How about an explanation in layman's english, too? Theanthrope 06:46 Jan 23, 2003 (UTC)
I'll try. However, while it's possible to sort of talk about a manifold in layman's english, the distinction between Riemann surfaces and 2-manifolds is very subtle and difficult to describe without using any math. Loisel 07:08 Jan 23, 2003 (UTC)
Axel, I saw you improved the opening paragraph. However, I'm concerned that perhaps we could serve Theanthrope's request (see above) by moving some of the jargon to a second paragraph, and keeping the opening paragraph as light on jargon as possible. I think it would also help if someone could give an interesting example of a 2-manifold that can't be viewed as a Riemann surface. This way, we can somehow say "Riemann surfaces are mainly what everyone thinks of as surfaces, however some surfaces such as X aren't Riemann surface, see below for technical details." Unfortunately, I'm not sure which 2 surface does not admit a Riemann surface structure. What do you think? Loisel 06:18 Feb 12, 2003 (UTC)
I moved the term "holomorphic" into the second paragraph, even though I think it's the whole point of the enterprise. We certainly need examples: open sets of the complex plane, the Riemann sphere, the two-sheet thingy you get from the square root function etc.
Everything2 has a nice writeup at http://everything2.com/index.pl?node_id=1007348
All Riemann surfaces are orientable, so the Moebius strip can't be made into a Riemann surface. AxelBoldt 16:56 Feb 12, 2003 (UTC)
I didn't know that. I added an argument outline to the article. I didn't want to infect the article with a lengthy proof. Delete it if you don't like it. Loisel 03:27 Feb 13, 2003 (UTC)
We need to de-TeX this article a bit, it takes far too long to load. -- Tarquin 16:39 Feb 12, 2003 (UTC)
Yes, I agree. AxelBoldt 17:15 Feb 12, 2003 (UTC)
Me too - plus it's horrible to read text where the font size jumps up and down by a factor of 2. Chas zzz brown 12:04 Feb 13, 2003 (UTC)
Alex, I'm not sure I have the correct idea for the charts of concrete riemann surfaces. I'm not completely sure that what I have in mind is a chart independent of the a_k. Can you clarify that for me? Loisel 22:07 Feb 17, 2003 (UTC)