Talk:Contact geometry

Hard to understand

I reached this article from Coefficient of friction hoping to get a basic understanding of the topic. However, the first sentence was completely impenetrable to a mere mortal such as myself. I'm not a math genius by any means, but I did get through Calculus 2 without too much trouble, so I'd expect to be able to glean something about the topic. The only thing I understood was parallel parking, but I still have no idea what it's got to do with contact geometry. Can at least the first paragraph be dumbed down to avoid terms like "manifold", "hyperplane", "tangent bundle", "non-degeneracy", "non-integrability", etc.? JesterXXV 20:42, 17 July 2007 (UTC)

Thank you for your comment! Indeed, this article is far from accessible, and can be improved. It's a long-standing tradition here to begin an article with a one-sentence definition of the subject of the title. You should skip that first sentence, which is necessarily technical, and read on. Be aware that contact geometry is a part of differential geometry, probably, 4 or 5 advanced mathematics courses beyond basic calculus, so quite a bit more mathematical knowledge is needed to understand this and similar articles. You may want to try learning analytical mechanics first, and come back to contact geometry later (not necessarily on wikipedia). Arcfrk 01:40, 18 July 2007 (UTC)

P.S. The link at Coefficient of friction was wrong, and I have fixed it.

Misleading

I find the following statement in the article misleading.

"One difference between contact and symplectic geometry is that every 3-manifold admits a contact structure while there are cohomological obstructions to the existence of symplectic structures."

There certainly are differences between contact and symplectic structures. The dimensions they exist for example. But this statement is about contact structures in dimension three and symplectic structures in any even dimension. If you restrict your attention to the two dimensional world, then there is no restriction on the existence of symplectic structures as well. On the other hand, there are cohomological restrictions to the existence of contact structures in dimensions above three !!

I would suggest to delete this sentence or write a complete section about what the cohomological restrictions are . Either way, I would not declare this statement to be a difference between symplectic and contact geometry.

best wishes. S.

I agree 100%. Consider it removed. VectorPosse 03:10, 23 April 2007 (UTC)

Not to be too pedantic about it, but there is sort of a cohomological restriction to the existence of symplectic structures, even for two-dimensional manifolds. They do need to be orientable. But as I said above, I certainly agree that the sentence was misleading. VectorPosse 05:11, 23 April 2007 (UTC)

Applications

I'm not so sure I would classify the result that all three-manifolds possess a contact structure as an application of contact geometry to low-dimensional topology. It's really an application of contact geometry to contact geometry. A true "application" should be used to prove something interesting about low-dimensional toplogy outside of the subset of facts already related to contact geometry. For example, Cerf's Theorem (that any diffeomorphism of the 3-sphere extends to the 4-ball) was reproven by Eliahsberg using contact techniques. VectorPosse 09:45, 13 July 2006 (UTC)

Here's my attempt. Orthografer 20:58, 13 July 2006 (UTC)

Nice job! These are even better than my suggestion since they were not already theorems before contact geometry came along. Of course, it doesn't hurt that Gompf is your advisor.  :) VectorPosse 00:24, 14 July 2006 (UTC)

Help

I'm trying to learn contact geometry, but I am having trouble with the section "Contact forms and structures". I understand few of the terms in the first part of the section, and there are no links to help with understanding. Terms like "kernel of a contact form", "hyperplane field", "symplectic bundle". What is the difference between a "contact structure ε on a manifold" and a "contact manifold"?

I was hoping the part beginning with "As a prime example" would orient me, but I am still having trouble. I understand the 1-form dz-ydx, but it then says the contact plane is spanned by vectors $x_{1}=\partial _{y}$ and $x_{2}=\partial _{x}+y\partial _{z}$. Where did the $\partial$ symbols come from and what do they mean? Are they related to dx and dy somehow? Notice that the x and z variables are interchanged in the definition of $x_{2}$ as compared to the definition of the 1 form. Is that correct, and if so, why?

Can anyone write this with a few more clues to follow? Thanks - PAR 02:09, 14 December 2006 (UTC)

There certainly are a few things that could be done to make some of the points more clear. It's good to have the perspective of someone new to the subject. I would recommend to you that you get a good book on differential geometry and study up if you are serious about leaning more about contact geometry. For example, $\partial _{y}$ and $\partial _{x}+y\partial _{z}$ represent vector fields (sections of the tangent bundle) and so since dz-ydx is a one-form, it can be evaulated on these vectors fields. In fact, dz-ydx is zero when evaluated on these vector fields, and so at a point of the manifold, these vectors span a plane in the "kernel" of the one-form. When I get some time, I'll put some of these things on my to-do list. As you correctly point out, at the very least, some of these items should be linked to places where more information is available. VectorPosse 20:34, 14 December 2006 (UTC)

Thanks - I'm big on having the right book. Are there one or two books on differential geometry that people agree are head and shoulders above the rest? PAR 23:23, 14 December 2006 (UTC)

You should be careful about trusting experts to recommend good books.  :) In all seriousness, many of the "standard" texts of differential geometry tend to be very heavy and difficult to use for learning the subject for the first time. I would put my money on John Lee's Introduction to Smooth Manifolds. I don't know your current level so I don't know how basic you need things. If you need a more undergraduate text as a prerequisite for Lee's book, you can try Do Carmo's Differential Geometry of Curves and Surfaces. Everything in this book is done in two and three dimensions so you can visualize the results. That would prepare you to read Lee's book from the "geometry" point of view. You might need a bit of basic topology as well. VectorPosse 04:12, 15 December 2006 (UTC)

Ok, Thanks. The bottom line is I am trying to understand this paper by Roger Balian. It involves contact geometry and symplectic geometry, so if that changes any recommendation, let me know. Thanks again for your help. PAR 05:32, 15 December 2006 (UTC)

I'm going to carry this conversation into your talk page. VectorPosse 08:58, 15 December 2006 (UTC)

Notation in Definition of Contact Form?

I have seen here on Wikipedia and elsewhere the definition for the contact 1-form written as

$\alpha \wedge (d\alpha )^{n}\neq 0.\$

Despite reasonable exposure to forms, I'm uncertain what the exponent is supposed to indicate here. Clearly it doesn't mean "apply $d$ n times," else the expression would vanish. What does it mean?Trevorgoodchild (talk) 05:25, 26 June 2008 (UTC)

It means "exterior multiply $d\alpha \$ by itself n times". Since $d\alpha \$ is a 2-form, the result is a 2n-form. Note that the exterior multiplication is commutative on even-degree forms, so the result is not necessarily 0 (unlike $\alpha ^{n}\$, which vanishes for n ≥ 2). "Apply n times d to $\alpha$" would have been denoted $d^{n}\alpha \$ and indeed would vanish for n ≥ 2. Arcfrk (talk) 00:52, 27 June 2008 (UTC)

Thanks. Looks like it's been clarified in the article as well.Trevorgoodchild (talk) —Preceding comment was added at 19:29, 28 June 2008 (UTC)

Local vs Global

I saw some books defined contact structure \alpha locally, i.e. \alpha need not exist globally.211.99.194.53 (talk) 11:46, 13 March 2009 (UTC)

http://books.google.com/books?id=RERR4zMDYRgC&pg=PA57&lpg=PA57&dq=Global+contact+structure&source=bl&ots=N9eS_srfup&sig=SJwHvyhXYArR8JWm_14FOfaiZtU&hl=en&ei=-Ua6SdfjDcPQkAX3g_ikCA&sa=X&oi=book_result&resnum=1&ct=result

A contact structure is a global structure, but a contact form need not exist globally. That is, there does not necessarily exist a globally defined 1-form whose kernel is the contact structure. However, if your contact structure is co-orientable then there is such a global 1-form. By co-orientable I mean that there is a nonzero transverse vector field or, equivalently, that the quotient of the tangent bundle by the contact hyperplane field is trivial. --- —Preceding unsigned comment added by 131.215.108.142 (talk) 22:54, 4 December 2009 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Contact geometry/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Needs section on tight vs. overtwisted, among other things. The history could be expanded. Needs illustrations, better history, concise description of applications, wikilinks to the concepts used in definitions... I've downrated it to 'Start' class. Arcfrk 06:02, 30 May 2007 (UTC)

Last edited at 06:02, 30 May 2007 (UTC). Substituted at 01:55, 5 May 2016 (UTC)