Talk:Orientability

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Don't you miss (the capacity of) orientation - a basic condition of a sane person?

Apogr 10:53, 2 Sep 2004 (UTC)

No. Elroch 18:12, 15 February 2006 (UTC)

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[edit] Orientable characters

The article earlier used an intuitive description of orientablility of surfaces based on moving a letter around the surface. To be rigourous, the letter needs to be considered as a graph made up of line segments. In most fonts, no letters (even ones like Courier R) are strictly suitable for this purpose, as they may be deformed as graphs in the plane into their mirror image (Check this!). I have now updated the article to refer to moving a small image Image:Small_pie.PNG (which has a clear handedness) around a surface. I hope this makes it clear to the reader, as well as more precise. Elroch 18:12, 15 February 2006 (UTC)

Wow, nice job! This was one of the first articles I edited anonymously years ago, and it's come a long way since. The article looks great! --C S (Talk) 04:23, 12 April 2006 (UTC)

I think using images in the text is a bad idea. I would like to change it back to text. Can you explain to me why you think that the letter R is not handed? It seems to me that you can profitably view the upper left corner of the "R" as a vertex, which I think will keep you from being able to deform it into its mirror. If that doesn't satisfy, why not consider rigid tranformations only? I really don't like at all filling up text with little pie charts. They don't scale with the font sizes, they can't be seen with text browsers, and they just look unprofessional. -lethe talk + 05:58, 13 April 2006 (UTC)

"R" (in the font I see on the uploaded version of this page) has three vertices (including the one you want) right? So it's actually topologically the same as "A". The top half is a triangle with two legs on the bottom half. Anyway, as for the rigid deformations, I believe that's probably what was intended with the very initial version of the page, and it's fine for intuition's sake, mostly. I personally didn't think it was that big a deal, but now that's it's been mentioned, since in most of that content, everything's being done topologically, I think it's a little confusing to talk about rigid moves. Especially since it's only "rigid" in a small neighborhood of the "R" anyway, and there's also the problem that this assumes the surface is geometric in some fashion.
As for the problem with accessibility, can't that be fixed somehow without removing the image? I don't know what "unprofessional" means here. I might use an illustration in the text like that in a paper, and certainly I've seen plenty of papers that use something like that. It's a matter of style. I don't think we're being less professional than the norm. --C S (Talk) 13:06, 13 April 2006 (UTC)
I'm inclined to agree that the pie doesn't look very "nice"; however, I agree that using a simple graphic is more intuitive than using a letter from the alphabet (and in particular, stands out nicely in the text). My proposal would be to use a simple black and white image instead of the pie (Maybe an umbrella? I don't know) so that it looks more professional. It would be a large contribution if somebody would take that image and produce a small animation showing how handedness can be changed on some non-orientable surface. Meekohi (talk) 17:16, 10 June 2008 (UTC)

[edit] Merge?

There has been a merge tag on Orientable manifold for a long time. I don't know the subject well enough to comment, but could someone who does start a discussion to resolve the proposition? Kcordina 12:29, 17 March 2006 (UTC)

Comment for readers: This has already been done; orientable manifold redirects here. --C S (Talk) 04:25, 12 April 2006 (UTC)

[edit] universe is closed

As far as I know, the difference from the critical density of the universe has not yet been measured, thus whether the universe is closed or not is completely up in the air. Anyway, it's not really relevant, so I'm removing it. -lethe talk + 06:03, 13 April 2006 (UTC)

It appears to me also that this is a controversial claim to make, so it's good to remove it. --C S (Talk) 13:09, 13 April 2006 (UTC)

[edit] tangent bundle always oriented?

I was also looking suspiciously at the recent anonymous edit that Oleg reverted, which changed "manifold orientable iff its tangent bundle is" to "tangent bundle always orientable". Oleg says that Mathworld's article agrees with the original. I myself am a little bit confused. I know for sure that the cotangent bundle of any manifold is always orientable, indeed even carries a choice of orientation. I think it's believable that the tangent bundle could also be orientable, Does not the orientation on the cotangent bundle also induce an orientation on the tangent bundle? I'm gonna poke in some books (mathworld isn't definitive). -lethe talk + 04:17, 22 May 2006 (UTC)

Yes, the cotangent bundle is always orientable, per planetmath, see at the bottom. I don't know details though. :) Oleg Alexandrov (talk) 04:22, 22 May 2006 (UTC)
The proof that the cotangent bundle is easy: there exists a canonical symplectic form on any cotangent bundle. Any symplectic manifold is oriented. How's this for a proof for tangent bundles: every second-countable Hausdorff manifold admits a Riemannian metric. Every Riemannian metric induces an isomorphism between the cotangent bundle and the tangent bundle. The cotangent bundle is always oriented, so the tangent bundle is too. Mathworld is wrong (haven't found it in my books yet though). -lethe talk + 04:26, 22 May 2006 (UTC)
My theory is that since the orientation of a manifold is defined as an orientation of tangent vector spaces, someone over at mathworld got confused and thought that the orientation of a manifold and its tangent bundle (which is after all a disjoint union of tangent vector spaces) were the same thing. This isn't right: the orientation of a manifold is defined in terms of its tangent bundle. The orientation of the tangent bundle is defined in terms of the tangent bundle of the tangent bundle, which is a different object. Someone at mathworld made that mistake and it propagated to us, that's my theory. Anyway, I like my easy proof above well enough to be convinced of its rightness. Comments welcome. -lethe talk + 04:35, 22 May 2006 (UTC)

A new anon changed it back. Without a real reference in hand, I'm unsure what to do, though I'm pretty confident in my argument above. -lethe talk + 19:46, 23 May 2006 (UTC)

If I may inject: I think there is some confusion here between orientation of a vector bundle and orientation of a manifold. A vector bundle is orientable if it admits a smooth choice of orientation (or if its structure group can be reduced to GL+(k)). A manifold is orientable iff its tangent bundle is orientable as a vector bundle. As Lethe points out, the tangent bundle is always orientable as a manifold, meaning the tangent bundle of the tangent bundle is orientable (as a vector bundle). Am I making any sense? The article should clarify this. -- Fropuff 21:04, 23 May 2006 (UTC)

Ah yes. That makes perfect sense, I probably should have known that right away. OK, yes, that needs to be made clear in the article. So anyway, the statement "a manifold is true iff its tangent bundle is orientable" is trivially true because it's the definition of orientability for manifolds. -lethe talk + 00:11, 24 May 2006 (UTC)
I've reworked that short section. Comments welcome. -lethe talk + 00:28, 24 May 2006 (UTC)