Symplectic Geometry: Foundations¶

Job: symplectic forms, Darboux charts, Hamiltonian vector fields and flows, equivariant moment maps, Moser stability, and the basic theory of Lagrangian submanifolds.

Core¶

Symplectic forms¶

A symplectic form on a smooth manifold $M$ is $\omega\in\Omega^2(M)$ with $d\omega=0$ and $\omega_p$ nondegenerate for all $p$ (equivalently $\omega^n\neq 0$ if $\dim M=2n$). Then $(M,\omega)$ is a symplectic manifold.

Fact	Statement
Dimension	$\dim M$ even
Volume	$\omega^n/n!$ is a nowhere-vanishing top form (Liouville orientation)
Linear model	$(\mathbb{R}^{2n},\omega_0)$, $\omega_0=\sum_{i=1}^n dq_i\wedge dp_i$
Cotangent	$(T^*Q,-d\theta)$ always symplectic
Kähler	closed positive $(1,1)$-form of a Kähler metric is symplectic

Almost complex structures. $J$ is compatible with $\omega$ if $g(u,v)=\omega(u,Jv)$ is a Riemannian metric and $\omega(Ju,Jv)=\omega(u,v)$. Space of compatible $J$ is nonempty and contractible — analytic input for pseudoholomorphic curves.

Darboux theorem¶

Theorem (Darboux). Every symplectic manifold is locally symplectomorphic to $(\mathbb{R}^{2n},\omega_0)$: about each $p$ there are coordinates $(q,p)$ with $\omega=\sum dq_i\wedge dp_i$.

No local curvature invariants of $\omega$ alone (contrast Riemannian geometry). Proof: Moser path method — connect $\omega$ to constant form in a chart and integrate an isotopy.

Hamiltonian flows¶

Given $H\in C^\infty(M)$, the Hamiltonian vector field $X_H$ is defined by $$ \iota_{X_H}\omega = -dH $$ (sign conventions vary; this matches many topology texts). Flow $\varphi_H^t$ (where defined) preserves $\omega$ (Cartan: $\mathcal{L}_{X_H}\omega=d\iota_{X_H}\omega+\iota_{X_H}d\omega=0$).

Poisson bracket: $\{F,G\}:=\omega(X_F,X_G)=X_G(F)=-X_F(G)$. Makes $C^\infty(M)$ a Lie algebra; $H\mapsto X_H$ is a Lie algebra antihomomorphism (with standard signs). Hamilton equations in Darboux coords: $\dot q=\partial H/\partial p$, $\dot p=-\partial H/\partial q$.

Object	Role
$\operatorname{Symp}(M,\omega)$	diffeomorphisms with $\varphi^*\omega=\omega$
$\operatorname{Ham}(M,\omega)$	time-1 maps of (compactly supported) Hamiltonian isotopies
Autonomous flow	level sets of $H$ invariant; energy conservation

Moment maps¶

A Lie group $G$ acts on $(M,\omega)$ by symplectomorphisms. The action is Hamiltonian if there is a moment map $\mu:M\to\mathfrak{g}^*$ with $$ d\langle\mu,\xi\rangle = -\iota_{\xi_M}\omega\qquad(\xi\in\mathfrak{g}), $$ $\xi_M$ the infinitesimal generator, and $\mu$ equivariant for the coadjoint action (or a cocycle-twisted variant).

Marsden–Weinstein reduction. If $0$ is a regular value and $G$ acts freely (properly) on $\mu^{-1}(0)$, then $$ M/!!/G := \mu^{-1}(0)/G $$ inherits a symplectic form $\omega_{\mathrm{red}}$ with $\pi^*\omega_{\mathrm{red}}=i^*\omega$.

Examples. Cotangent lifted $G$-action on $T^*Q$; coadjoint orbits with KKS form; $S^1$-actions and symplectic quotients (toric geometry when $(S^1)^n$ acts effectively on $2n$-manifolds).

Moser theorems¶

Theorem (Moser stability). Compact $M$, path $\omega_t$ of cohomologous symplectic forms ($[\omega_t]$ constant in $H^2_{\mathrm{dR}}$): there is an isotopy $\varphi_t$ with $\varphi_t^*\omega_t=\omega_0$.

Relative Moser / Weinstein tubular nbhd. Neighborhood of a Lagrangian $L$ is symplectomorphic to a neighborhood of the zero section in $T^*L$. Neighborhood of a symplectic submanifold modeled on a symplectic normal bundle.

Consequence. Symplectic forms in a fixed cohomology class on compact $M$ form a connected (in fact, often contractible-mod-Diff) moduli story controlled by Moser — deformations are isotopic when cohomologous.

Lagrangian submanifolds¶

$L\subset(M^{2n},\omega)$ is Lagrangian if $\dim L=n$ and $\omega|_L=0$ (equivalently $TL=\operatorname{ann}_\omega(TL)$).

Example	Notes
Zero section $Q\subset T^*Q$	$\theta
Conormal $N^*S$	to submanifold $S\subset Q$
Graph of $df$	exact Lagrangian in $T^*Q$ near zero section
Graph of symplectomorphism	$\Gamma_\varphi\subset(M\times M,(-\omega)\oplus\omega)$ Lagrangian iff $\varphi$ symplectic
Real locus	fixed set of anti-symplectic involution (when smooth of dim $n$)

Exact Lagrangians: $\lambda|_L=df$ when $\omega=-d\lambda$. Nearby exact Lagrangians in $T^*Q$ are Hamiltonian isotopic to the zero section under many hypotheses (nearby Lagrangian conjecture — open in general; settled in special cases, e.g. $Q=S^1$, $T^n$ partial results, Abouzaid–Kragh, etc.).

See it¶

Coadjoint orbits. For $G$ compact connected, coadjoint orbits $\mathcal{O}\subset\mathfrak{g}^*$ carry the KKS symplectic form $$ \omega_\mu(\operatorname{ad}^_\xi\mu,\operatorname{ad}^_\eta\mu)=\langle\mu,[\xi,\eta]\rangle. $$ Moment map for the $G$-action is inclusion $\mathcal{O}\hookrightarrow\mathfrak{g}^*$. Geometric quantization of integral orbits recovers irreducible representations (Borel–Weil / Kirillov orbit philosophy) — pure symplectic input to representation theory.

Moser in action. On a compact surface, area forms of total area $1$ are all diffeomorphic (Moser + $[\omega]$ determines the class in $H^2\cong\mathbb{R}$) — classification of symplectic structures on surfaces reduces to volume + topology.

Check¶

Prove that Hamiltonian flows preserve $\omega$, using Cartan’s magic formula.

$\mathcal{L}_{X_H}\omega=d(\iota_{X_H}\omega)+\iota_{X_H}(d\omega)=d(-dH)+0=0$. Thus $(\varphi_H^t)^*\omega=\omega$ on the flow domain.

Why does reduction at a nonzero regular value $c\in\mathfrak{g}^*$ typically use the coadjoint stabilizer $G_c$ rather than all of $G$?

Level set $\mu^{-1}(c)$ is preserved by $G_c$, not necessarily by $G$. Equivariance $\mu(g\cdot p)=\operatorname{Ad}^*_g\mu(p)$ moves the value; reduce $\mu^{-1}(c)/G_c$, or shift by equivariant identification with reduction at $0$ for a modified action.

Pitfalls¶

Sign conventions for $X_H$, $\theta$ vs $-\theta$ on $T^*Q$, and left vs right actions on moment maps — check the paper’s convention before quoting formulas.
Presymplectic forms (closed, constant corank) need constraint reduction (Dirac) — not the same as symplectic reduction of nondegenerate $\omega$.
Compact support for $\operatorname{Ham}_c$ vs global $\operatorname{Ham}$ on noncompact $M$; Liouville vector fields and convexity at infinity change the group.
Isotropic / coisotropic / Lagrangian: isotropic $\omega|_{TL}=0$; coisotropic $(TL)^\omega\subset TL$; Lagrangian = both + half-dimensional.

Refs¶

McDuff–Salamon, Introduction to Symplectic Topology
da Silva, Lectures on Symplectic Geometry
Abraham–Marsden, Foundations of Mechanics
Guillemin–Sternberg, Symplectic Techniques in Physics

Next¶

Symplectic index · Floer homology · Contact foundations · CG / contact homology & SFT — the SFT the contact / symplectic · Differential geometry · Complex geometry · Mathematical physics · Lie groups · Hamiltonian & KAM — KAM theory as the dynamical-systems side of Hamiltonian / symplectic

Example	Notes
Zero section \(Q\subset T^*Q\)	$\theta
Conormal \(N^*S\)	to submanifold \(S\subset Q\)
Graph of \(df\)	exact Lagrangian in \(T^*Q\) near zero section
Graph of symplectomorphism	\(\Gamma_\varphi\subset(M\times M,(-\omega)\oplus\omega)\) Lagrangian iff \(\varphi\) symplectic
Real locus	fixed set of anti-symplectic involution (when smooth of dim \(n\))