# Tagged Questions

Contact Geometry studies what in some sense is a diametric opposite to integrable distributions. A contact structure is a maximally non-integrable hyperplane field on an odd-dimensional manifold.

90 views

### contact metric structure on squashed spheres

My goal to write down an explicit (and simplest) contact metric structure on squashed $S_\omega^{2n + 1}$ defined as S_\omega ^{2n + 1} = \left\{ {\left( {{z_i}} \right) \in ...
104 views

### Comparing Contact Structures: What do we Mean when we Say that two Contact Structures are Homotopic/Eliashbergs Class. of OT structures

Please forgive me if this is too simple for MO; most of my posts on anything contact-structure-related in Math Stack, other sites, have barely received answers (maybe because I'm not an expert in the ...
34 views

### Contact structures transverse to a given line bundle

Let M be a orientable three dimensional manifold with a 1 dimensional continuous distributon $F$ (i.e 1 dimensional subbundle of TM). It is known that every orientable three dimensional manifold ...
47 views

### Extendability of Contact Structures; Foliations of $S^2$

I am currently reading Eliashberg's paper on the classification of overtwisted contact structures (http://bogomolov-lab.ru/G-sem/eliashberg-tight-overtwisted.pdf). In it, there is the following ...
80 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 ...
134 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 ...
144 views

### question on Thurston-Bennequin number

I have three questions actually: 1- is it true that in a sufficiently small neighborhood of Legendrian knot in a 3-manifold we can find another Legendrian knot? 2- If the above is true, suppose we ...
98 views

### Homotoping a 2-plane field on a closed orientable 3-manifold to a contact structure

I am a beginner in Contact Geometry. To prove that the inclusion $\text{Cont}(M)\hookrightarrow \text{Dist}(M)$ induces surjection at $\pi_0$ level, the closest I got was based on Ko Honda's notes. ...
146 views

### Contact structures and adjunction inequality in 3-manifolds

It is a theorem of Eliashberg that in a tight contact 3-manifold $(M, \xi)$ we have the adjunction inequality $|\langle e(s),[\Sigma] \rangle| \leq -\chi(\Sigma)$ where $s=s(\xi)$ is the ...
375 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 ...
119 views

### The shape operator and an almost contact structure of a real hypersurface in $\mathbb{C}^n$

Let $S$ be an immersed real hypersurface in the Euclidean $\mathbb{C}^n$ with the standard complex structure $J$. Let $A:T(S)\rightarrow T(S)$ be the shape operator of $S$ (e.g. w.r.t. the outer ...