Contact Geometry: Foundations¶

Job: contact forms and structures, the Reeb vector field, Gray stability, open-book support (Giroux pointer), and the dictionary between contact boundaries and symplectic fillings.

Core¶

Contact forms and structures¶

Let $Y$ be a smooth oriented manifold of dimension $2n+1$. A contact form is $\alpha\in\Omega^1(Y)$ with $$ \alpha\wedge(d\alpha)^n \neq 0 $$ everywhere. The contact structure is the hyperplane field $\xi=\ker\alpha$. Two forms define the same $\xi$ iff $\alpha'=f\alpha$ for some $f\neq 0$ (usually take $f>0$ to fix coorientation).

Equivalent: $(d\alpha)|_\xi$ is a symplectic form on the bundle $\xi\to Y$. Maximal nonintegrability: no integral hypersurface of $\xi$ through a point (Frobenius fails maximally).

Object	Definition / role
Cooriented contact structure	$\xi=\ker\alpha$ with oriented line $TY/\xi$
Contactomorphism	diffeomorphism $\varphi$ with $\varphi_\xi=\xi$ (equiv. $\varphi^\alpha=f\alpha$)
Legendrian submanifold	$L^n\subset Y^{2n+1}$, $TL\subset\xi$ (half-dimensional isotropic)
Transverse knot (dim 3)	knot everywhere transverse to $\xi$

Standard models.

$(\mathbb{R}^{2n+1},\alpha_{\mathrm{st}}=dz+\sum x_i dy_i)$
$(S^{2n+1},\xi_{\mathrm{st}})$ as boundary of the ball in $\mathbb{C}^{n+1}$
1-jet space $J^1(Q)=T^*Q\times\mathbb{R}$ with $\alpha=dz-\theta$

Darboux (contact). Every contact manifold is locally contactomorphic to $(\mathbb{R}^{2n+1},\alpha_{\mathrm{st}})$ — no local invariants of $\xi$ alone.

Reeb vector field¶

Given contact form $\alpha$, the Reeb field $R=R_\alpha$ is the unique vector field with $$ \iota_R\alpha=1,\qquad \iota_R d\alpha=0. $$ Flow preserves $\alpha$ ($\mathcal{L}_R\alpha=0$), hence preserves $\xi$. Dynamics of $R$ depend on the choice of $\alpha$ within the conformal class — different forms yield different Reeb flows for the same $\xi$.

Weinstein conjecture. On a closed contact manifold, every contact form has at least one closed Reeb orbit. Proved in dim 3 (Taubes, via Seiberg–Witten = ECH); open in full generality in higher dimensions (many cases known: Weinstein fillable, $S^1$-invariant, etc.).

Action spectrum. Periods of closed Reeb orbits; filtered contact homology / cylindrical contact homology (when defined) package this data.

Gray stability¶

Theorem (Gray). Let $\xi_t$, $t\in[0,1]$, be a smooth path of contact structures on a closed manifold $Y$. Then there is an isotopy $\varphi_t$ of $Y$ with $(\varphi_t)_*\xi_0=\xi_t$.

Proof idea. Write $\xi_t=\ker\alpha_t$; solve for a vector field $X_t\in\xi_t$ with $\mathcal{L}_{X_t}\alpha_t + \partial_t\alpha_t = \mu_t\alpha_t$ (Moser-type equation using nondegeneracy of $d\alpha$ on $\xi$). Integrate $X_t$.

Contrast with symplectic Moser: no cohomology constraint — contact structures in a connected component of the space of contact hyperplane fields are isotopic on closed manifolds.

Open books (pointer)¶

Definition sketch. An open book $(\mathcal{B},\pi)$ on closed $Y$ is a binding $\mathcal{B}\subset Y$ (codim 2 oriented submanifold) and fibration $\pi:Y\setminus\mathcal{B}\to S^1$ with pages $\pi^{-1}(\theta)$ compact with boundary $\mathcal{B}$.

Theorem (Giroux). In dimension 3: contact structures up to isotopy $\leftrightarrow$ open books up to positive stabilization. Higher dimensions: every contact structure is supported by an open book (Giroux–Mohsen); uniqueness fails without stabilization caveats.

Supporting contact form: $\alpha>0$ on binding (oriented) and $d\alpha$ symplectic on pages. Open books connect contact topology to mapping class groups of the page (monodromy) — monodromy factorization encodes surgery presentations.

Symplectic fillings¶

A compact symplectic manifold $(W,\omega)$ is a strong filling of $(Y,\xi)$ if $\partial W=Y$, $\omega$ is exact near the boundary via a Liouville field $V$ transverse outward ($\mathcal{L}_V\omega=\omega$), and $\xi=\ker(\iota_V\omega)|_Y$. Weak filling: $\omega|_\xi$ symplectic (no Liouville control). Stein / Weinstein fillings: handle decompositions with isotropic cores; Stein from complex/J-convex geometry.

Type	Strength
Stein / Weinstein	strongest geometric control; handlebody theory
Strong (Liouville)	Liouville vector field at boundary; completion to Liouville manifold
Weak	only $\omega

Symplectization. $(\mathbb{R}_t\times Y, d(e^t\alpha))$ is the model infinite cone; strong fillings glue to positive ends of symplectizations. Contact homology and symplectic field theory (SFT) are built from holomorphic curves in symplectizations and completed fillings.

Obstructions. Contact structures with overtwisted disks (dim 3) or plastikstufen / loose charts (higher dim) are not fillable. Gromov–Eliashberg: Stein fillable $\Rightarrow$ tight in dim 3.

See it¶

Tight vs overtwisted $S^3$. Unique tight contact structure on $S^3$ up to isotopy ($\xi_{\mathrm{st}}$); overtwisted structures classified by homotopy classes of plane fields (Eliashberg). Only the tight one is fillable (by the 4-ball). Lutz twists produce overtwisted structures from any contact 3-manifold.

Pseudoconvex boundaries. Boundary of a strictly pseudoconvex domain in a complex manifold carries a natural contact structure (CR distribution $\xi=TY\cap JTY$). Stein fillings arise as complex domains — the bridge complex geometry $\leftrightarrow$ contact.

Check¶

Show that $R$ is uniquely determined by $\iota_R\alpha=1$ and $\iota_R d\alpha=0$.

Nondegeneracy: $\alpha\wedge(d\alpha)^n\neq 0$ implies the map $v\mapsto(\alpha(v),d\alpha(v,\cdot))\in\mathbb{R}\oplus\xi^*$ is a bundle isomorphism $TY\to\mathbb{R}\oplus\xi^*$. Hence exactly one $R$ hits $(1,0)$.

Why does a Liouville vector field $V$ near $\partial W$ induce a contact form on $\partial W$?

Set $\alpha=(\iota_V\omega)|_{\partial W}$. Cartan: $\mathcal{L}_V\omega=d\iota_V\omega=\omega$ (since $d\omega=0$), so $d\alpha=\omega|_{\partial W}$ on $T\partial W$. Transversality of $V$ plus $\omega^n\neq 0$ give $\alpha\wedge(d\alpha)^{n-1}\neq 0$ on the $(2n-1)$-boundary when $\dim W=2n$.

Pitfalls¶

Same $\xi$, different Reeb dynamics — periodic orbit statements need a form, not only $\xi$.
Strong vs weak fillability diverge in higher dimensions; in dim 3 they are closer but still distinct notions historically intertwined with taut foliations.
Legendrian $\neq$ Lagrangian: Legendrian lives in contact $(2n+1)$; Lagrangian in symplectic $2n$. Suspension/cone constructions relate them in the symplectization.
Contact homology requires orbit nondegeneracy, cylindrical almost complex structures, and often virtual perturbations — naive counts can fail.

Refs¶

Geiges, An Introduction to Contact Topology
Cieliebak–Eliashberg, From Stein to Weinstein and Back
Etnyre, “Lectures on open book decompositions and contact structures”
Eliashberg–Givental–Hofer, SFT outline papers

Next¶

Contact index · Legendrian & Transverse · Contact Homology & SFT · h-Principle & Flexibility · ECH & Weinstein · Symplectic foundations · Floer homology · Differential geometry · Complex geometry · Mathematical physics