Talk:Riemann curvature tensor

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I started new page Curvature of Riemannian manifold, I think it is bit better than this one, I plan to remove from this page everything except things directly connected with curvature tensor and link it to page above.

Tosha 04:09, 13 May 2004 (UTC)

But please write Lie ... with a capital L, because Lie refers to the norwegian mathematician Sophus Lie. Hannes Tilgner

It will be hard for me, but I will try

Tosha 05:40, 15 May 2004 (UTC)

If you remove this article OK, but still I doubt that you catch the most general aspect of curvature in the sense of Nomizu, Kulkarni and other modern writers. Look at the reference http://www.EarningCharts.NET/ipm/ipmWaves.htm where you find more references. In the references there (look also at that one in Lecture Notes in Mathematics) you find a decomposition of the space of all curvature structures in terms of Lie and Jordan algebras. And you find how elegantly electrodynamics and gravitational waves fit into the curvature play, look at the basic work of Lichnerowics. As an ,algebraiker' I like to write the curvature structure in the following triple form, generalizing the concept of Lie triples (the book on Symmetric Spaces of Otmar Loos is a nice generalization of Lie theory): [x,y,z]=R(x,y)z. This concept generalizes the notion of a Lie triple to that one of a curvature triple, where only the Jacobiidentity is missing, but a reference to the bilinear form <,> is added in such a way, that R(x,y) is an element of the pseudoorthogonal Lie algebra. Note that the complete work of Ricci, Einstein and Weyl can be summarized as a decomposition of the space of curvature structures of Levi type (for Lie algebras). All this shows, that we do not yet understand this curvature space completely. Especially the gravitational wave aspect needs clarification. Hannes Tilgner

It seems that you want to include some basic identites with curvature plus Pseudo-Riemannian case (is it?) I think it is a good idea.

Tosha 05:40, 15 May 2004 (UTC)

Yes, I'm considering the following: Instead of writing a full publication in some mathematical journal (I have done that too often, it didn't pay out), use the Wikipedia for publication. By the rules of the scientific world, everything written down here, is published. Actually you can start with an outline of the idea, putting it step by step into a full scientific article. This cannot be done with a scientific journal. Writing an scientific article is time consuming. In this way everybody can see how - and immediately comment. The (my)problem is time - since I work hard on my webpage, mentioned above. Hannes Tilgner

Please have a look at Wikipedia:No original research. This is a good place for 'survey articles'; but not for new results. Charles Matthews 08:02, 17 May 2004 (UTC)

[edit] the "typo"

Hey, I just reverted an edit. It was a subtle point (but perhaps worth writing down) but

and so the original formula was correct as written by Geometry guy. Cheers, Wesino 21:47, 14 April 2007 (UTC)

[edit] Merge proposal

The articles Riemann curvature tensor and Curvature of Riemannian manifolds cover much the same sort of material. I propose merging them. Does anyone have any objections? Silly rabbit 20:15, 21 May 2007 (UTC)

I agree to merge those two articles. But how about the opposite way? Currently, it is proposed to merge Curvature of Riemannian manifolds to Riemann curvature tensor. However, as Riemann curvature tensor is only one of the ways to define curvature on Riemannian manifold, I think Riemann curvature tensor should be a part of the other article. --Acepectif 08:43, 24 August 2007 (UTC)

Actually, I don't think we need to have each article for every kind of curvatures. For example, Weyl tensor article should better be a part of this article. --Acepectif 08:46, 24 August 2007 (UTC)

I think that it is valuable to have separate articles on the various types of curvatures (e.g., Weyl tensor). A great deal can be said about the Weyl tensor vis-a-vis conformal geometry, for example, that would be decidedly out of place in a general article on the curvature of Riemannian manifolds. The same sort of remark applies to the Ricci tensor, where people study things like the positivity of the eigenvalues (although in my field it's mostly the Weyl-Schouten tensor that is of interest). Again, much can be said which is not relevant here.

Point taken about merging the opposite way. Obviously this is a "slow merge". Silly rabbit (talk) 02:02, 23 February 2008 (UTC)

I don't think a merge is a really good idea. The articles do cover much of the same material but Curvature of Riemannian manifolds is meant to be an overview of the various ways of understanding curvature in the Riemannian setting. The Riemann curvature tensor is one way, but it is not the only one. Sectional curvature is an equivalent method. There are many other inequivalent methods. I think having a separate overview article (written in summary style) is helpful. -- Fropuff (talk) 06:03, 23 February 2008 (UTC)

You know, I am beginning to have the same feeling. However, the both articles could probably use some refocusing and trimming, with summary style in mind. Silly rabbit (talk) 14:03, 23 February 2008 (UTC)