# Questions tagged [contact-geometry]

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

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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 ...
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6 votes
1 answer
232 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 ...
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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 ...
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2 votes
2 answers
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2 votes
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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-...
• 459
5 votes
0 answers
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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
334 views

### 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 ...
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4 votes
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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 ...
• 459
2 votes
1 answer
195 views

### $S^3$ 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 -...
3 votes
1 answer
130 views

### Transverse knots with knot types of strongly quasi-positive knots

In 2008, Etnyre and Van Horn-Morris proved that if $L$ is a fibered strongly quasi-positive link, there is a unique (up to transverse isotopy) transverse link with the knot type of $L$ in the standard ...
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2 votes
1 answer
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### Intuition for the Liouville one-form restricted to the unit cotangent-bundle

So, it seems to be a fairly classical result that the Liouville one-form restricts to the unit cotangent bundle of a Riemann surface equipped with a Riemannian metric. I know that the flow of Reeb ...
• 949
4 votes
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137 views

### Are these two arguments incompatible?

I want to understand why (if so) these two arguments are not incompatible. And if that's the case, which one is wrong. First we have this paper (by Honda, Kazez and Matic). We look at the last Lemma ...
• 1,379
1 vote
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154 views

### Is every hyperplane distribution in $\mathbb{R}^n$ given by a nowhere vanishing one form?

From wikipedia: Contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-...
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### «Euclidean» local systems

The moduli space of G-local systems on a surface is a fundamental object in mathematics. The cases $G=SU(2)$ and $G=SL_2(\mathbb{R})$ are of particular interest. Consider the group $E$ of isometries ...
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7 votes
2 answers
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### 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 ...
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1 vote
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### What does it mean for two natural numbers to be *approximately equal*?

This is related to this other question of mine about a paper of Colin and Honda. I'm trying to follow the proofs line by line. I found the following piece of notation that is not explained in the ...
• 1,379
12 votes
1 answer
868 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 ...
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3 votes
0 answers
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### Reference for "holomorphic contact geometry"

Just like holomorphic symplectic geometry is a complexification of real symplectic geometry, I am wondering is there any good survey paper or book talking about holomorphic version of real contact ...
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