Questions tagged [contact-geometry]
Contact manifolds, contact structures, contact forms, Reeb dynamics, Legendrian knots, contact homology, symplectic field theory
174
questions
21
votes
3
answers
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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 ...
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 ...
17
votes
1
answer
1k
views
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 ...
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}:\...
16
votes
1
answer
2k
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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 $...
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 ...
15
votes
3
answers
2k
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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?
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}...
13
votes
6
answers
2k
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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 ...
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,\...
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 ...
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 ...
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 ...
10
votes
2
answers
2k
views
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 ...
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)$$
for any Seifert surface $\Sigma$ for $K$, where $\mathrm{sl}$ is the self-...
10
votes
1
answer
3k
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'Contactization' and Symplectization
Given any manifold $M$, we can get a symplectic manifold by taking the cotangent bundle $T^\ast M$ with symplectic form $\omega=\sum dp_i\wedge dq_i$. Given any manifold $M$, we can get a contact ...
9
votes
3
answers
3k
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Tight vs. overtwisted contact structure
I know the familiar differences between tight and overtwisted contact structures. For example, each homotopy class of plane-bundles on a three-manifold has an overtwisted representative but tight ...
9
votes
3
answers
2k
views
When does a hypersurface have contact-type?
In a symplectic manifold $(X^{2n},\omega)$, a hypersurface $Y\subset X$ has contact-type if there is a contact form $\lambda$ such that $d\lambda=\omega|_Y$. Recall that a contact form is a 1-form ...
9
votes
2
answers
613
views
Contact distributions on $(G_2,P)$-type Cartan geometries in dimension 5
Up to topology, the 5D homogeneous space
$$
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)\}
$$
of scalar functions in two ...
9
votes
1
answer
614
views
Reeb flows on $S^3$ versus volume preserving flows
Is there an example of a smooth vector field $v$ on $S^3$ such that $v$ preserves a volume form and $v$ is not a Reeb vector field?
Recall that $v$ is a Reeb vector field if there exists a contact $...
9
votes
1
answer
229
views
Physical motivation for tight/overtwisted dichotomy
I'm learning about tight vs. overtwisted contact structures in contact geometry. I understand that we care about the existence/nonexistence of overtwisted disks in a contact structure in part because ...
9
votes
0
answers
546
views
Schoenflies and symplectic topology
The final report from a workshop on Morse theory in low-dimensional and symplectic topology includes the following question, posed by Michael Hutchings: Can we apply symplectic geometry to solve the ...
8
votes
2
answers
975
views
strong contactomorphism group inside contactomorphism group
Let $(M, \xi)$ be a closed contact manifold with co-oriented contact structure $\xi = \ker \alpha$. Let $\mathrm{Cont}(M, \alpha)$ be the group of diffeomorphisms that preserve the contact form $\...
8
votes
1
answer
592
views
Physical intuition behind prequantization spaces
Given a symplectic manifold $(M,\omega)$ with integral symplectic form, that is $$\omega \in \text{Im}(H_2(M,\mathbb{Z}) \to H_2(M,\mathbb{R})),$$ one can form a so-called prequantization space, that ...
8
votes
1
answer
2k
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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 $...
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 $\mathbb{R}^n$. ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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_{...
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 ...
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 ...
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-...
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 ...
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 ...
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 ...
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,...
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 ...
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 ...
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 ...
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 ...
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 ...