Contact Geometry¶
Job: odd-dimensional geometry of maximally nonintegrable hyperplane fields — Reeb dynamics, stability theorems, and the boundary theory of symplectic fillings / open books.
Core¶
A contact structure on a \((2n+1)\)-manifold \(Y\) is a hyperplane field \(\xi\subset TY\) that is maximally nonintegrable: locally \(\xi=\ker\alpha\) with \(\alpha\wedge(d\alpha)^n\neq 0\). Equivalent: \((d\alpha|_\xi)^n\neq 0\). Dual Reeb field \(R_\alpha\) solves \(\iota_{R}\alpha=1\), \(\iota_R d\alpha=0\).
| Layer | Content |
|---|---|
| Contact forms | \(\alpha\), \(\xi=\ker\alpha\); conformal class |
| Reeb dynamics | periodic orbits; Weinstein conjecture |
| Stability | Gray: isotopic contact structures through contact |
| Open books | supporting open books (Giroux) |
| Fillings | strong / weak / Stein / Liouville symplectic fillings |
Reading map¶
| Page | Focus |
|---|---|
| Foundations | contact forms, Reeb, Gray stability, open books pointer, symplectic fillings |
| Legendrian & Transverse | Legendrian knots / classical invariants (tb, rot, Bennequin); Chekanov DGA; EFH loose vs rigid; transverse / self-linking / HF |
| Contact Homology & SFT | contact homology / cylindrical / Conley–Zehnder; full SFT / \(A_\infty\) / BV / pair-of-pants; polyfold (HWZ); string topology |
| h-Principle & Flexibility | Gromov \(h\)-principle; overtwisted (3D Eliashberg) / loose (BEM higher-D); Weinstein conjecture / Taubes 3D; flexible Weinstein |
| ECH & Weinstein | ECH (Hutchings) / Hutchings–Taubes \(\cong\) SW Floer; Weinstein via ECH; ECH capacities / symplectic embedding; contact invariant \(c(\xi)\) |
Dependency order¶
- Differential forms and distributions — differential geometry.
- Symplectic geometry of filling / symplectization — symplectic geometry.
- 3-manifold topology for tight/overtwisted dichotomy (dim 3 special).
- Floer-type invariants — Floer homology, ECH/HF pointers.
Interfaces¶
| Area | Contact |
|---|---|
| Symplectic geometry | symplectization \((\mathbb{R}\times Y,d(e^t\alpha))\); convex boundaries |
| Differential geometry | hyperplane fields, Frobenius integrability (contact = opposite) |
| Mathematical physics | time-dependent Hamiltonians; thermodynamics contact models |
| Complex geometry | strictly pseudoconvex CR boundaries of Stein domains |
| Topology | 3-manifold contact topology; Heegaard Floer links |
Notation (paper-standard)¶
- \(\xi=\ker\alpha\), \(R_\alpha\), \((\xi,d\alpha)\) symplectic bundle
- \((\mathbb{R}\times Y,d(e^t\alpha))\) symplectization; contactomorphism group \(\operatorname{Cont}(Y,\xi)\)
- Tight / overtwisted (dim 3); Weinstein / Liouville domain fillings
See it¶
Standard contact \(S^{2n+1}\). \(\alpha_0=\frac{1}{2i}\sum(z_j d\bar z_j-\bar z_j dz_j)|_{S^{2n+1}}=\sum(x_j dy_j-y_j dx_j)\) (up to scale) has Reeb flow the Hopf \(S^1\)-action. Symplectization recovers \(\mathbb{C}^{n+1}\setminus\{0\}\) (up to Liouville completion). This is the model convex boundary of the ball — the contact sphere bounds a Stein filling.
Check¶
How does Frobenius fail for a contact distribution \(\xi=\ker\alpha\)?
Frobenius integrability would require \(\alpha\wedge d\alpha=0\) (ideal differentially closed). Contact requires \(\alpha\wedge(d\alpha)^n\neq 0\), the maximal nonintegrability condition — integral submanifolds have dimension at most \(n\) (Legendrian bound).
Why is Gray stability for contact structures on closed manifolds stronger than Moser for symplectic forms?
Contact structures are open conditions on hyperplane fields (nonintegrability); Gray produces an isotopy through contactomorphisms whenever \(\xi_t\) is a path of contact structures. Symplectic forms need fixed cohomology class for Moser — contact has no analogous \(H^2\) obstruction on closed \(Y\).
Pitfalls¶
- Contactomorphism vs diffeomorphism preserving a fixed \(\alpha\): \(\xi\) determines \(\alpha\) only up to conformal factor \(f\alpha\), \(f>0\).
- Overtwisted vs tight is a 3D dichotomy (Eliashberg); higher-dimensional looseness (Borman–Eliashberg–Murphy) is the analog.
- Filling \(\neq\) unique: a contact manifold may admit many non-deformation-equivalent symplectic fillings, or none.
Refs¶
- Geiges, An Introduction to Contact Topology
- Etnyre, lecture notes on contact geometry
- Cieliebak–Eliashberg, From Stein to Weinstein and Back
- McDuff–Salamon, Introduction to Symplectic Topology (contact chapters)
Next¶
Foundations · Legendrian & Transverse · Contact Homology & SFT · h-Principle & Flexibility · ECH & Weinstein · Symplectic geometry · Floer homology · Differential geometry · Complex geometry · Mathematical physics