- Tags
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Refs:
- https://www.ams.org/journals/notices/202204/rnoti-p515.pdf #resources/notes/expository
- Geiges’ An Introduction to Contact Topology #resources/books
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Links:
- open book decomposition
- hyperplane field
- foliate
- contact manifold
- Unsorted/Giroux correspondence
Contact structures
Motivation
Pasted image 20211118161612.png
Historically, the study of periodic orbits 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= d(f\lambda 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\inT_p(Y 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.
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, by Michael Hutchings to define an invariant of smooth three manifolds, and by Lenhard Ng to define invariants of knots. It was also used by Yakov Eliashberg to derive a topological characterization of Unsorted/Stein of dimension at least six.