Talk:Diffeomorphism

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: Start Class Mid Priority  Field: Geometry

Contents

[edit] Resolved question concerning linking 2003

HELP : can't get the links to Manifold#Differentiable_manifolds to work. When I click (try atlases for instance) I get to the top of the Manifold page. What's wrong ? Pascalromon 14:19, 27 Nov 2003 (UTC)

[edit] Unresolved question concerning unexplained mathematical symbols

I think I followed the definition at the beginning through and including this statement:

Two manifolds M and N are diffeomorphic (symbol being usually \simeq) if there is a diffeomorphism f from M to N. For example

.... So far, so good, but wait ... The very next line has this expression:

\mathbb{R}/\mathbb{Z} \simeq S^1.

Let me take some guesses at what this means in layperson's terms:

  • "The set of real numbers without the integers is diffeomorphic to S1 (whatever S1 means)" ... or is it something like:
  • "The quotient set formed by the real numbers with the integers (see also word salad (mental health)) is diffeomorphic to S1 (whatever S1 means)"

If in about a week or so I see no disambiguation either here or in the article, then I plan to be bold and update the article with one of the above two definitions.

Please be gentle on me, my WikiStress is higher than usual. Vonkje 16:12, 1 Jun 2005 (UTC)

It's the quotient group. You can think of it this way: take a line segment and bend it round into a circle, identifying the end points. Of course that gives you one special point, and there should be no special points at all. Still, it is the correct topological picture. Charles Matthews 16:40, 1 Jun 2005 (UTC)

set minus is usually written with a backslash (or at least looks like one), not a forward slash. So the reals minus the integers would be \mathbb{R}\setminus \mathbb{Z}. As Charles points out, what we have here is\mathbb{R}/\mathbb{Z}, the quotient group of the additive group of real numbers modulo the normal subgroup of integers. This quotient group is set of all cosets of \mathbb{Z}. Cosets will look like {..., -2.5, -1.5, -0.5, 0.5, 1.5, 2.5,...}, which is the coset 0.5+\mathbb{Z}. Notice that the cosets 0+\mathbb{Z} and 1+\mathbb{Z} are actually the same coset, so that this quotient space eventually gets back to where it started, just like a circle. Speaking of circles, Sn or sometimes \mathbb{S}^n is the n-dimensional sphere. A 1-dimensional sphere is also called a circle. So the observation that the quotient group above comes back to where it starts is the starting point of the observation that that quotient group is like a circle. If you see your circle as the set of complex numbers of modulus 1, then your diffeomorphism can be given by the formula x\mapsto e^{2\pi ix}. Then you just have to check that this is a homeomorphism, is smooth, and has smooth inverse. It will also turn out to be an isomorphism of groups, so that it is actually an isomorphism of Lie groups. Hope that helps. -Lethe | Talk 18:36, Jun 1, 2005 (UTC)
Thank you so much ... your added explanation in the article, and your patient explanation on this page helped. ... So where the cosets 0+\mathbb{Z} and 1+\mathbb{Z} are the same would refer to that "special point" that Charles was mentioning where the two ends of the line segment forming the circle meet? Vonkje 15:50, 6 Jun 2005 (UTC)
Charles's description has a special point because he started with a line segment. A line segment has two special points: its endpoints. When you join the two endpoints with glue, they become a single point, and a real circle doesn't have any special points, from a topological standpoint. But the true construction R/Z doesn't suffer this defect. Like I mentioned, 0+Z and 1+Z are the same cosets, and so the same points of R/Z, however, they are not special. It's also true that 0.5+Z and 1.5+Z are the same points, and 0.9+Z and 1.9+Z are the same points, etc. Since I started with an infinite line R, instead of a line segment, there are no special points anywhere, at least not from a topological point of view. When you recall that R and Z are groups, both have a special point, 0, the identity. R/Z then has a special point as well, but this is OK for groups. You could get rid of this special point by viewing R as an affine space with a Z action, and R/Z the set of Z-orbits. -lethe talk 21:23, 29 November 2005 (UTC)

Newcomer to the subject here, but isn't the diffeomorphism in the example above rather given by x \mapsto e^{2\pi i x}? Thanks, 155.198.196.187 11:03, 31 May 2006 (UTC)

Yes. -lethe talk + 12:25, 31 May 2006 (UTC)

[edit] Any chance of a lay example?

I'm linking into here from guage theory, trying to wrap my head around some of these topics. However, IMHO, all of them are written by math nerds for other math nerds. Can't anyone provide a single example of these concepts for us non-math-nerds? The second sentance of ANY article should not start "the mathematical definition is as follows"!

Maury 22:26, 28 November 2005 (UTC)

The definition is very simple. Just read carefully, it says a function which is invertible, and both the function and the inverse are smooth. I assume you know what the words "invertible function" and "smooth" mean. Otherwise, this article will not be helpful to you. Oleg Alexandrov (talk) 22:33, 28 November 2005 (UTC)
And there is an example there, and a good discussion at ==Local description==. Oleg Alexandrov (talk) 22:34, 28 November 2005 (UTC)

it says a function which is invertible, and both the function and the inverse are smooth

But it's buried. If this is the definition, it should be the second sentance of the article. The definition should be blocked off into its own para. Like this... Maury 13:06, 29 November 2005 (UTC)

I don't think it was buried. It was the third sentence in the article (if you count the wording "The mathematical definition is as follows." as a sentence). But I do agree that it looks better now. Oleg Alexandrov (talk) 20:18, 29 November 2005 (UTC)

R/Z : Divide any real number by any integer; throw away the integer part of the answer; what remains is a number, x, between -1 and +1: -1 < x < +1; this set of numbers is R/Z. Now use the formula, z(x) = exp( i*pi*x ); "exp" = "e to the"; "*" = multiply ); for any value of x in our set, R/Z, z(x) is a point on the unit circle in the Complex Plane. This is the circle centered on 0 (the origin), and having radius, 1. For example, the x-value 0 maps onto the point z=1; the x-value 0.5 maps onto the point z=i, the x-value 1 (we'll allow it to sneak in) maps onto the point z=-1, and the x-value -0.5 maps onto the point z=-i. So using this sequence of x-values we have traversed the unit circle counterclockwise, E, N, W, S. If we allowed the x-value, x=-1, it would also map to z=-1, since exp( i*pi*1) = exp( -i*pi*1) = -1 (Euler's identity, q.v.). This unit circle is called S1, because a circle is a 1-dimensional "sphere" (embedded in 2-dimensional space), a sphere (S2) is a 2-dimensional object (embedded in 3-dimensional space), etc. The integer part of the cosets (the "Z" part), when inserted into the formula, sends you to z=+1, z=-1, (E and W), again and again, contributing nothing new. So we have mapped a line segment (-1 to +1) on the Real axis, onto the unit circle, of which it is a diameter. If this segment is traversed from -1 to +1, then the unit circle will be traced from -1, counterclockwise, all the way around to -1 again.

[edit] Manifolds or just normed vector spaces?

The current definition of a diffeomorphism is given in terms of a map between manifolds, but a diffeomorphism can be defined over much more general spaces (namely, normed vector spaces - see e.g. Marsden, Ratiu, & Abraham, "Manifolds, Tensor Analysis, and Applications"). Why is the more specific case used here? Trevorgoodchild (talk) 05:23, 29 January 2008 (UTC)

I think the most general case is even more general than that, you can have diffeomorphisms over infinite-dimensional manifolds (which look locally like a normed vector space). But I guess going that general could make the article hard to understand. The most used cases are in manifolds over R^n anyway. If you want, you can add somewhere at the bottom a section on generalizations, but I'd prefer that at least the first several sections in the article be about the "usual" R^n manifolds case. Oleg Alexandrov (talk) 15:44, 29 January 2008 (UTC)

[edit] Relationship with homoeomorphisms

The "model example" section says that the diffeomorphism f "happens to be" a homoeomorphism. From the definition, it seems to me that all diffeomorphisms are homoeomorphisms, since all differentiable functions are continuous. If that is the case, "happens to be" seems misleading. LachlanA (talk) 22:45, 6 May 2008 (UTC)