Talk:Riemannian manifold/Archive 1

Archive 1

Riemann for Anti-Dummies

Deleted external link to http://wlym.com/tiki/tiki-index.php?page=Pedagogicals Riemann for Anti-Dummies. The site was of questionable authority and had a very non-neutral point of view.

You were right to delete the link, thanks. I found two more articles with the same link and also deleted it there. However, for future reference, you accidentally deleted a bit more than just the link: categories like [[Category:Riemannian geometry| ]] and interwiki links like [[de:Riemannsche Mannigfaltigkeit]] . Please leave them next time. Anyway, I fixed it now. -- Jitse Niesen (talk) 03:14, 19 February 2007 (UTC)

For The Lay(er) Man

Is there any chance that one could have a paragraph (or two) about What this means with less of the math? I realize that one would have to make assupmtions etc. in order to explain this to others. E.Lefraw 23:19, 23 February 2007 (UTC)

Untitled question

I've come to terms with the fact that I will never understand this.

You know, the article is much too short. It could stand to have many examples and explanatory sentences added. Perhaps you could help us decide how best to improve it, with focused questions. at what point does the exposition lose you? if you ask for more, you might receive it... - Lethe

I added a paragraph to the article. I should probably look it over again tomorrow to make sure it's coherent, but I want to make the basic idea of a Riemannian manifold transparent; an inner product gives you a notion of length on a vector space, and a Riemannian metric gives you a notion of length on a smooth manifold by defining an inner product on each tangent space, pointwise, in a smooth way. to pass from a notion of length of a tangent vector to length of a curve on the manifold, all you have to do is integrate. There is a lot more you can do with Riemannian geometry, but if you can grasp that starting notion, then the rest should fall into place. I guess what I'm saying is, if you think the article is incomprehensible, you may be right. Asking for clarification might be a good idea.... Lethe

-- I think my new paragraph here kinda sticks out like a sore thumb. I don't like it, and perhaps it belongs in a different article (differential geometry, perhaps?) I'm looking for feedback, i guess. - Lethe 00:17, Jul 17, 2004 (UTC)

Hi Lethe, relatively new contributor here. What I'm thinking would be useful, is some initial mention for motivation -- why did Riemann introduce this concept, what could it effectively model? Including a few sentences outlining such information, optimally as a continuance of the first paragraph, would substantially improve the overall quality of this article. Geo.per 19:21, 26 June 2006 (UTC)

I went back and added a few section headers. Lumping everything under Introduction seems inappropriate. Actually, a more thorough list of properties should probably be added. Geo.per 19:26, 26 June 2006 (UTC)

I agree with the idea that this article was too small and not enough documented. That's why I have added some further things regarding the metric and the metric space ... Hope this is coherent for everybody --Zadigus (talk) 10:14, 29 January 2008 (UTC)

Question about the first sentence

I have not yet given up the hope to understand this article. Let me start with the beginning: what confuses me in the first sentence is that g is used for two different purposes: once it is the metric tensor and second time it is the inner product.

I know from classical differential geometry that one needs the vector product in order to calculate the metric tensor. But these two things are surely not one and the same. TomyDuby (talk) 02:48, 7 September 2008 (UTC)

Yes, g is the inner product and the metric tensor. They are one and the same thing, by the canonical isomorphism

$L^{2}(TM)\to T^{*}M\otimes T^{*}M$

from the space of bilinear mappings on the tangent bundle to the tensor product of the cotangent bundle with itself. More details can be found in the metric tensor article. siℓℓy rabbit (talk) 03:02, 7 September 2008 (UTC)

Thanks. TomyDuby (talk) 10:15, 7 September 2008 (UTC)

Riemannian metrics: Question

What is ν(M) (see main article)? TomyDuby (talk) 02:56, 7 September 2008 (UTC)

I think you mean ${\mathcal {V}}(M)$. This is a somewhat idiosyncratic notation for the sheaf of vector fields on M. I have removed it, since the article already suffers from too much unexplained notation. siℓℓy rabbit (talk) 03:07, 7 September 2008 (UTC)

Yes I meant ${\mathcal {V}}(M)$. Thanks for clearing this. TomyDuby (talk) 10:15, 7 September 2008 (UTC)

Another two questions from the same section:

Let $\{\left({\frac {\partial }{\partial x_{i}}}\right)_{p}\}_{i}$ be a basis of tangent vectors over $p\in M$. What are the x_i-s? Are these the coordinates of point p in Rⁿ?

Is the derivative ∂ / ∂ x_i the derivative of coordinates of point p = (x₁, ..., x_n) giving (0, ..., 1, ..., 0)?

TomyDuby (talk) 10:15, 7 September 2008 (UTC)

I have made some changes to the offending section. Let me know if it is clearer. Thanks, siℓℓy rabbit (talk) 12:49, 7 September 2008 (UTC)

I think that this is much better. Thanks! TomyDuby (talk) 18:06, 7 September 2008 (UTC)