Talk:Contact geometry

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[edit] 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)

[edit] 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 and . Where did the 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 x₂ 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, and 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)