The study of topological invariants of contact manifolds, the odd-dimensional counterpart of symplectic manifolds. Classical mechanics has a 2n-dimensional phase space and one time dimension and the definition of contact manifolds axiomatizes this structure by positing a maximally non-integrable hyperplane distribution. Fields of application include low-dimensional topology, 3-manifolds and knots, geometrical optics, thermodynamics and control theory.
Questions tagged [contact-topology]
96 questions
14
votes
3 answers
Non-integrability of distribution arising from 1-form and condition on 1-form
Suppose $M$ is a $(2k+1)$-dimensional manifold on which a 1-form $\alpha$ is defined. $M$ is termed as a contact manifold if the distribution arising from $\alpha$ is nowhere integrable, i.e. if:
$$\xi_q=\{v\in T_qM:\alpha(v)=0\}$$
is a distribution…
MickG
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13
votes
1 answer
Convex boundary of a symplectic manifold
Given a symplectic manifold $(M,\omega)$, suppose that $\partial M$ is of contact type. A Liouville field on a symplectic manifold is a vector field $X$ such that $\mathcal L_X \omega = \omega$. We say that $\partial M$ is convex if there is an…
user98602
10
votes
1 answer
An analogue of the Poisson bracket in contact geometry?
McDuff and Salamon define an analogue of the Poisson bracket in contact geometry on page 135 in the third version of Introduction to Symplectic Topology. The definition is the following.
Let $(M,\alpha)$ be a contact manifold with a globally…
Chris Kuo
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9
votes
1 answer
Coorientation of contact structures
When reading about contact geometry one quickly encounters the notion of a cooriented contact structure/form. But I do not seem to be able to find a definition of "coorientation".
In some places they define a cooriented contact structure as one…
NDewolf
- 1,979
9
votes
2 answers
Why are contact structures studied from a cohomological, rather than homological, perspective?
As far as I know, contact structures are studied from a "cohomology/dual" perspective, meaning mostly from the perspective of contact forms and their respective kernels, instead of from a "homological" version using either contact vector fields or…
gary
- 4,167
7
votes
0 answers
An imcomprehensible proof on Arnold's Lectures on PDE, contact diffeomorphism
In Lectures on Partial Differential Equations, Arnold gives a theorem in the chapter Huygens' Principle in the Theory of Wave Propagation:
Theorem 1 (the theory of support functions). The manifold of 1-jets of functions from the sphere $S^{n-1}$…
Yai0Phah
- 10,031
7
votes
1 answer
How to Interpret "Lie Derivative of a Form Equals 0"(Contact-Reeb Vector Field )
Everyone:
I'm trying to understand better the meaning of a diff. form being constant along a flow;
more specifically:
One of t properties of a Reeb vector field X associated to a contact form w
(meaning that the contact distribution is…
FBD
- 392
6
votes
1 answer
Linking number of specific Reeb orbits in a toric domain ($S^3$ diffeomorphic)
Consider a toric domain defined by the region bounded on the first quadrant by a function $f:[0,a]\mapsto [0,b]$ with $a,b>0;f(0)=b,f(a)=0,f(x)>0 \hspace{2mm} \forall x\in [0,a)$. We know that $\forall x$ s.t. $f'(x)\in \mathbb{Q}$ and $x\in (0,a)$,…
kvicente
- 399
6
votes
0 answers
closed orbits in the flow of the Reeb vector field of a contact manifold
This is once again from Hansjorg Geiges' introduction to contact topology. In Gray Stability Theorem in $\S2.2$ asserts that one can achieve stability of contact structures. However, one can't in general achieve stability of contact forms. An…
Karthik C
- 2,627
6
votes
1 answer
Geometric meaning of the contact condition?
I am trying to understand contact structures. The definition of a contact manifold is this:
Let $M$ be a $2n + 1$-manifold and let $\omega$ be a differential $1$-form such that $\omega \wedge (d\omega)^n \neq 0$ pointwise. Then $M$ is a contact…
self-learner
- 1,217
5
votes
1 answer
Singular Vector Fields as Basis Elements in Contact Planes.
I'm trying to generate specific examples of contact structures in $\mathbb R^3$, but I'm running into something strange. It seems like the contact planes associated to the structure (using cylindrical coordinates in $\mathbb R^3$)
EDIT/CORRECTION:…
Contactoid
- 141
- 5
5
votes
2 answers
Darboux coordinate for contact geometry
I'm reading Geiges' notes. (https://arxiv.org/pdf/math/0307242.pdf) In the proof of Theorem 2.44 on page 17, the existence of the contact version Darboux coordinate is reduced to solving $H_t$ for each $t$, the PDE near the origin of…
Chris Kuo
- 1,849
- 11
- 14
5
votes
1 answer
Reference request in contact geometry.
I am looking for an introductory book to contact geometry, as clear and detailed as possible.
Thank in advance.
Thalanza
- 183
5
votes
1 answer
How can I draw plane distributions in $\mathbb{R}^3$?
I see so many nice pictures of contact structures, integrable plane distributions, etc., in manuscripts and online and I have absolutely zero idea how they're made. For example, the following image was posted on Tumblr (source unknown):
I was able…
cstover
- 360
5
votes
1 answer
Trouble understanding proof of this proposition on contact type hypersurfaces
I have finally started working on my thesis. I am reading the first reference book, which is Dusa McDuff, D. Salamon, Introduction to Symplectic Topology. It's a tough struggle, given my not-too-great experience with differential forms. I will…
MickG
- 9,085