--- date: 2022-02-23 18:45 modification date: Friday 18th March 2022 12:40:42 title: Contact aliases: ["contact", "contact topology", "contact manifold", "contact structure", "contact geometry", overtwisted, tight] --- --- - Tags - #geomtop/contact-topology - Refs: - #resources/notes/expository - Geiges' _An Introduction to Contact Topology_ #resources/books - Links: - [open book decomposition](Unsorted/open%20book%20decomposition.md) - [hyperplane field](Unsorted/hyperplane%20field.md) - [foliate](Unsorted/foliate.md) - [contact manifold](Unsorted/Contact.md) - [Unsorted/Giroux correspondence](Unsorted/Giroux%20correspondence.md) --- # Contact structures - [Contact project Draft 1](Unsorted/Contact%20project%20Draft%201.md) ![](attachments/Pasted%20image%2020220318123923.png) ![](attachments/Pasted%20image%2020220411141800.png) ![](attachments/Pasted%20image%2020220411141821.png) # Motivation ![Pasted image 20211118161612.png](Pasted%20image%2020211118161612.png) Historically, the study of [periodic orbits](periodic%20orbits) motivated the definition of contact structures. Convention: all manifolds discussed will be smooth, real, Hausdorff, second-countable, connected, not necessarily closed/compact, possibly with boundary. # Definitions **Definition (Hyperplane Field):** A *hyperplane* field $\xi$ is a codimension 1 sub-bundle ${\mathbb{R}}^{n-1} \to \xi \to M$ of the tangent bundle ${\mathbb{R}}^n \to TM \to M$. > See examples. **Definition (Contact Manifold)** A smooth manifold with a hyperplane field $(M^{2n+1}, \xi)$ is *contact* iff $\xi = \ker \alpha$ for some $\alpha \in \Omega^1(M)$ where $\alpha \wedge (d\alpha)^n$ is a top/volume form in $\Omega^{2n+1}(M)$ > Note that $\lambda \wedge (d\lambda)^n = 0$ defines a foliation? **Definition (Reeb Vector Field):** There is a canonical vector field on every contact manifold: the [Reeb vector field]() $X$. This satisfies $\lambda(X) = 1$ and $\iota_x d\lambda = 0$. > *Remark:* [Contact manifold) are cylinder-like boundaries of symplectic manifolds; namely if $M$ is contact then we can pick any $C^1$ increasing function $f: {\mathbb{R}}\to {\mathbb{R}}^+$ (e.g. $f(t) = e^t$) and obtain an exact symplectic form \$`\omega `{=tex}= d(f`\lambda`{=tex}](Contact manifold) are cylinder-like boundaries of symplectic manifolds; namely if $M$ is contact then we can pick any $C^1$ increasing function $f: /RR /to /RR^+$ (e.g. $f(t) = e^t$) and obtain an exact symplectic form $/omega = d(f/lambda)$ on $M_C \coloneqq M \times{\mathbb{R}}$. > Any such $f$ induces a Hamiltonian vector field on $M_C$, and the Reeb vector field is the restriction to $M \times\left\{{0}\right\}$. Definition (Contact Form): ? **Definition (Contact Type):** For $(W, \lambda)$ an exact [transverse) to $Y$, i.e. for every $p\in Y$, we have \$X(p) `\not`{=tex}`\in `{=tex}T_p(Y](transverse) to $Y$, i.e. for every $p/in Y$, we have $X(p) /not/in T_p(Y)$. We say $Y$ is of **contact type** iff there is a neighborhood $U \supset Y$ and a one-form $\lambda$ with $d\lambda = {\left.{{\omega}} \right|_{{U}} } $ making $(U, \lambda)$ of restricted contact type. Remark: $(U, \lambda)$ is of restricted contact type iff $ {\left.{{\lambda}} \right|_{{U}} } $ is a contact form. **Definition (Hypersurface of contact type):** For $(X, \omega)$ a symplectic manifold, a hypersurface $\Sigma \hookrightarrow X$ is of **contact-type** iff there is a contact form $\lambda$ such that $d\lambda = {\left.{{\omega}} \right|_{{Y}} } $. - Not every compact 3-manifold $M$ admits a [Legendrian framing](Legendrian%20framing). Contact geometry also has applications to low-dimensional topology; for example, it has been used by Kronheimer and Mrowka to prove the [property P conjecture](property%20P%20conjecture), by Michael Hutchings to define an invariant of smooth [three manifolds](three%20manifolds), and by Lenhard Ng to define invariants of knots. It was also used by Yakov Eliashberg to derive a topological characterization of [Unsorted/Stein](Unsorted/Stein.md) of dimension at least six.