Symplectic Geometry¶

Job: even-dimensional geometry of nondegenerate closed 2-forms — Hamiltonian dynamics, Lagrangian submanifolds, and the analytic/topological toolkit (Moser, Floer) for symplectic manifolds.

Core¶

A symplectic manifold $(M,\omega)$ is a smooth $2n$-manifold with $\omega\in\Omega^2(M)$ closed ($d\omega=0$) and nondegenerate (the map $v\mapsto\iota_v\omega$ is an isomorphism $TM\to T^*M$). Local model: $(\mathbb{R}^{2n},\omega_0=\sum dq_i\wedge dp_i)$ by Darboux.

Layer	Content
Forms / Darboux	$\omega$ closed + nondeg.; local standard form
Hamiltonian flows	$X_H$ via $\iota_{X_H}\omega=-dH$; Poisson bracket
Moment maps	Hamiltonian $G$-actions; reduction
Lagrangian geometry	$\dim L=n$, $\omega
Floer package	periodic orbits / HF; Arnold conjecture

Reading map¶

Page	Focus
Foundations	symplectic forms, Darboux, Hamiltonian flows, moment maps, Moser, Lagrangians
Floer homology	Floer sketch, PSS, Arnold conjecture applications

Dependency order¶

Smooth manifolds, forms, de Rham — differential geometry, topology.
Lie groups / coadjoint orbits — representation theory.
Foundations $\to$ Floer; contact boundary via contact geometry.
Complex/Kähler special cases — complex geometry; physics — mathematical physics.

Interfaces¶

Area	Contact
Differential geometry	cotangent bundles, almost complex $J$ tamed by $\omega$
Contact geometry	convex boundaries, symplectizations, fillings
Complex geometry	Kähler $\Rightarrow$ symplectic; $\omega$ closed positive $(1,1)$
Mathematical physics	classical mechanics, geometric quantization
Hamiltonian & KAM	Hamiltonian dynamics, KAM tori, action-angle, Arnold diffusion
Mirror symmetry	Fukaya $A_\infty$ / HMS
Representation theory	coadjoint orbits, geometric quantization of orbits

Notation (paper-standard)¶

$\omega$, $\omega^n/n!$ Liouville volume; $\operatorname{Symp}(M,\omega)$, $\operatorname{Ham}(M,\omega)$
$X_H$, $\{F,G\}=\omega(X_F,X_G)$; moment map $\mu:M\to\mathfrak{g}^*$
Lagrangian $L$; Hamiltonian isotopy $\varphi_H^t$; action functional $\mathcal{A}_H$

See it¶

Cotangent bundle. $T^*Q$ with $\omega_{\mathrm{can}}=-d\theta$, $\theta=\sum p_i\,dq_i$ (tautological 1-form): the universal Hamiltonian phase space of a configuration manifold $Q$. Zero section and conormal bundles $N^*S\subset T^*Q$ are Lagrangian; geodesic flow is Hamiltonian for the dual metric kinetic energy. Generating functions and generating hypersurfaces live naturally here — the model for Weinstein’s Lagrangian neighborhood theorem.

Check¶

Why does $d\omega=0$ matter for the Poisson bracket to satisfy Jacobi?

Nondegeneracy alone defines a bivector and a bracket, but the Jacobi identity for $\{\,,\,\}$ is equivalent to $[\pi,\pi]_S=0$ (Schouten), which for the inverse bivector of $\omega$ is equivalent to $d\omega=0$. Closedness is the integrability condition for the Hamiltonian formalism.

Darboux says there are no local symplectic invariants in finite dimensions. Where do global invariants live?

In global topology/dynamics: $\pi_2$, Gromov width, Hofer geometry of $\operatorname{Ham}$, Floer homology, symplectic capacities — all invisible in a Darboux chart.

Pitfalls¶

Almost symplectic ($d\omega$ may fail) vs symplectic ($d\omega=0$); nondegeneracy alone is not enough for Moser/Darboux as usually stated.
Kähler $\Rightarrow$ symplectic, not conversely — many symplectic manifolds admit no compatible integrable complex structure.
Hamiltonian vs symplectic isotopy: $\operatorname{Ham}\subset\operatorname{Symp}_0$; flux/Calabi-type invariants measure the gap.

Refs¶

McDuff–Salamon, Introduction to Symplectic Topology
da Silva, Lectures on Symplectic Geometry
Arnold, Mathematical Methods of Classical Mechanics
Cannas da Silva / Weinstein notes on Lagrangian geometry

Next¶

Foundations · Floer homology · Contact geometry · Differential geometry · Complex geometry · Mathematical physics · Mirror symmetry