Questions tagged [contact-geometry]

Contact manifolds, contact structures, contact forms, Reeb dynamics, Legendrian knots, contact homology, symplectic field theory

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Proof of Giroux's correspondence

It is extensively used and cited the following statement due to Giroux: Given a closed $3$-manifold $M$, there is a $1:1$ correspondence between oriented contact structures on $M$ up to isotopy and ...
Paul's user avatar
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20 votes
4 answers
3k views

What is the role of contact geometry in the hamiltonian mechanics?

Let us assume someone is interested in the study of Hamiltonian mechanics. What are good examples to illustrate him of the usefulness of contact geometry in this context? On one hand the Hamiltonian ...
agt's user avatar
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17 votes
1 answer
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What is the mirror of symplectic field theory?

Mirror symmetry is, very roughly, a relation between symplectic geometry on one side and complex/algebraic geometry on the other side. For example, counts of pseudoholomorphic spheres in a closed ...
user25309's user avatar
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16 votes
2 answers
2k views

The Lefschetz operator

Let $\omega=\sum_{i=1}^n dx_i\wedge dy_i\in\bigwedge^2(\mathbb{R}^{2n})^*$ be a standard symplectic form. The following result is due to Lefschetz: For $k\leq n$, the Lefschetz operator $L^{n-k}:\...
Piotr Hajlasz's user avatar
16 votes
1 answer
2k views

Weinstein neighborhood theorem for Lagrangians with Legendrian boundary

$\require{AMScd}$ Weinstein's neighborhood theorem says that every Lagrangian has a standard neighborhood. The more precise statement goes like this. Theorem 1: (Lagrangian Neighborhood Theorem) Let $...
Julian Chaidez's user avatar
16 votes
0 answers
326 views

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 ...
John Pardon's user avatar
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15 votes
3 answers
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Examples of odd-dimensional manifolds that do not admit contact structure

I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure. Can someone provide me with some examples?
Overflowian's user avatar
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14 votes
3 answers
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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}...
Giovanni Moreno's user avatar
13 votes
6 answers
2k views

Polynomial contact structures on $RP^3$

Let us consider polynomial contact structures on $\mathbb RP^3$, i.e. contact structures on $\mathbb R^3$ defined by a form $w=Pdx+Qdy+Rdz,\ P,Q,R\in \mathbb R[x,y,z]\ $ in an affine part and then ...
Nikita Kalinin's user avatar
12 votes
3 answers
1k views

Do contact and CR structures have corresponding $G$-structures?

For an $n$-dimensional manifold $M$, almost complex and almost symplectic structures on $M$ correspond to reductions on the structure group of the tangent bundle, introducing a $\operatorname{GL}(n/2,\...
E. Addison's user avatar
12 votes
2 answers
746 views

Solving ODE via contact geometry

I have been reading H. Geiges' "A Brief History of Contact Geometry and Topology". According to him contact transformations were introduced as a geometric tool to study systems of differential ...
Lukas S's user avatar
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12 votes
1 answer
869 views

On a corollary of a paper by Colin and Honda

The question is about the last sentence of the last corollary of Stabilizing the monodromy of an open book decomposition by Vicent Colin and Ko Honda. This question is also related to this other ...
Paul's user avatar
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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 ...
Juan OS's user avatar
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10 votes
2 answers
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Is there a table of (fibred knot) monodromies?

Background/motivation I'm working on contact topology (in dimension three): a fundamental theorem of Giroux gives us a bijection between contact structures (up to isotopy) and open books (up to ...
Marco Golla's user avatar
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10 votes
2 answers
537 views

Is the Lisca-Matic bound (aka slice-Bennequin bound) strictly stronger than the Bennequin bound?

The Bennequin bound [1] says that, for a transverse knot (or later link) $K$ in $S^3$, $$\mathrm{sl}(K) \le - \chi(\Sigma)$$ for any Seifert surface $\Sigma$ for $K$, where $\mathrm{sl}$ is the self-...
Dylan Thurston's user avatar
10 votes
1 answer
3k views

'Contactization' and Symplectization

Given any manifold $M$, we can get a symplectic manifold by taking the cotangent bundle $T^\ast M$ with symplectic form $\omega=\sum dp_i\wedge dq_i$. Given any manifold $M$, we can get a contact ...
Chris Gerig's user avatar
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9 votes
3 answers
3k views

Tight vs. overtwisted contact structure

I know the familiar differences between tight and overtwisted contact structures. For example, each homotopy class of plane-bundles on a three-manifold has an overtwisted representative but tight ...
nikita's user avatar
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9 votes
3 answers
2k views

When does a hypersurface have contact-type?

In a symplectic manifold $(X^{2n},\omega)$, a hypersurface $Y\subset X$ has contact-type if there is a contact form $\lambda$ such that $d\lambda=\omega|_Y$. Recall that a contact form is a 1-form ...
Chris Gerig's user avatar
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9 votes
2 answers
613 views

Contact distributions on $(G_2,P)$-type Cartan geometries in dimension 5

Up to topology, the 5D homogeneous space $$ G_2/P $$ of the (real form of the) 14D exceptional Lie group $G_2$ is the 5D jet space $$ M:=J^1(2,1)=\{(x,y,u,p,q)\} $$ of scalar functions in two ...
Giovanni Moreno's user avatar
9 votes
1 answer
614 views

Reeb flows on $S^3$ versus volume preserving flows

Is there an example of a smooth vector field $v$ on $S^3$ such that $v$ preserves a volume form and $v$ is not a Reeb vector field? Recall that $v$ is a Reeb vector field if there exists a contact $...
aglearner's user avatar
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9 votes
1 answer
229 views

Physical motivation for tight/overtwisted dichotomy

I'm learning about tight vs. overtwisted contact structures in contact geometry. I understand that we care about the existence/nonexistence of overtwisted disks in a contact structure in part because ...
boink's user avatar
  • 213
9 votes
0 answers
546 views

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 ...
Kyle Hayden's user avatar
8 votes
2 answers
975 views

strong contactomorphism group inside contactomorphism group

Let $(M, \xi)$ be a closed contact manifold with co-oriented contact structure $\xi = \ker \alpha$. Let $\mathrm{Cont}(M, \alpha)$ be the group of diffeomorphisms that preserve the contact form $\...
Sam Lisi's user avatar
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8 votes
1 answer
592 views

Physical intuition behind prequantization spaces

Given a symplectic manifold $(M,\omega)$ with integral symplectic form, that is $$\omega \in \text{Im}(H_2(M,\mathbb{Z}) \to H_2(M,\mathbb{R})),$$ one can form a so-called prequantization space, that ...
BrianT's user avatar
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8 votes
1 answer
2k views

What is knot contact homology?

Recently, it was conjectured by the paper of Aganagic and Vafa that the $Q$-deformed $A$-polynomials can be identified with the augmentation polynomials of the knot contact homology. The $Q$-deformed $...
Satoshi  Nawata's user avatar
8 votes
1 answer
396 views

From convex geometry to contact topology

Here is a problem in contact topology that was suggested by Petya's answer to this mathoverflow question of mine. Let $S^* \mathbb{R}^n$ be the space of cooriented contact elements of $\mathbb{R}^n$. ...
alvarezpaiva's user avatar
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7 votes
2 answers
415 views

Symplectisation as a functor between appropriate categories

Let $(M,\xi)$ be a transversally orientable contact manifold, that is, there exists a form $\alpha \in \Omega^1(M)$ such that $\xi = \ker \alpha$. Then we can associate to $(M,\xi)$ its ...
TheGeekGreek's user avatar
7 votes
1 answer
542 views

"Explicit" perturbations of Morse-Bott functions

There are explicit perturbations of Morse-Bott functions $f:X\to\mathbb{R}$ used in the literature (ex: Austin-Braam, Banyaga-Hurtubise, Bourgeois) to help solve various problems (ex: building Morse ...
Chris Gerig's user avatar
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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 ...
contakto's user avatar
7 votes
1 answer
460 views

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 ...
PVAL's user avatar
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7 votes
0 answers
286 views

On exotic symplectic structures of smooth closed 4-manifolds

What are some known techniques and examples of exotic symplectic structures on a fixed smooth closed 4-manifolds [by exotic I mean two symplectic structures that are not symplectomorphic]. This sounds ...
Anubhav Mukherjee's user avatar
7 votes
0 answers
444 views

Correct notion of chain homotopy for linearized homology of augmented DGAs?

$\require{AMScd}$ Preliminaries: Let $(A,\partial)$ be a differential graded $k$-algebra with an augmentation $\epsilon$. That is, $\epsilon$ is a DGA map $\epsilon:(A,\partial) \to k$ where $k$ is ...
Julian Chaidez's user avatar
7 votes
0 answers
205 views

When do geodesics reconverge?

Say I stand at the north pole and talk; in sufficiently frictionless conditions, one imagines that someone standing at the south pole could listen. More generally, say $M$ is a compact Riemannian ...
Vivek Shende's user avatar
  • 8,663
7 votes
0 answers
201 views

Biholomorphic neighborhoods of the boundary of Stein domains

Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...
PVAL's user avatar
  • 773
6 votes
2 answers
434 views

Symplectic orthogonality and projective duality: how do they work together?

Let $(V,\omega)$ be a $2n$-dimensional linear symplectic space, and $(\mathbb{P}V,\theta_\omega)$ the corresponding $(2n-1)$-dimensional contact manifold. Given a smooth $(n-1)$-dimensional smooth ...
Giovanni Moreno's user avatar
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 ...
Jerry's user avatar
  • 63
6 votes
1 answer
237 views

Can differential forms be exact and positive on a distribution?

Let $M$ be a manifold of dimension $d$, and let $\mathscr D$ be a distribution of rank $d - 1$ on $M$ (I would also be interested in lower rank distributions, but mainly I am interested in codimension ...
Aidan Backus's user avatar
6 votes
1 answer
423 views

stabilization of Legendrian knots

There are two ways to stabilize a Legendrian knot $k$ in standard contact sphere $(S^3,\xi_{st})$ i.e. adding right cusps or left cusps, let's call these two stabilized Legendrian knots $k_R$ and $k_{...
nikita's user avatar
  • 1,335
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 ...
alvarezpaiva's user avatar
  • 13.2k
6 votes
1 answer
178 views

Positive vs negative Dehn twist monoids

Given a bounded surface $S$ and a mapping class $h$, construct the open book 3-manifold $(S,h)$. If $h$ lies in the positive monoid of the mapping class group, then $(S,h)$ supports a tight contact ...
Josh Barnard's user avatar
6 votes
1 answer
393 views

Non-Reeb vector fields on the three-sphere

Let $X$ be the Hopf vector field on the three-sphere. Is there a smooth nowhere zero function $f$ so that the modified vector field $fX$ is not the Reeb vector field of any contact form on the three-...
alvarezpaiva's user avatar
  • 13.2k
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 ...
Math1016's user avatar
  • 369
5 votes
2 answers
593 views

contactomorphism of $S^{2n+1}$ for n>1

Is there some result about contactomorphism groups of $S^{2n+1}$ or $T^{2n-1}$ for n>1? For example, do we know the rank of $\pi_{i}(Cont(S^{2n+1})) \otimes \mathbb{Q}?$ where "Cont" means the ...
cccmjj's user avatar
  • 51
5 votes
1 answer
454 views

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 ...
shestipalov's user avatar
  • 1,000
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,...
OBD's user avatar
  • 51
5 votes
1 answer
561 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 ...
agt's user avatar
  • 4,246
5 votes
1 answer
267 views

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 ...
Seonhwa  Kim's user avatar
5 votes
3 answers
591 views

Thom polynomial for contact algebraic structures

Let's consider a algebraic contact structure $P$ on $\mathbb CP^3$ and a algebraic curve $C$ degree $d$ and genus $g$. Let's assume that contact structure has degree $p$ (see Polynomial contact ...
Nikita Kalinin's user avatar
5 votes
1 answer
866 views

Question about the dimension of a Contact (Symplectic) manifold

I am reading about contact geometry and I have a question: Why do we only consider contact structure of an odd-dimension manifold? and the same question for definition of symplectic geometry? I think ...
Phi Le's user avatar
  • 103
5 votes
2 answers
347 views

Ozsváth-Szabó's contact invariant on the Brieskorn sphere $\Sigma(2,3,6m+1)$

According to Theorem 1.7 of Mark-Tosun's paper, the Brieskorn sphere $\Sigma(2,3,6m+1)$ admits two tight contact structure $\xi_{i}\ (i=0,1)$. They are both Stein fillable and they are contactomorphic ...
user44651's user avatar
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