Questions tagged [contact-geometry]
Contact manifolds, contact structures, contact forms, Reeb dynamics, Legendrian knots, contact homology, symplectic field theory
174
questions
2
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1
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Superlevel sets of a parametrized quadratic forms
Let $N$ be an odd integer, $n\in\mathbb{N}$, and $-\frac{2T}{NR^2}\leq a\leq0$ for given $R,T\in\mathbb{R}$ with $\frac{T}{NR^2}\leq\frac{\pi}{2}$.
Now consider the quadratic form $\Omega(a)=\sum_{l\...
1
vote
0
answers
63
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Looking for examples of 3rd-order contact transformations
In the Herglotz Lectures on Contact Transformations and Hamiltonian Systems, after going through contact transformations of the form $X=X(x,y,p)$, $Y=Y(x,y,p)$, $P=P(x,y,p)$ it is stated:
As a final ...
6
votes
1
answer
237
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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 ...
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 ...
2
votes
0
answers
103
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Almost contact structures on spheres
I will write $(M,\xi)=\mathbf{OB}(P,\phi)$ to denote that $M$ admits an open book decomposition with page $P$ and monodromy $\phi$ supporting a contact structure $\xi$. I will focus on the case where $...
3
votes
0
answers
78
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Characterization of contact vector fields
Let $H$ be a subbundle of the tangent bundle $TM$ of a smooth manifold $M$.
A vector field $K$ on $M$ is contact if its flow $\Phi_K^t$ preserves $H$.
I found in many references the following ...
1
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0
answers
57
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Lagrangian cobordisms from a Legendrian knot to its scaled version
Having a Legendrian knot L in $\mathbb R^3$ and its scale aL (the length of Reeb chords of it are scaled by a>0), are these two Legendrians Legendrian isotopic? Maybe weaker, is there an exact ...
1
vote
1
answer
185
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Neighborhood theorem for conical Lagrangian
Let $(M,\omega)$ be a compact $2n$ dimensional symplectic manifold and $T$ be a compact smooth $(n-1)$ dimensional manifold.
Let $CT$ be the cone over $T$, i.e. $CT = T\times [0,1] / \sim $ where $\...
2
votes
2
answers
158
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What is the motivation of contact Hamiltonian equation
I've just checked that this is constructed to mimic the ordinary Hamiltonian equation in symplectic geometry. There are several literatures, and they use
$$
\eta(X_H) = -H\\
\mathrm{d}\eta(X_H,-) = \...
12
votes
1
answer
869
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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 ...
4
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0
answers
125
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Overtwisted contact structures on $S^3$
All the isotopy classes of overtwisted contact structures are classified by the Hopf invariant. Are any of these contact structures contactomorphic?
Suppose $d_{3}(\xi_{n}) = n$, then my guess is that ...
2
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0
answers
79
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Weinstein fillings of a unit cotangent bundle
Given a closed, orientable manifold M, and its unit cotangent bundle $ST^{\ast}M$. I wonder under which conditions $ST^{\ast}M$ admits a subcritical Weinstein filling?
3
votes
0
answers
115
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Smooth handle attachment vs Weinstein handle attachment
Given a closed smooth manifold $M$ of dimension $n$, to which we attach a $k$-handle $H_k$.
Take $T^{\ast} M$, can one realize $T^{\ast} (M\cup H_k)$ as a result of symplectic or Weinstein handle ...
2
votes
1
answer
306
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Standard contact forms on the torus
Is there a standard or "simple" contact structure on the $3$-dimensional torus $T^3$, like there are for example for the Eucliden space and the $3$-sphere?
My first thought was to consider a ...
4
votes
2
answers
267
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Realizing closed manifolds as Legendrian submanifolds of the standard contact vector space
I started learning some basic contact geometry, in particular its flexible side, and I got stuck with the following naive question. Given a closed manifold of dimension $n$, we can always embed it ...
5
votes
2
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590
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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,...
3
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0
answers
142
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Chekanov-Eliashberg Legendrian DGA with positive grading?
I was just looking back to some notes that I took a few years ago, when I was reading Etnyre's notes on Legendrian Contact Homology in $\mathbb R^3$ and I happened upon the following question that I ...
7
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0
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286
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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 ...
6
votes
1
answer
178
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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 ...
4
votes
1
answer
183
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Are there a geometry behind the singularity formation in solutions to nonlinear ODE's?
Disclaimer: This question was originally posted in math.stackexchange.com and, after 30 days with no answers, I followed the instructions of this topic.
If we take two apparently simple first order ...
1
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0
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159
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Does the blow-up preserve symplectic structure?
Let $X \subset \mathbb P(\mathfrak g)$ be an adjoint variety for a simple complex Lie algebra $\mathfrak g$ appearing in the last line of the Freudenthal Magic Square, that is $\mathfrak g \in \{F_4,...
3
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1
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172
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Stein fillable tight contact structures on the 3-torus
Kanda classified tight contact structures on the 3-torus. Which of them is Stein fillable? Is there any good reference?
3
votes
1
answer
224
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Change of Reeb orbits after scaling the contact form
Let $M$ be a contact manifold with a contact form $\theta$ with Reeb vector field $X$ and $f$ be a positive function on $M$. If $\mathcal{L}_X f\neq 0$ the Reeb vector field $X'$ of $\theta'=f \theta$ ...
1
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0
answers
67
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In a contact Lie algebra, when is the Reeb vector a semisimple element?
Below is a question I have come across in my research, and it seems like a question that has been answered (or at least asked) in the past; however, I have been unable to find any references that ...
1
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0
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67
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Isotopy of open book supporting same contact structure
In dimension 3, the Giroux correspondence gives us a bijection between contact structures (up to isotopy) and open book decompositions (up to positive stabilisation). Moreover, Giroux shows that two ...
20
votes
4
answers
3k
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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 ...
3
votes
1
answer
132
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Effect of a Lutz twist on Euler number
I already asked this question on the Math Stack Exchange but did not get an answer.
I am currently working through Geiges proof of the Martinet-Lutz theorem, which can be
found here, and am trying to ...
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,\...
0
votes
2
answers
102
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Tightness/Overtwistedness of genus one open book decomposition
Suppose we have an open book decomposition $(P,\phi)$ of a 3-manifold $Y$, where $P$ is a punctured torus and $\phi$ is the monodromy. We know $\phi$ can be represented by a matrix in $SL(2,\mathbb{Z})...
1
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1
answer
257
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An analogue of the Poisson bracket in contact geometry?
I was looking at this old question and thought it might get more attention at this site. In summary, the OP asks the following question:
McDuff and Salamon define an analogue of the Poisson bracket ...
4
votes
2
answers
491
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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 $\...
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
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0
answers
170
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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\...
3
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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 ...
6
votes
1
answer
423
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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_{...
2
votes
1
answer
108
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$2$-Form inducing a non-degenerate form on $\Gamma(T\mathbb{R}^{2n+1})$
Every $2$-form $\omega\in \Omega^2(\mathbb{R}^{2n+1})$ induces a skew-symmetric map
$$
\omega(-,-)\colon\Gamma(T\mathbb{R}^{2n+1})\otimes \Gamma(T\mathbb{R}^{2n+1}) \to C^\infty(\mathbb{R}^{2n+1})
$$
...
2
votes
0
answers
94
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How does the Maslov index of a loop `project’ to the rotation number?
I’m trying to learn some Legendrian contact homology and the grading of the generators of the DGA are given by computing a fractional rotation number. In the symplectisation, this number is the Conley-...
5
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0
answers
231
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Overtwisted contact forms on open manifolds
I tried first at Math Stack Exchange but got no answers, so I thought maybe this question belongs here.
It is known that on closed $3$-manifolds the Reeb vector field of any contact form inducing an ...
3
votes
2
answers
340
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Why is the standard contact structure on $\mathbb R^{2n+1}$ called "standard"?
The standard contact structure on $\mathbb R^{2n+1}=(x_1,y_1,\dots,x_n,y_n,z)$ is given by $\ker\alpha$, where $\alpha=dz-\sum_{i=1}^ny_idx_i$. But is there a reason why this contact structure is ...
4
votes
0
answers
240
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Maximal Thurston--Bennequin number of boundary knot classes in contact handlebodies
Let $H$ be a contact handlebody. In other words, $H$ is a small regular neighborhood of a Legendrian graph in a contact $3$-manifold (wlog $\mathbb R^3$). Equivalently, $H=(\Sigma\times[0,1],dt+\...
9
votes
1
answer
229
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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 ...
1
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0
answers
164
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Maximal dimension guaranteed for integral manifolds of hyperplane distributions
To KSackel and anyone else has viewed this: I'm sorry my edits have been all over the place. I've tried to cut it down to my remaining curiosities, so there's less to wade through (and hopefully fewer ...
4
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0
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132
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Sheaves with specified singular support at infinity coming from hyperplane arrangements
Given a manifold $M$, we consider its cotangent bundle $T^*M$, and its cocircle bundle $T^\infty M$, quotienting out by the scaling action of the positive reals. Given a Legendrian submanifold $\...
16
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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}:\...
1
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0
answers
210
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Genericity of contact structures all of whose closed Reeb orbits are nondegenerate
First, a contact form $\alpha$ on $M$ with Reeb vector field $R$ is said to be non-degenerate if, for any point $p$ such that $\phi_T^R(p) = p$, we have $\det{(\textrm{id}_{T_pM} - (d \phi_T^R)_p)} \...
16
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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 $...
1
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0
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59
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How does the knot contact homology augmentation polynomial change under a surgery on the base manifold?
So for every knot $K \in S^3$, there is a knot contact homology of $K$, and we can find the augmentation variety for this homology. The defining polynomial is known as the augmentation polynomial $A(K)...
8
votes
1
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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 $...
2
votes
0
answers
124
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Convex surfaces with transverse boundary (contact geometry)
Suppose I have a compact surface $\Sigma$ in a contact 3-manifold, where the boundary $\partial\Sigma$ is transverse to the contact structure. Am I able to perturb $\Sigma$ rel boundary so that it is ...
2
votes
1
answer
195
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$S^3$ as a Sasakian Manifold
Reading about Sasakian manifolds one come across two slogans:
A) "A Sasakian manifold is an odd-dimensional analogue of a Kahler manifold."
B) "A Sasakian manifold sits between two Kahler manifolds -...