Contact geometry is a subfield of differential geometry, focusing on contact manifolds, which are differentiable manifolds endowed with some codimension 1 distribution that is maximally non-integrable.
Questions tagged [contact-geometry]
37 questions
6
votes
1 answer
Constructing a contact form for which a given contact vector field is Reeb
Given a contact manifold $(M, H)$ and a smooth vector field $X$ on $M$, I'm trying to show that $X$ is the Reeb field of some contact form for $H$ if and only if it's a contact vector field that's nowhere tangent to $H$.
First, the definitions as I…
Itserpol
- 523
- 2
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4
votes
2 answers
Effect of a Lutz twist on Euler number
I am currently working through Geiges proof of the Martinet-Lutz theorem, which can be
found here, and am trying to figure out the effect of a half Lutz twist on the Euler class
of the contact structure. During the proof of Propositioin 3.15, he…
4
votes
1 answer
Integrability of an "almost" CR-structure defined thanks to an almost contact metric structure.
Let $(M,\eta,\xi,\varphi)$ be an almost contact metric manifold, that is:
$M$ is a smooth manifold
$\eta$ is a contact 1-form on $M$, i.e $\eta$ is a 1-form and $\mathrm{d}\eta|_{\ker \eta}$ is non-degenerate
$\xi$ is the Reeb vector field of…
Didier
- 20,820
4
votes
1 answer
Contact transformations (definition)
An orthogonal transformation preserves a symmetric bilinear form. A symplectic transformation can be defined as a linear transformation that preserves a skew-symmetric bilinear form on a $2n$-dimensional vector space. Is there a similar definition…
Andrey Sokolov
- 1,614
3
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0 answers
Linear Legendrian relations
There is a "partial category" of Lagrangian relations, otherwise known as canonical relations, whose objects are symplectic vector spaces and whose morphisms are Lagrangian submanifolds.
By changing the objects to symplectic vector spaces, and the…
Cole Comfort
- 83
3
votes
0 answers
Tubular neighborhood of a surface in a 3-manifold
Consider a smooth $3$-manifold $M$, a submanifold $P$ of dimension $2$ and a non-vanishing vector field $V$ transverse to $P$. Does there exist a neighborhood of $P$ diffeomorphic to $P\times (-\varepsilon, \varepsilon)$, such that $V$ is pushed…
OSBM
- 131
3
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1 answer
A couple of basic questions on contact structures
Consider the following definition of contact structures:
A contact structure on an n dimensional smooth manifold is a codimension 1 tangent distribution $\xi$ whose first curvature $\beta:\xi\times \xi\to TM/\xi$, defined by $\beta_p:=[-,-]_p\mod…
Watanabe
- 1,081
3
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0 answers
Is there a geometry behind the singularity formation in solutions to nonlinear ODE's?
Disclaimer: I have posted this question on mathoverflow.net following the instructions of this topic.
If we take two apparently simple first order ODE's like $y'=y$ and $y'=y^2$ we find that:
For the first one the general solution is $y=C\exp(t)$…
Diego Santos
- 1,283
3
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0 answers
Calculating a contact hamiltonian vector field
I'm currently trying to study contact geometry and stumbled upon the section talking about contact vector fields. Contact vector fields are constructed in a similar manner as the symplectic vector fields so i tried to do the derivation myself, but…
mttdang
- 31
3
votes
1 answer
Contact form in Polar Coordinates
While going through Etnyres Lectures on Contact Topology (which can be found here) Example 2.8 two questions came up
He uses cylindrcal coordinates $(r,\theta, z)$ to define a 1-form $\alpha_2 = dz +r^2d\theta$ on $\mathbb{R}^3$ and it is not clear…
2
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The maximally non-integrable condition vs. $\alpha \wedge (d\alpha)^{n-1} \neq 0$ on contact manifold
I’m reading Anahita Eslami Rad’s book Symplectic and Contact Geometry: A Concise Introduction. In Definition 4.1 (p. 116), she gives a definition of contact structure:
Let $M$ be a $(2n-1)$-dimensional manifold. A contact structure $\xi \subset TM$…
Springeer
- 47
2
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0 answers
What is the intuition of the Conley-Zehnder index of a Reeb Orbit?
Given a path of symplectic matrices starting at the identity, the Conley-Zehnder index is essentially defined as twice the winding number plus some "correction" term if the path is not a closed path, but I do not understand where is the correction…
diffgeo
- 91
2
votes
1 answer
Example where rotation number of Legendrian knot depends on choice of Seifert surface
In Surgery on Contact Manifolds and Stein Surfaces there is the following exercise [below $Y$ is a 3-manifold, and $e(\xi)$ is the Euler class of $\xi$]:
Exercise 4.2.6. Find a contact structure $(Y,\xi)$, a Legendrian knot $L\subset (Y,\xi)$ and…
Hrhm
- 3,555
2
votes
1 answer
Characterization of contact vector fields
I am trying to prove Theorem 22.33 in Lee's Introduction to Smooth Manifolds, whose proof was left as a Problem:
If $(M, H)$ is a contact manifold and $\theta$ is a contact form for $H$, then a smooth vector field on $M$ is a contact vector field…
Tob Ernack
- 5,319
2
votes
0 answers
Action of $GL(4)$ on $SO(3)$
There is a natural way to identify $\mathbb{R}P^3:=S^3/\mathbb{Z}_2$ to $SO(3):=\{ A\in GL(3)\ | A^tA=I_3,\ \det(A)=1\}$. To a point $[(w,x,y,z)]\in\mathbb{R}P^3$ associate the $3D$ rotation through the axis generated by $(x,y,z)\in\mathbb{R}^3$ of…
