Talk:Immersion (mathematics)

Definition Correctness

Am I wrong in thinking that f having constant rank equal to dim M implies f is an immersion only if M is a manifold without boundary? Seems this condition should be mentioned (assuming I'm not being dumb). —Preceding unsigned comment added by 210.1.197.1 (talk) 11:12, 24 February 2009 (UTC)

Difficult to Understand

An english student who got an A in two different math's gcse's cannot even begin to make sense of this. Please can someone deal with the jargon. I am very curious as to what the hell this page is about. Thankyou. —Preceding unsigned comment added by 81.99.106.40 (talk • contribs)

This is a very nice article that explains the idea very well. The use of links - the words in blue - will take you to other pages that will explain all of the other words. It would be impossible to define every single part of an article. Wikipedia is a body of knowledge; the idea is for you to cross reference, i.e. use the links that are highlighted in blue. There are two other points I would like to make. There are two accepted abbreviations for mathematics. The first is "maths" and the second is "math"; the former is accepted British English, the latter is accepted American English. The phrase "math's gcse's" means that the "gcse's" belong to math. That is not what you want to say. Secondly, in England the secondary school (i.e. ages 11 to 16) examinations are GCSEs (General Certificates of Secondary Education). To write "GCSE's" means that whatever follows belongs to the GCSE. The plural form of an acronym (like almost every other word) is formed my adding an s and not 's . Allow me to correct your sentence: "An english English student who that got an A [grade] in two different math's gcse's maths GCSEs cannot even begin to make sense of this." Even though this sentence is now grammatically correct it still doesn't make sense: when I completed my GCSEs there was a single mathematics exam. Only at A-level are their two mathematics exams: maths and further maths. This artice covers an idea that I did not meet until my third year of undergraduate studies. It's no wonder that a GCSE student doesn't understand. That's why the links are there: to help you understand. Research mathematicians don't sit around all day solving GCSE problems. We solve difficult problems that require many years of study to understand. Dharma6662000 (talk) 03:21, 25 August 2008 (UTC)

RE: Examples and Properties

In the section Examples and Properties it is claimed that the sphere can be immersed onto a closed disk in the plane. Could someone please give an explicit mapping to do this. It is not obvious how to show this, for example: a simple orthogonal projection fails to be an immersion along a curve. The critical points being the equator of the sphere and the critical values being the boundary of the closed disk.  Δεκλαν Δαφισ   (talk)  09:47, 23 November 2008 (UTC)

It can't be. A *closed* surface cannot immerse into the plane. So I removed the error. A formal proof would use something along the lines of: by Brouwer's invariance of domain, a small enough open set must map to an open set in the plane. ("small" here mean small enough so that the immersion is 1-1 on the set). This means the image is open in the plane, but the image of a closed surface in the plane is closed and cannot be open also. So the result isn't true for a topological immersion, let alone a differentiable one.

Addendum: your example is not differentiable along the equator so strictly speaking, critical point doesn't make sense there. —Preceding unsigned comment added by 64.180.3.20 (talk) 06:01, 24 November 2008 (UTC)

I'm sorry, but your addendum is incorrect. The example is fine provided the right coordinate chart is used. Since the sphere is not a parametrisable manifold (there is no global differentiable parametrisation) the choice of coordinate chart is key. Consider the circle in the yz-plane of xyz-space. Form the sphere given by the equation x² + y² + z² = 1 by rotating this circle about the z-axis. This gives a differentiable parametrisation of the sphere away from the poles (0,0,-1) and (0,0,1). This parametrisation is (θ,t) → (sin(θ)cos(t), cos(θ)cos(t),sin(t)) where t is the parameter around the circle and θ the rotational parameter about the z-axis. We assume that 0 ≤ t < 2π, t ≠ π/2, 3π/2 (i.e. we're not at the poles), and 0 ≤ θ < π. Projecting this sphere onto a plane orthogonal to the z-axis gives, in local coordinates, (θ,t) → (sin(θ)cos(t), cos(θ)cos(t)), and this is clearly differentiable! This mapping, which is a mapping from the plane to the plane, is singular when cos(t) = 0 or sin(t) = 0. Since we are working away from the poles we assume that cos(t) ≠ 0. The singular points are then given by sin(t) = 0, i.e. the equator. In fact the equator consists of fold points. This type of point is stable (a small deformation of the mapping will not destroy a fold point), as was shown by Hassler Whitney's famous list classifying stable singularities of differentiable mappings from the plane to the plane, there are diffeomorphisms, fold points, and cusp points. These three types of singularity can come from projections. The normal forms (i.e. equivalence class representatives, where the equivalence relation is diffeomorphic changes of variable in both the source and the target) for these singularities are (x,y) → (x,y), (x,y) → (x,y²) and (x,y) → (x, xy + y³) respectively.  Δεκλαν Δαφισ   (talk)  10:55, 24 November 2008 (UTC)

Table of Contents

The article needs to be restructured slightly with more section headers. I have forced a table of contents with the command __FORCETOC__ and this show the problem. The "introduction" rambles on and on. There is a lot of information included in the "introduction" which should be in its own section.  Δεκλαν Δαφισ   (talk)  09:49, 23 November 2008 (UTC)

Someone seems to have taken care of this without saying so on this page. Anyway, at least it's been done.  Δεκλαν Δαφισ   (talk)  10:43, 28 November 2008 (UTC)

Proposed Change

I suggest that the section on multiple points needs to be moved: we already have an article on embeddings. Also the section on regular homotopies needs to be moved: we already have an article on homotopies. Any comments?  Δεκλαν Δαφισ   (talk)  10:39, 25 November 2008 (UTC)

It is not clear where the section on multiple points should be moved to, unless someone is willing to write an article about multiple points in singularity theory and algebraic geometry. In fact I do not think it should be moved at all. It does not help that there is already an article on embeddings, since embeddings are precisely the immersions without multiple points. The nature of the multiple points classifies immersions; for example, immersions of a circle in the plane are classified up to regular homotopy by the number of double points. In surgery theory immersions are used to decide to decide if double points can be cancelled. As to moving regular homotopies to homotopy theory this is not such a good idea either: homotopy theory is a very big subject, whereas the concept of a regular homotopy is only of interest as a classification tool for immersions. ranicki (talk) 12:52, 25 November 2008 (UTC)

If this is the case ranicki then the multiple points section needs to be filled out with this very information. As it stands it is a pointless aside. If more detail were to be added then in would be worthwhile. Maybe you would like to make the changes ranicki?  Δεκλαν Δαφισ   (talk)  22:32, 26 November 2008 (UTC)

I have expanded the multiple point section.ranicki (talk) 06:17, 27 November 2008 (UTC)

Good work! By the way, it's the norm to put one extra colon when replying to a post as to indent your reply in relation to the previous comment.  Δεκλαν Δαφισ   (talk)  21:58, 27 November 2008 (UTC)

Immersions of surfaces into the plane

I have reverted[1] the addition of the statement that the sphere can be immersed into the plane since that statement is mathematically incorrect. No closed surface (that is, a compact surface without boundary) can be immersed into the plane, not even in topological sense (that is via a continuous locally injective map). This is a basic fact that follows from the Invariance of domain theorem: by that theorem any continuous locally injective map $f:S\to \mathbb {R} ^{2}$ from a closed surface S to the plane is an open map. Hence its image is open, but it is also compact (since f is continuous and S is compact). That is a contradiction since there are no compact open nonempty subsets in the plane. Nsk92 (talk) 17:25, 8 December 2008 (UTC)

This problem was already mentioned in the section RE: Examples and Properties. What happened, was it not removed in the first place, or was it added again after being removed?  Δεκλαν Δαφισ   (talk)  18:54, 8 December 2008 (UTC)

It was re-added[2] after it had been removed[3]. Nsk92 (talk) 18:56, 8 December 2008 (UTC)

Non-differentiable Immersions?

Is immersion a topological concept or a concept of differential geometry? I have seen immersion used in non-differentiable settings, in particular, in polyhedral settings (like a self-crossing polygonal curve as an immersion of the circle). So I believe the very first sentence is already misleading. --Günter Rote, 130.133.8.114 (talk) 11:29, 30 January 2009 (UTC)

Immersion is a differential idea. The definition of an immersion is a differentiable map whose differential is injective.  Δεκλαν Δαφισ   (talk)  13:06, 1 February 2009 (UTC)

Move?

Since there is the concept of immersion in algebraic geometry, I think this article should be moved to Immersion (differential geometry) or Immersion (differential topology) with the current article title becoming a disambiguation page for the uses of the the term immersion in mathematics. Thoughts? RobHar (talk) 16:37, 11 December 2010 (UTC)

Sounds reasonable to me. We should make sure nothing important is lost in the move, but I figure that's common-sense and whomever does it will be careful about that. Rybu (talk) 04:11, 13 December 2010 (UTC)

Definition of regular homotopy wrong?

I think the map h: M x[0,1] --> N need not be smooth for it to be a regular homotopy. I think that it is called a regular homotopy if: (a) it is continuous; (b) each restriction h_t : M --> N is a C^1 immersion; (c) the induced map of tangent bundles (h_t)_* : TM x [0,1] --> TN is continuous. Notice that no differentiability in the t variable is assumed here. I got this from the article of S. Smale "A classification of immersions of the two-sphere". — Preceding unsigned comment added by Gigou (talk • contribs) 20:11, 28 May 2012 (UTC)