Talk:Tangent bundle

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You mean tangent bundle is a term used in auto mechanics, Michael Hardy? Phys

Not that I've ever heard of, but you can get from cars to tangent bundles in three easy steps: Automobile → differential → derivative → tangent bundle. Yes, I'm bored :) Fropuff 00:54, 2004 Mar 15 (UTC)

This is a very nice and modern description. Note that the purely algebraic description of curvature and especially its group theoretical invariance discussion leads via tangent bundling to examples of cosmoligical models, which general relativity needs for an global approach. Hannes Tilgner

Should have some import of sections, the hairy ball theorem, etc. Jake 22:07, 2 Jun 2005 (UTC)

Contents

[edit] discussion at Wikipedia talk:WikiProject Mathematics/related articles

This article is part of a series of closely related articles for which I would like to clarify the interrelations. Please contribute your ideas at Wikipedia talk:WikiProject Mathematics/related articles. --MarSch 14:08, 12 Jun 2005 (UTC)

[edit] Description in the introduction

Oleg, when I read your example I was led to visualize the tangent space as the subspace in the plane consisting of a line tangent to the given point on the circle. I do not know whether this is what you intended, but this does not strike me as a good way of looking at things. Instead, I think of a vector space attached to each smooth point, representing all possible tangent vectors to this point. I don't really think about the space being embedding in anything, which is perhaps why I was disturbed by this. I changed it to how I see things for now. Also, I thought the cylinder construction could be clarified... it seems vague right now. Thoughts? - Gauge 05:05, 19 September 2005 (UTC)

Hi Gauge. You are the differential geometry guy, so I will defer to you in everything in that region. Let me explain my motivation however. The reason I created that example, and the reason I still prefer it as originally stated, is because I found the tangent bundle page very hard to understand for anybody who is not proficient in differential geometry. That's why I have the silly example with tangent lines to the circle, lines which need to be rotated in 3D so that they don't intersect and so that they align smoothly, eventually having no choice than to settle into a cyllinder. I know this is not rigurous, but I have a question: can one at least intuitively think of the tangent bundle that way? Oleg Alexandrov 15:25, 19 September 2005 (UTC)
The tangent bundle to a circle is (isomorphic to) a cylinder, not just intuitively, but in actuality. Viz. TS^1=S^1 \times \mathbb{R} (using the product topology, I suppose, if one wanted to get fancy.) linas 00:50, 20 September 2005 (UTC)

I caught the comment in the last edit; wanted to point out that locally, every tangent bundle is a simple product. That is, one can always find a subset U \subset M where dim U = dim M = n, such that TU \simeq U \times \mathbb {R}^n (for a real manifold, C for a complex manifold). (And that futhermore, a whole collection of these can be found so as to cover all of M. And when they intersect, the intersections are of dimension n as well). That's the whole point of charts: to flatten out the manifold into these rectangular bits. Its how the charts stitch together that makes the global topology of the thing non-trivial. This article should absitively posolutly, mention this. What I can't think of is a good simple example of a manifold with a non-trival tangent bundle; I guess mobius band doesn't really cut it. I'm also thinking it might be good to explain why the mobius band is not the tangent bundle to the circle. linas 04:05, 20 September 2005 (UTC)

Err, well, tangent bundle to S2, but you have to explain a whole pile of stuff, the whole "combing the hair" bit. Somewhere on WP, there's an article on combing the hair ... maybe in line bundle ? linas 04:18, 20 September 2005 (UTC)

Yes, every tangent bundle is locally a product, but not globally. About combing the hair, how about the hairy ball theorem? Oleg Alexandrov 04:23, 20 September 2005 (UTC)
Yep that's the one, but I see that article is less than illuminating as well. Sigh. Parts of WP are a mess. linas 04:25, 20 September 2005 (UTC)

[edit] To do

The state of this article was deplorable, so I rewrote it. Still to do:

  • Expound upon the simplest nontrivial example, T(S2).
  • Comment on the conditions for the tangent bundle to be trivial (existence of a global frame) and the definition of parallizability.
  • Give the relationship between the tangent bundle of M and the frame bundle of M.
  • Mention functorality of the tangent bundle construction and the pushforward map.

I may not get around to these soon, so I invite others to help. -- Fropuff 06:51, 20 September 2005 (UTC)

The rewrite made the article much better, thanks! It is still not accessible for people not knowing anything at all about differential geometry, but that is to be expected in a sense, as this is not a trivial concept. Oleg Alexandrov 15:21, 20 September 2005 (UTC)

[edit] union

why don't you use \bigcup instead of \coprod for disjoint union I didn't do it myself, as I why would rather ask first (is this the right way to handle such a question?)

\coprod is the standard symbol for a disjoint union. See that article for reasons. -- Fropuff 19:48, 22 December 2005 (UTC)
The idea is tbat the tangent spaces at two different points of the manifold may contain the same tangent vector. In this case, it has to be included twice in the tangent bundle, once as a tangent vector at the first point and once as a tangent vector at the second point. -- Jitse Niesen (talk) 22:10, 22 December 2005 (UTC)

[edit] Disputed statement

MarSch, Is there any particular reason why you removed this statement from the tangent bundle article?

When this happens the tangent bundle is said to be trivial. Just as manifolds are locally modelled on Euclidean space, tangent bundles are locally modelled on M × Rn.

-- Fropuff 15:31, 18 January 2006 (UTC)

Yes.
1) "when this happens" is misleading/ambiguous.
2) TM does NOT locally look like M x R^n.
--MarSch 12:19, 19 January 2006 (UTC)

Depends on what you mean by "locally looks like". Certainly, small patches of TM look like small patches of M × Rn. I think this is sufficiently useful intuition to include in the article. -- Fropuff 20:19, 19 January 2006 (UTC)

Of course small regions on the tangent bundle look like small regions on M × Rn; that's how the charts are defined. So agree with Fropuff. Oleg Alexandrov (talk) 21:11, 19 January 2006 (UTC)
Agree with Oleg and Fropuff. This is a standard way to introduce trivial bundles versus arbitrary bundles; see Milnor and Stasheff, for example. There they speak of "local triviality", which can be made precise. - Gauge 21:40, 20 January 2006 (UTC)

[edit] Examples

It's taking me a while to figure all of this bundle business out. Suppose I have an idealized particle somewhere in space. Its position can be modeled by a point in an affine space. Is it correct then that its complete state (position and velocity) represents a point in (member of the) tangent bundle of that affine space? Similarly, would a bound vector (a vector with a defined tail position as well as direction and magnitude) really be a member of a vector bundle? —Ben FrantzDale 06:20, 28 January 2007 (UTC)

Yes and yes. -- Fropuff 16:27, 28 January 2007 (UTC)
Great. Thanks. —Ben FrantzDale 21:46, 28 January 2007 (UTC)

[edit] There is no Canonical Vector Field

Upon opening this article I was shocked to see that it claims the existence of a canonical vector field living on TM. There is no such thing. You can try write it down in a particular coordinate system (see the current article) but it is *not* a well defined (i.e. coordinate independent) geometrical object. For a reference, see Burke p. 93. I recommend just deleting this section, though I didn't want to just do it before giving people an explanation. Milez 00:26, 16 September 2007 (UTC)

There seem to be a misunderstanding. Even if the canonical vector field V is not as well known as the canonical 1-form on the cotangent bundle, there definitely is a canonical vector field on every tangent bundle. This is sometimes called the Liouville vector field or the radial vector field. I don't have Burke so I can not check his exact claim. Does he maybe say there there is no canonical vector field on M? In any case, I added one reference that uses V to characterize the tangent bundle. Essentially, V can be characterized using 4 axioms, and if a manifold has a vector field satisfying these axioms, then the manifold is a tangent bundle and the vector field is the canonical vector field on it. Haseldon 06:15, 20 September 2007 (UTC)
Burke's Applied Differential Geometry only says that (using coordinates as in the article)
\sum_i v_i \frac{\partial}{\partial x_i}
is not a coordinate-independent vector field. -- Jitse Niesen (talk) 07:17, 20 September 2007 (UTC)
Ok, my bad. I actually just misread the old entry due to the notation (I was thinking that the y's were coordinates on the underlying manifold). I totally agree that the quantity appearing on the current page is a well defined geometric quantity. Also, thanks for the info - I came to this page in the first place to learn more about the Liouville field. Milez 18:14, 21 September 2007 (UTC)