Questions tagged [contact-geometry]
Contact manifolds, contact structures, contact forms, Reeb dynamics, Legendrian knots, contact homology, symplectic field theory
174
questions
5
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Legendrian surgery and invertible elements in zeroth degree symplectic cohomology
Is there anything known about the relation between Legendrian handle attachment and invertible elements in $\mathit{SH}^0(M)$?
As the simplest interesting case, take $M_0$ to be the cotangent bundle $...
2
votes
0
answers
182
views
Stabilizing an open book with Anosov piece
It was proven by Colin and Honda in Stabilizing the monodromy of an open book decomposition that any diffeomorphism can be made pseudo-Anosov and right-veering after a series of positive ...
3
votes
0
answers
102
views
Contact 3-manifolds with hyperkahler Stein fillings?
Is there any classification result on (homeomorphism type) of contact 3-manifolds $\Sigma$ that have Stein filling $W$ that is
1. Hyperkahler (s.t. Stein structure is the Kahler part of it)
2. not ...
1
vote
0
answers
64
views
Evolute hypersurfaces
Do you have any references on studies examining the evolute (or focal) hypersurface to a given hypersurface in dimension greater than 3 ?
The evolute can for instance be defined as the envelope of ...
15
votes
3
answers
2k
views
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?
3
votes
2
answers
437
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Symplectic vector fields everywhere transverse to a co-dimension one hypersurface
Usually when speaking about vector fields transverse to a hypersurface in a symplectic manifold, we talk about Liouville vector fields, i.e. vector fields $X$ with the property that $\mathcal{L}_X\...
2
votes
0
answers
308
views
First Chern Class of Contact Structure which is not Torsion
Let $(M,\xi)$ be a closed connected $3-$dimensional contact manifold with contact structure $\xi$. It is known that the first Chern class $c_{1}(\xi)$ defines an element in $H^{2}(M;\mathbb{Z})$ and ...
5
votes
0
answers
375
views
Is there any known relationship between sutured contact homology and Legendrian contact homology?
On one hand, Colin-Ghiggini-Honda-Hutchings' construction (https://arxiv.org/abs/1004.2942) provides an invariant of a Legendrian $L$ in a closed contact manifold $(M,\xi)$ via the sutured contact ...
1
vote
0
answers
164
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On different definitions of a prequantization space
Geometric quantization associates to a symplectic manifold $(M,\omega)$ a hermitian line bundle $L \to M$ with connection $\nabla$ whose curvature is $\omega$ (up to some constant).
Without talking ...
2
votes
1
answer
141
views
On the existence and classification of prequantization spaces over a closed symplectic manifold
Let $(M,\omega)$ be a closed symplectic manifold. If the cohomology class $[\omega]$ is rational, that is if it lies in the image of the natural homomorphism $H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R})$, ...
16
votes
2
answers
2k
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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}:\...
4
votes
0
answers
123
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Bound on critical points of Lefschetz fibration over the disk with prescribed monodromy
Let $\phi$ be a right-veering diffeomorphism of a surface $\Sigma$ of genus $g$ and $r$ boundary components. Suppose that the diffeomorphism is freely periodic so if $M$ is the associated open book ...
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 $...
5
votes
0
answers
196
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Contact geometry: approximation of Legendrian mappings
Let $\alpha$ be a standard contact form on $\mathbb{R}^{2n+1}$. We say that a map $f:\mathbb{R}^k\to\mathbb{R}^{2n+1}$ contact if $f^*\alpha=0$.
Question 1. Is it true that a $C^1$-contact ...
8
votes
1
answer
592
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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 ...
3
votes
0
answers
236
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Where can I find good surveys on Symplectic and Contact geometry
Are there any good survey articles in symplectic and contact geometry, which focus on the "big picture", i.e how this discipline fits into the mathematical world ?
In the symplectic case : I am ...
21
votes
3
answers
2k
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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 ...
10
votes
2
answers
537
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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-...
4
votes
2
answers
485
views
Non-trivial examples of overtwisted contact structures
Are there any non-trivial examples of overtwisted contact structures on closed contact $3$-manifolds? By non-trivial I mean any examples besides the trivial one $\xi = \ker (\cos(r)dz - r \sin(r)d\...
5
votes
0
answers
99
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Is there a simply connected contact manifold, "non-exactly" fillable, cappable, such that the whole thing is symplectically aspherical?
Is there an example of a simply connected contact manifold W, with a non exact symplectic filling $M_1$, (that is, $M_1$ is a symplectic manifold, with contact boundary $W$ and a Liouville vector ...
7
votes
0
answers
444
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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 ...
3
votes
0
answers
217
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Lutz twist and open book decompositions
Let $M^3$ be a closed oriented 3-manifold, endowed with an open book decomposition. Consider a section of the open book, that is a knot $K \subset M$ disjoint from the binding and meeting every page ...
4
votes
0
answers
184
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Right-veering periodic automorphisms of surfaces that are not composition of right-handed Dehn twists
Basically the title of the question. For the sake of completeness I state an introduction to the question.
In "Right-veering diffeomorphisms of compact surfaces with boundary I" and II, the authors ...
5
votes
2
answers
347
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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 ...
12
votes
3
answers
1k
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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 ...
2
votes
1
answer
163
views
An equivalent definition for contact pair manifolds
َContact manifold
A $(2n + 1)$-dimensional manifold $M$ is said to be a contact manifold if it
admits a global 1-form $\eta$ such that $\eta\wedge (d\eta)^n\neq 0$.
There is an equivalent ...
3
votes
0
answers
149
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The isotopy class of a Boothby-Wang contact structure
Let $p:\Sigma\to M$ be a non-trivial principal $S^1$-bundle over a closed orientable surface $M$. Let's call $V$ the vector field generating the $S^1$-action. It is known that there exists a Boothby-...
7
votes
1
answer
544
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"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 ...
5
votes
1
answer
246
views
Almost complex structures on a 4-ball that are not tamed
Recall that an almost complex structure $J$ on a manifold $M^{2n}$ is called tamed if there exists a symplectic form $\omega$ on $M^{2n}$ such that $\omega(v,Jv)>0$ for any non-zero tangent vector $...
1
vote
0
answers
170
views
Confusion about the definition of a formal Legendrian isotopy
We say two Legendrian embeddings $f_0,f_1:L^n\rightarrow (Y^{2n+1},\xi)$ are formally isotopic if there is a smooth isotopy $f_t$ connecting $f_0$ and $f_1$ and a bundle monomorphism $F_t^s:TL\...
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-...
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 $...
3
votes
1
answer
475
views
When is the gradient of a Hamiltonian function a Liouville vector field?
Let $(M, \omega)$ be a symplectic manifold, $H$ a Hamiltonian function on $M$, $Y = H^{-1}(c)$ for a regular value $c$, and $J$ a compatible almost complex structure.
If $X_H$ is the Hamiltonian ...
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 ...
3
votes
1
answer
269
views
Degenerate Reeb orbits
I am reading about contact homology and ECH, and realized that I do not see what goes wrong with the definition of these theories, if one takes into count degenerate Reeb orbits. In general, I would ...
1
vote
1
answer
315
views
Conformal Killing vector field on contact manifolds
An interesting class of contact manifolds is the class of $K$-contact manifolds ($\mathcal{L}_\xi g=0$) which have been studied by many authors. It is natural to study conformal Killing-contact ...
4
votes
0
answers
79
views
Neighborhood of (singular) Legendrian with convex boundary
Let $(M, \xi)$ be a contact manifold, and $\Lambda \subset M$ a closed connected Legendrian with a tubular open neighborhood $W$. The question is that, can one find a smaller tubular neighborhood $U \...
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 ...
0
votes
1
answer
130
views
On generalized Tanaka connection
Many authors used the Tanaka connection in their papers such as
[1]
to define new Tanaka connection so-called Generalized Tanaka connection $^*\nabla$ on a contact Riemannian manifold $(M,\eta,\xi,\...
4
votes
0
answers
236
views
Contact manifolds and pseudodifferential operators
By way of background, I am currently trying to understand the theory of pseudodifferential operators in the context of contact geometry. I have some knowledge of pseudodifferential operators on ...
3
votes
1
answer
276
views
Decreasing the binding number of an open book while increasing the genus of the pages
Let $(B,\pi)$ be an open book decomposition of a closed, connected, oriented 3-manifold $M$ with odd (even) binding number and with pages of Euler characteristic $\chi$. Is it possible to define ...
4
votes
2
answers
219
views
Is $\mathbb{P}T^*M$ a sub-Riemannian manifold if $M$ is Riemannian?
(this question is about a particular aspect of a previous question, which was not duly stressed)
Let $(M,g)$ a Riemannian $n$-dimensional manifold, and let
$$
\widetilde{M}:=\mathbb{P}T^*M
$$
be the $...
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 ...
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 ...
3
votes
1
answer
212
views
How many second-order PDEs can be obtained from a contact EDS?
Let $(M,[\theta])$ be a contact manifold, $\dim M=2n+1$, and denote by $\mathcal{I}^\theta$ the differential ideal generated by the contact form $\theta$.
An exterior differential system on $M$ of ...
1
vote
2
answers
630
views
Contact and CR Examples
What is an example of a manifold such that:
(A) It is both a contact manifold and a CR manifold
(B) It is a contact manifold but not a CR manifold
(C) It is not a contact manifold but not a CR ...
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
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_{...
4
votes
1
answer
200
views
Does a $G$-structure on $M$ automatically descend to a contact $G$-structure on $\mathbb{P}T^*M$?
If $M$ is a real smooth manifold of dimension $n+1$, by $D\in\mathbb{P}T^*M$ I mean a tangent hyperplane at some point of $M$. I denote by $b$ the canonical projection of the $(2n+1)$-dimensional ...