Questions tagged [contact-geometry]
Contact manifolds, contact structures, contact forms, Reeb dynamics, Legendrian knots, contact homology, symplectic field theory
174
questions
2
votes
1
answer
264
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Ricci soliton on contact manifolds
Recently I am studying Ricci flow and its self-similar solution called Ricci soliton. In this respect I found some papers which focuses Ricci soliton in the setting of various kind of contact ...
2
votes
0
answers
210
views
contact structure on double branched covers of $S^3$
We can assign a natural contact structure to double branched cover of a transverse knot $k$ in the $3$-sphere with its standard contact structure $(S^3,\xi_{st})$, as described in:http://arxiv.org/pdf/...
4
votes
1
answer
410
views
contact surgery diagram on Brieskorn manifolds
For the Brieskorn manifold $\Sigma(p,q,r)=\{z_1^p+z_2^q+z_3^r=0\} \cap S^5 \subset C^3$, replacing zero with $\epsilon$ in the above, realizes $\Sigma$ as boundary of a Stein domain which induces a ...
1
vote
0
answers
81
views
Smoothness of the twistor space of a lorentzian manifold, or “convexity wrt null geodesics”
Null lines in Minkowski space form a 5-dimensional manifold, represented as a (real) quadric $\mathbf{PN}\subset\mathbb{C}\mathbf{P}^3$. This is a well-known fact, on which R. Penrose’s twistor ...
4
votes
1
answer
218
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Are all bidimensional second-order PDE at most quadratic in the top derivatives of Monge-Ampère type?
The general Monge-Ampère equation in $n$ independent variables is a quasi-linear combination of all the possible minors of the $n\times n$ Hessian matrix
$$
\left\|\frac{\partial^2u}{\partial x^i\...
7
votes
1
answer
460
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Homology 3-sphere with a unique Stein-fillable contact structure
Are there any known examples of oriented integer homology 3-spheres $Y$ (besides $S^3$) which have exactly one Stein-fillable contact structure up to isotopy? Failing that, what are the known examples ...
3
votes
1
answer
179
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Non-vanishing contact vector field transverse to contact codimension-2 submanifold with trivial normal bundle?
So, I have a codimension 2 contact submanifold of a closed contact manifold which has (topologically) trivial normal bundle. The question is, can I find a non-vanishing contact vector field which ...
16
votes
0
answers
326
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Is tightness decidable?
Given a contact structure on a three-manifold, is there an algorithm to decide whether or not it tight?
For concreteness' sake, let's agree to represent the given contact three-manifold via an open ...
4
votes
1
answer
286
views
Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk
Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise.
For any $x \in E$, there is a ...
4
votes
2
answers
491
views
Legendrian knot in 3-sphere
We are given a Legendrian knot, fixed up to Legendrian isotopy, in $(S^3,\xi)$ ($\xi$ is the standard contact structure). Does it necessarily bound a symplectic surface in $(B^4,\omega)$ (again $\...
9
votes
0
answers
546
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Schoenflies and symplectic topology
The final report from a workshop on Morse theory in low-dimensional and symplectic topology includes the following question, posed by Michael Hutchings: Can we apply symplectic geometry to solve the ...
2
votes
1
answer
293
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What is the "type" of a contact vector field?
Let $(M,\theta)$ be a $(2n+1)$-dimensional contact manifold, $\mathcal{C}=\ker\theta$ the contact distribution, and $X\in\mathcal{C}$ a vector field belonging to $\mathcal{C}$.
In a couple of minor ...
2
votes
1
answer
198
views
Symplectic (contact) structure on $M_{n}(\mathbb{R})$
Assume that $n$ is an even number. What is a natural symplectic structure on $M_{n}(\mathbb{R})$, such that for every $1\leq k \leq n$ the manifold of $k$-rank matrices would be invariant under ...
3
votes
2
answers
1k
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Boundary geometry of a contact manifold
Let $(M, \xi = \text{ker}\,\alpha)$ be a compact contact manifold with non-empty boundary. Vaguely asked, is there any natural geometric structure on the boundary $\partial M$ induced from the contact ...
4
votes
2
answers
1k
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Modifying the Reeb vector field by multplying by a function
Given a contact 3-manifold $(M,\omega)$ and its Reeb vector field $R$ and contact structure $\Delta$, I want to understand in some sense 'how large' is the set of Reeb vector fields supported by $\...
1
vote
0
answers
130
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Regularity of the taut foliation
In Eliashberg-Thurston's famous paper "Confoliations" Corollary 3.2.11, they proved that Irreducible three manifold with $b_{1}>0$ admits semi-fillable contact structure using Gabai's theorem in ...
1
vote
0
answers
80
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why is there such a 1-form on a planar open book?
Suppose $B$ is the binding (with more than one component) of a planar open book on a 3-manifold $Y$ and let $L\subset B$ be the complement of a single component of the binding. Now we perform page-...
5
votes
1
answer
454
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Are there spaces in which there are no fibered knots?
I am looking for orientable closed 3-manifolds in which there are no fibered knots. Although I know little about this, I think for links the answer to the question above is "no", and the result is ...
5
votes
1
answer
267
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Is there a known Legendrian simple link?
Several knots like unknot, $4_1$, $3_1$ are known to be Legendrian simple, i.e., Thurston-Bennequin number and rotation number determine Legendrian type completely.
How about the same notion for link ...
1
vote
1
answer
254
views
Reeb orbit and open books
Weinstein conjecture is about existence of a closed orbit of the Reeb vector field on every contact manifold. On the other hand, we know every contact 3-manifold admits a compatible open book, which ...
1
vote
1
answer
469
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Symplectic Submanifolds of Contact Manifolds and Contact Submanifolds of Symplectic Manifolds
We know every contact manifold admits a symplectic submanifold, e.g., by Giroux's bijection : if $\omega$ is a contact form for a 3-manifold $M^3$ , then $d \omega$ is symplectic on the fibers of the ...
5
votes
1
answer
674
views
Generalization of Giroux's Theorem for Higher Dimensions?
Just wanted to know if Giroux's theorem for 3-dimensional contact manifolds can be generalized:
In contact geometry for manifolds of dimension 3 , we have Giroux's theorem , stating that for any ...
3
votes
1
answer
330
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Embedded Contact Homology and Manifold Decompositions
Embedded Contact Homology (ECH) defines an invariant for contact 3 manifolds. It does this by considering certain J-holomorphic curves in $\mathbb R\times Y$ and "counting" them.
In the symplectic ...
6
votes
0
answers
586
views
Gompf's invariant of $2$-plane fields
I am interested in low dimensional contact topology. These days I read "Handlebody construction of Stein surfaces" written by R. E. Gompf, and study an invariant $\theta (\xi)$. This invariant is ...
3
votes
2
answers
313
views
non-isotopic but homotopic tight contact structure
By a theorem of Eliashberg, two overtwisted contact structures on a 3-manifold which belong to the same homotopy class (as plane fields), are also isotopic (through contact structures). Is there an ...
1
vote
0
answers
100
views
This weaker version of CR-structure: is it studied somewhere
When I study 5-dimensional $\mathcal{N} = 1$ supersymmetry, I came across such structure as follows.
$(R, \kappa, \Phi, M)$ is an almost contact 5-manifold, such that
\begin{equation}
\kappa \...
14
votes
3
answers
2k
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Is there a "unique" homogeneous contact structure on odd-dimensional spheres?
Let $S^{2n-1}\subset\mathbb{C}^{n}$, and denote by $\langle\,\cdot\,,\,\cdot\,\rangle$ the Hermitian product. Then
$$
\mathcal{C}_p:=\{\xi\in T_pS^{2n-1}\mid\langle p,\xi\rangle=0\},\quad p\in S^{2n-1}...
11
votes
2
answers
518
views
The importance of differentiable dynamics from outside dynamics? (mainly topology)
I'm looking for examples that highlight how dynamical systems (particularly, Hamiltonian and Reeb dynamics) can be used to shed light in other areas of mathematics. This could potentially include ...
3
votes
1
answer
176
views
Legendrian knots on pages of a compatible open book
Suppose we have a Legendrian knot embedded on a page of an open book compatible with the given contact structure on the 3-manifold. Is it true that the page framing and Thurston-Bennequin framing of ...
2
votes
1
answer
508
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Meaning of " Open Book cannot be Stabilized Further"?
I'm going over some old notes on Giroux's theorem on the equivalence ( bijection, actually) between open books ( up to positive stabilization) for 3-manifolds and contact structures ( up to isotopy.) ...
1
vote
1
answer
140
views
Factor of 2 In the Definition of Metric Contact Structure
In Blair's book and many many literatures, I see definition of a contact metric manifold which involves a relation
\begin{equation}
d\kappa \left( {X,Y} \right) = g\left( {X,\Phi Y} \right)
\end{...
3
votes
1
answer
209
views
3d-analog of "every 2d oriented manifold is complex"
Is there an analog of the statement of "every 2d oriented surface is a complex manifold"?
I saw a theorem in Blair's book, that "every 3d contact metric manifold is a strongly pseudo convex CR ...
0
votes
0
answers
72
views
a section invariant under Reeb flow
Let $M$ be a compact contact manifold with $R$ the Reeb vector field. Let $E$ be some vector bundle of gauge group $G$, say, $G = SU(2)$ and $E$ is the adjoint vector bundle.
So my question is:
If $\...
3
votes
1
answer
352
views
Extending Reeb field from contact submanifold to ambient contact manifold
Let $(Y,\lambda)$ be a contact manifold, with a codimension-2 contact submanifold $(S,\lambda|_S)$ (this requires $TS\pitchfork\text{Ker}\lambda$). On $Y$ there is a natural vector field, the Reeb ...
1
vote
1
answer
185
views
contact metric structure on squashed spheres
My goal to write down an explicit (and simplest) contact metric structure on squashed $S_\omega^{2n + 1}$ defined as
\begin{equation}
S_\omega ^{2n + 1} = \left\{ {\left( {{z_i}} \right) \in \...
4
votes
1
answer
410
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Comparing Contact Structures: What do we Mean when we Say that two Contact Structures are Homotopic/Eliashbergs Class. of OT structures
Please forgive me if this is too simple for MO; most of my posts on anything contact-structure-related in Math Stack, other sites, have barely received answers (maybe because I'm not an expert in the ...
4
votes
0
answers
459
views
Transversality in Bourgeois Oancea's non-equivariant contact homology
In their paper, "AN EXACT SEQUENCE FOR CONTACT- AND SYMPLECTIC HOMOLOGY", Bourgeois and Oancea defined, whenever there is sufficient transversality, a "non-equivariant contact homology". Essentially, ...
1
vote
0
answers
133
views
Contact structures transverse to a given line bundle
Let $M$ be an orientable $3$-dimensional manifold with a $1$-dimensional continuous distribution $F$ (i.e a $1$-dimensional subbundle of $TM$).
It is known that every orientable $3$-dimensional ...
2
votes
0
answers
86
views
Extendability of Contact Structures; Foliations of $S^2$
I am currently reading Eliashberg's paper on the classification of overtwisted contact structures (http://bogomolov-lab.ru/G-sem/eliashberg-tight-overtwisted.pdf). In it, there is the following ...
4
votes
1
answer
1k
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How to compute Conley-Zehnder indices on prequantization spaces?
My question is pretty much as in the title. On page 100 of his thesis, Bourgeois gives a computation of the CZ(or I suppose more correctly this is the Robbin-Salamon index) index of the Reeb orbits ...
6
votes
1
answer
263
views
Determining Tightness/Overtwistedness of Contact Structure using Lift of Structure to the Universal Cover
All:
I would appreciate any ideas, refs., etc. on the following:
Let $M^3$ be a contact 3-manifold, and let $X$ be its universal cover. Then the
contact structure, say $\eta$ on $M^3$ lifts to a ...
5
votes
2
answers
590
views
Automorphisms of surfaces, open books and contact structures
Let $S$ be a surface with non-empty boundary $\partial S$ and let $f$ be an element of $\mathrm{MCG}(S)$, the mapping class group of $S$, i.e., the group of self-diffeomorphisms of $S$ up to isotopy,...
1
vote
1
answer
190
views
Group of CR automorphisms
Let $(M, D, J)$ be a strictly pseudoconvex hypersurface type CR manifold with $J$ integrable.
Let $D$ be the kernel of a $1$-form $\eta_0$.
As known the automorphism group is defined to be
$$
CR = \{ \...
7
votes
1
answer
220
views
Why is the dividing set nonempty when a convex surface has Legendrian boundary?
I am an undergrad and curious about the following question. Let $(Y,\xi)$ be a contact manifold, and $L\subset (Y,\xi)$ be a Legendrian knot which is the boundary of a convex surface $\Sigma$ embedded ...
5
votes
1
answer
560
views
Is there a Legendrian Neighbourhood Theorem also for non-cooriented contact manifolds?
Context
According to Arnol'd, a contact structure on a smooth manifold $M$ is given by a corank 1 tangent distribution $C$ which is maximally non-integrable; this means that, for any local $1$-form ...
2
votes
1
answer
757
views
pre-quantization of Jet bundle
We know that the notion of Jet bundle $J^kM×\mathbb{R}$, is generalization of cotangent bundle. What is the prequantization of $J^kM×\mathbb{R}$?
3
votes
2
answers
1k
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Legendrian Tubular Neighborhood Theorem
Given a Lagrangian submanifold $L\subset(M,\omega)$ of a symplectic manifold, we have Alan Weinstein's celebrated Lagrangian tubular neighborhood theorem. I now look for the analog on Legendrian ...
1
vote
0
answers
213
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Classification of (almost) contact structures on $S^3$
Question: Is there a classification of almost contact or contact structures on $S^3$? What is it and references?
The motivation of this question is as follows:
(1) There is one paper showing that a ...
2
votes
2
answers
725
views
question on Thurston-Bennequin number
I have three questions actually:
1- is it true that in a sufficiently small neighborhood of Legendrian knot in a 3-manifold we can find another Legendrian knot?
2- If the above is true, suppose we ...
6
votes
1
answer
813
views
Shortest geodesic loop vs. shortest periodic geodesic
Are there simple conditions on a Riemannian metric on the two-sphere that imply that a geodesic loop of minimal length is actually a periodic geodesic?
For example, is this true for small ...