chance to look at relevant papers nor to read the CUP book of Nomizu & Sasaki. Thanks, Mathsci (talk) 22:41, 16 January 2022 (UTC)

These sources and content would be great for a wiki page on projective equivalence. Please try to focus on the topic at hand, which is their relevance to this particular page. You are wasting people's time. Gumshoe2 (talk) 23:00, 16 January 2022 (UTC)

If there is anyone reading this coming from my complaint on the edit war noticeboard, I would like to point out that none of mathsci's comments here address the issue at hand, which is the relevance of the material for this page. Gumshoe2 (talk) 23:10, 16 January 2022 (UTC)