Talk:Tangent bundle

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 &rarr; differential &rarr; derivative &rarr; 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)

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)

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)

Tangent bundles are a special type of fibre bundle (when the manifold is connected as required in the definition of a fibre bundle) with fibre homeomorphic to a tangent space of the manifold. Also, since we are considering the direct sum (i.e disjoint union), of the tangent spaces, each point has a neighbourhood U such that the direct sum of all tangent spaces on U is actually homeomorphic to the product space U X T. In fibre bundles, each point is required to have a neighbourhood whose preimage is homeomorphic to a product space so the example with the circle is not surprising.

You have raised a good point; I think that it should be mentioned in the article that the tangent bundle is a special type of fibre bundle.

Topology Expert (talk) 04:01, 28 August 2008 (UTC)

Err, well, tangent bundle to $$S^2$$, 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)

To do
The state of this article was deplorable, so I rewrote it. Still to do: I may not get around to these soon, so I invite others to help. -- Fropuff 06:51, 20 September 2005 (UTC)
 * 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.
 * 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)

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)

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 &times; 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 &times; 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 &times; 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)

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)

It seems weird in the "Examples" section that it says that "The tangent bundle of the circle is also trivial and isomorphic to $S^1\times R$." Given the definition of trivial previously, that is redundant (and hence confusing, because the reader might think that he has misremembered the definition of trivial.) Thomaso (talk) 16:05, 13 February 2013 (UTC)

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)


 * There seems to be some confusion resulting from Burke's claim. The expression that Jitse Niesen wrote above is not a coordinate expression for the canonical vector field V of TM. Additionally, I have just reverted an erroneous "correction" which somebody made to the page using the above expression. The correct expression is
 * $$ V = \sum_i v^i \frac{\partial}{\partial v^i}. $$
 * I recommend ignoring Burke's book, as it simply confuses matters. Leo C Stein (talk) 21:16, 6 October 2017 (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)


 * IMO the so-called canonical vector field would be better described if you avoided local coordinates. See double tangent bundle for how to do this. Rybu (talk) 05:11, 10 September 2008 (UTC)


 * The canonical vector field lurks in several articles with this same precise meaning, and I would like to point out that it is not in any way special to the tangent bundle. Namely, for any vector bundle (E,p,M) one can define the canonical vector field V as the infinitesimal generator of the Lie-group action
 * $$ \mathbb R\times (E\setminus 0) \to (E\setminus 0) \quad ; \quad (t,v)\mapsto e^tv $$
 * on the slit vector bundle (E/0,p|E/0,M), which is obtained from (E,p,M) by removing the zero section. IMO this construction should be moved into the article on vector bundles. Lapasotka (talk) 09:58, 4 March 2010 (UTC)

Higher order Tangent bundle deleted by TopologyExpert
TopologyExpert deleted the higher order tangent bundle section saying it was completely incorrect. What exactly do you see as incorrect, because it looks largely correct to me and the people who created it. The article clearly states the tangent bundle is an object that exists for a smooth manifold, moreover, later in the article it clearly states that the tangent bundle is itself a smooth manifold, so the concept of the tangent bundle of the tangent bundle is perfectly well-defined. The bit about the tensor is a little uninformative but it was otherwise totally on the mark. So what do you perceive as totally wrong? Rybu (talk) 04:24, 3 October 2008 (UTC)


 * I brought higher order tangent bundles back. I moved them ahead of the canonical vector field on the tangent bundle since the double tangent bundle is needed to make sense of that section.  I also added a "role" section where it's explained that the point of the tangent bundle is to serve as domain and range for the derivative of a smooth map of manifolds. Rybu (talk) 03:28, 4 October 2008 (UTC)


 * Yes, it is true that the total space of the tangent bundle of a smooth manifold is a smooth manifold. I deleted that section because at the time there were no reasons as to why these 'higher order tangent bundles' are of interest. Also, the section claimed (wrongly) that higher order tangent bundles are like 'higher order derivatives' which is meaningless. Of course, higher order derivatives of a smooth function between smooth manifolds can be induced on higher order tangent bundles. But the section is good know so I have no intentions of removing it again.

Topology Expert (talk) 03:34, 9 October 2008 (UTC)

Request to clarify intro and intro figure
The intro, in my view, is frustratingly opaque, possibly for want of a few very minor stepping stones. I'll try to point out some sticky points.

First, could someone please annotate the initial "TM = ..." definition a bit more thoroughly? Specifically:

-- What is TM? Is that two separate symbols, with T and M having individual meanings, or a single combined symbol denoting "entire tangent space for manifold M". And if so, where does tangent bundle appear in this algebra? Or does TM mean "tangent bundle"?


 * "TM" denotes the tangent bundle of the manifold M (in a vague sense, one could view T as an "operator" sending a manifold to its tangent bundle, "analogous" to the "operator" D sending x to its differential "Dx"). The tangent space of a manifold at a particular point is denoted by "TxM", where the point in question is x. -- PS T  01:54, 9 December 2009 (UTC)


 * Thanks, that was useful. So "T" = "Tangent bundle of". And overall the algebra is saying that the tangent bundle of the entire manifold (eg: circle) comprises the disjoint union set of the tangent bundles at each individual point on the manifold. Gwideman (talk) 03:47, 9 December 2009 (UTC)

-- What the heck is that upside-down pi? How would a reader go about looking it up? I looked at the disjoint union link and didn't find it there. Or is this expression intended to introduce and define upside-down pi?


 * The "upside-down pi" (although one usually does not refer to it in that manner)

serves to write the tangent bundle as the disjoint union of its tangent spaces (as a set). This means that an element of the tangent bundle is in one and only one tangent space (indeed, it is a tangent vector to the manifold at some point). Topologically, however, the tangent bundle is not necessarily the disjoint union of its tangent spaces (that is, its topology is not necessarily the "disjoint union topology") (do not worry about understanding this, if you do not wish to; it is not necessary). -- PS T  02:00, 9 December 2009 (UTC)
 * Don't worry, I don't understand the distinction being made here :-). But as a consolation it would be helpful to know how one does refer to the upside-down pi symbol. Gwideman (talk) 03:47, 9 December 2009 (UTC)
 * One usually refers to the "upside-down pi" as a disjoint union. Sorry for not mentioning that. -- PS T  10:49, 9 December 2009 (UTC)

OK, now moving on to the informal circle example, which could be the way into this topic for those stumped by upside-down pis. Though the example has a plain english description, the description doesn't relate the figure to the introduction.

Perhaps the correspondence is as follows: -- The blue circle is the manifold (as the caption notes.)


 * Correct. -- PS T  02:21, 9 December 2009 (UTC)

-- The red lines in the top figure are the tangents at a few example points around the circle, and the concept extends to the complete set of tangents at every point around the circle.


 * Clarification: Each red line is the tangent space of the circle at a particular point (and not merely a tangent vector). The totality of all such red lines is the tangent bundle.
 * Got it. In this case one of these tangent spaces is one dimensional, occupying the same space as the infinite-line tangent. Hence containing all possible tangent vectors for that point on the manifold.Gwideman (talk) 03:47, 9 December 2009 (UTC)

-- Murky: Is each red line the "tangent space at x" described in the intro? Ie: in this instance, the tangent space is 1-D? Or is each red line actually the edge view of a tangent plane?


 * Since the circle is one-dimensional (locally it "looks like a curved line segment"), the tangent bundle only encompasses one-dimensional tangent spaces (that is tangent lines). If it were a sphere instead of a circle, one would consider two-dimensional tangent spaces (that is, tangent planes). The top of the image shows the tangent bundle horizontal to the circle. However, since one aspect of mathematics is the development of different views of particular phenomena, one seeks for another view of the tangent bundle which is simpler. The bottom part of the image, that is the totality of all red lines pointing vertically, shows the tangent bundle as an infinite cylinder (a far simpler description than the top part of the image). -- PS T  02:21, 9 December 2009 (UTC)
 * OK, got the one-D part. But here's where I come off the rails. What makes it OK to regard each tangent space as equivalent to a one-D space in a completely different direction? (Because apparently I was wrong about the cross operation, below.) Gwideman (talk) 03:47, 9 December 2009 (UTC)
 * I would just like to say that Lie groups are parallelizable but that would not be right ;). Before you read what follows, just note that profound mathematicians such as John Milnor, Raoul Bott and co. have done major work in trying to determine which spheres (in higher dimensions) have tangent bundles that can be represented in the "vertical manner" that is done for the circle (so you are not the only one who is trying to figure that out (seriously, it was a tough problem; people eventually determined that the circle (one-dimensional sphere), the three-dimensional sphere and the seven-dimensional sphere are the only ones)). Intuitively, the circle is "uniform" in that given any two points on the circle, there is a smooth map carrying one point to the other (rotation does this, for example). Therefore, the tangent spaces of the circle "free-flow" in that they spread around the circle in a "circle-like manner" (cylindrical) (the fact that there is a smooth map carrying one point of the circle to any other implies that the map induces a linear isomorphism of tangent spaces at the respective points, more precisely). In particular, one can arrange the tangent bundle in a vertical manner, while essentially retaining its internal structure. If you did not understand that which I have said at first glance, don't worry too much. This particular point is not too important in a first look at tangent bundles (but when you gain more experience, you will see that this is indeed trivial ;)). -- PS T  11:07, 9 December 2009 (UTC)

-- Taking each of the red line tangents, and substituting a line that's perpendicular to the plane of the circle corresponds (I'm assuming) to the cross (X) operation in the third part of the TM= expression, and results in the lower figure. This produces lines that are non-overlapping but together form a smooth surface (here a cylinder).


 * The "cross" (see below) serves to represent an element of the tangent bundle (which is of course a set) as an ordered pair; it can be uniquely described by knowing the tangent space to which the element belongs. Equivalently, one must know the point to which the tangent space is associated, as well as the particular element of the tangent space considered (that is, the particular tangent vector). Any element of the tangent bundle can be described as (x, v), therefore, where x is in the manifold and v is some vector in the tangent space at x. -- PS T  02:21, 9 December 2009 (UTC)

-- Murky: What kind of operation is this cross? It doesn't seem to be ordinary vector cross product, because U{x} doesn't seem to be a vector, but rather a set. (The figure caption describes this as "considering and joining", so maybe cross is the "considering" operator.)


 * See cartesian product. -- PS T  02:21, 9 December 2009 (UTC)


 * Ah-hah. OK, so cross (cartesian product) here just assembles the x points with the v vectors to make a set whose members are (x,v). So cross does not explain (in the example) the move from tangent lines/spaces to lines/spaces arranged perpendicular to the plane of the circle. (As I noted above.) Gwideman (talk) 03:47, 9 December 2009 (UTC)
 * Quick clarification: The cartesian product represents elements of the tangent bundle as rdered pairs; the first "coordinate" represents the point at which one is considering the tangent space, and the second "coordinate" represents the particular tangent vector within the tangent space. For instance, if I was considering an element of the tangent bundle of the circle, I would have to consider which "red line" the element is in (according to the image); so I would have to know which point the red line is tangent to, as well as the corresponding vector on the red line. These two pieces of information would describe the element of the tangent bundle uniquely. If I just said "unit vector", I would not be able to determine the particular element of the tangent bundle since each and every red line has a unit vector; to determine the element uniquely, I would have to know which red line to choose. Likewise, if I only knew the red line to which an element of the tangent bundle belongs, I would not be able to determine the element since there are "many" vectors on the red line; I would have to know which vector to choose (in addition to the given red line). -- PS T  10:47, 9 December 2009 (UTC)

Clearly at this point I don't understand this subject enough to make complete sense of it, but perhaps my description will allow someone to troubleshoot. Gwideman (talk) 14:39, 8 December 2009 (UTC)


 * Your comments are much appreciated for they show that the lead could be explained in a simpler fashion. However, this subject is usually referred to as differential topology (or differential geometry); one must usually have a background of point-set topology and linear algebra to appreciate it. I am not saying that you do not have the background to appreciate the subject, but rather that a complete acquiantance with these subjects aids one in appreciating differential topology. Hope this helps (note that I have responded to your points in detail above). -- PS T  02:21, 9 December 2009 (UTC)
 * Thanks for your patient explanations. I feel pretty close to a light-bulb moment on this one, with the remaining mystery being the jump from tangent vector spaces to a bunch of vector spaces conveniently arrange all parallel to each other. Gwideman (talk) 03:47, 9 December 2009 (UTC)
 * No problem - I am happy to help. If you are interested in furthur questions about tangent bundles (or any other math-related topic, for that matter), a good place to visit would be Reference Desk/Mathematics. Cheers, -- PS T  11:09, 9 December 2009 (UTC)

-- It seems to me that the first equation in this article is a bit odd, particularly the last expression. First of all, is $$\{x\}$$ supposed to refer to the set of all $$x\in M$$, or simply to the set containing only the single point $$x$$? If the former, then there is no need for union symbol since this is already part of the definition of the Cartesian product, and also in that cast we could just refer to $$x\in M$$ as simply $$M$$ (since we're only defining a space rather than elements of that space). If the latter, why not just refer to $$x$$ rather than $$\{x\}$$? It would seem to me that the clearest way to define the tangent bundle would be just $$TM \equiv \coprod_{x\in M} T_x M = \{ (x,v) \mid x \in M, v \in T_x M \}$$ similar to on Wolfram.

Also isn't it true that each $$T_x M$$ is mathematically just $$\mathbb{R}^n$$, and is only unique in its interpretation/significance? Given thats the case, wouldn't it also be correct to identify $$T_x M$$ as $$M \times \mathbb{R}^n$$? Somebody please me know if I'm going astray here since I'm new to differential geometry. AnIsometry (talk) 11:25, 21 December 2009 (UTC)

Request for clarification, cont'd
The previous section was very helpful in clarifying some problems. Here are a few items that the discussion has prompted:

1. The upside-down pi symbol is used here to stand for "disjoint union". However, the disjoint union article, which this article references, uses the \bigsqcup, not the coproduct symbol \coprod. I think it would help to change to the bigsqcup, unless coproduct is intended. (A previous talk section "union" suggested the plain bigcup symbol, which the other article would argue against, though I do see bigcup used that way in many other web sources.)

2. Footnote note 1 is a bit mangled on the way to almost explaining a point I'm about to make below. I think the phrase "align them in a plane perpendicular plane" doesn't make sense. Possibly it should be "align them perpendicular to the plane of the circle".

3. From the "Request for clarification" discussion of the last few days, and my finally understanding the notation of the "TM=..." algebra, I think I see where the intro got me way off track, and how the intro could be improved.

Not understanding the notation, I tried to make it make sense as a statement of how to calculate the arrangement of the tangent bundles in space. I.e.: if we were to apply that algebra somehow to the circle example, it would "calculate" the answer: a set of "vertically" arranged 1-D spaces. Instead, I now see that the algebra is merely a statement of the requirements: a set of vector spaces, one vector space for each point on the manifold, with each vector space disjoint from the others (and make them collectively contiguous and smooth? -- though I don't see that in the requirements). The actual solution -- 1-D vector spaces perpendicular to the circle's plane, and arranged around the circle so as to form a cylinder -- was devised by separate thinking not shown here, and is "the answer" because it satisfies the requirements.

Now, if I'm right about this, it would be helpful for the intro to point this out: That having stated "Tangent bundle" in terms of requirements, we are looking for solutions for those requirements, not calculating the shape of the tangent bundle.

Thanks again for the discussion Gwideman (talk) 12:54, 9 December 2009 (UTC)

Canonical vector field: tangent space is naturally a product
Clearly, "the tangent space of a vector space W is naturally a product, $$TW \cong W \times W$$" should read "the tangent bundle of a vector space...", which raises the question why we can apply this product structure to the tangent bundle $$T(T_x M)$$ at each point. How do we get $$T(T_x M)$$ from $$T(TM)$$? Also, is it obvious that the given "informal" definition gives a coordinate-independent vector field? If not, what is the point of this lengthy explanation? Doesn't the map $$(x, v) \mapsto (x, v, 0, v)$$ given below achieve the same less cryptically? I would also like to see an explanation of the phrase "splitting the map via". Theowoll (talk) 00:07, 12 March 2018 (UTC)