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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)
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'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)
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 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)
I have seen here on Wikipedia and elsewhere the definition for the contact 1-form written as
Despite reasonable exposure to forms, I'm uncertain what the exponent is supposed to indicate here. Clearly it doesn't mean "apply n times," else the expression would vanish. What does it mean? Trevorgoodchild ( talk) 05:25, 26 June 2008 (UTC)
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)
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)
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
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)
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'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)
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 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)
I have seen here on Wikipedia and elsewhere the definition for the contact 1-form written as
Despite reasonable exposure to forms, I'm uncertain what the exponent is supposed to indicate here. Clearly it doesn't mean "apply n times," else the expression would vanish. What does it mean? Trevorgoodchild ( talk) 05:25, 26 June 2008 (UTC)
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)
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)