Talk:Tubular neighborhood

Question

A close up of the figure above. The curve is in blue, and its tubular neighborhood T is in red. With the notation in the article, the curve is M′, the space containing the curve is M, and T=j(N).

A schematic illustration of the normal bundle N, with the zero section N₀ in blue. The transformation j maps N₀ to the curve M′ in the figure above, and N to the tubular neighbourhood of M′.

By GFDL, I adapted this entry to PlanetMath (with due acknowledgement, of course). Now they have the question:

what is i exactly? how does it establish a bijective correspondence between N₀ and $M'$?

How should I respond to that? PrimeFan 22:40, 13 June 2007 (UTC)

One should read the article zero section for that. Intuitively, if you look at the top picture, you see the tubular neighbourhood T and the curve M'. If you look at the bottom picture, you see the normal bundle, with the zero section in blue. Each of those red lines is a vector space, and its 0 is on the blue curve. The mapping i maps the blue curve at the bottom to the blue curve on top, and each of the infinite lines on the bottom, to each of the finite lines (they can also be curves) on top. Does that answer the question? Oleg Alexandrov (talk) 03:06, 14 June 2007 (UTC)

Since you did not copy the pictures to the planetmath article, the intuitive explanation there does not make much sense, perhaps. Oleg Alexandrov (talk) 15:30, 14 June 2007 (UTC)

Oh, of course. Thank you very much, you've been a heck of a lot more helpful and understanding than anyone over there. We've got a good thing going here at Wikipedia.

If the pictures are also GFDL I will copy them to EPS and upload them over there. Otherwise I'll try to reproduce them with Mathematica. PrimeFan 23:43, 14 June 2007 (UTC)

The pictures are in the public domain, together with their code (at the picture). You can run the matlab codes to get better quality eps pictures than converting them from png. If you don't have Matlab installed, I can email you the eps versions (you could send me an email first so that I know your email address, if you want them). Cheers, Oleg Alexandrov (talk) 03:05, 15 June 2007 (UTC)

How about the tubular neighbourhood theorem?

Isn't there a theorem that asserts that tubular neighbourhoods always exist? I'd add it myself, but I'm not sure of the exact hypotheses needed. --345Kai (talk) 18:27, 13 August 2011 (UTC)

3rd image

is File:Tubular neighborhood3.png really a tubular neighborhood? The red lines there aren't normals unlike in say File:Tubular neighborhood2.png. In 2D geometry a simple translation of a (progenitor) curve does not, in general, produce a curve parallel to its progenitor (except in trivial cases like translating a line). JMP EAX (talk) 04:56, 15 August 2014 (UTC)

Too confusing: assessment

I have removed the warning at the top of the lead which says the article [may be] "too confusing." Given the obscurity of the topic, it is not overly technical, and any interested parties should have a relatively intuitive understanding of it as is presented. Additionally, considering this objection is nearly a decade old, it has been removed. SpiralSource (talk) 02:36, 6 July 2022 (UTC)