---
date: 2022-04-05 23:42
modification date: Saturday 9th April 2022 23:29:38
title: "symplectic"
aliases: [symplectic, "symplectic manifold", "symplectic geometry", "symplectic topology", "symplectic form", "symplectic", symplectic basis]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #todo/learning/definitions #projects/notes/reading #projects/my-talks #geomtop/symplectic-topology #MOC 
- Refs:
	- Crash course in manifolds: <https://people.maths.ox.ac.uk/ritter/lie-groups/Ritter-Lie-Groups-HT2015.pdf#page=5> #resources/notes 
	- [Mike's course notes](https://usherugamath.files.wordpress.com/2019/05/8230-s19.pdf) #resources/notes/lectures 
	- [Harvard Course Notes 2021](https://people.math.harvard.edu/~jeffs/SymplecticNotes.pdf) #resources/notes/lectures 
	- Cannas da Silva *Lectures on Symplectic Geometry*  #resources/books 
	- Links to physics: <https://people.maths.ox.ac.uk/ritter/ritter-survey-2.pdf#page=1> #resources/notes
	- Hutchings' course: <https://web.ma.utexas.edu/users/vandyke/notes/242_notes/> #resources/full-courses 
- Links:
	- [Talbot Talk 2](Unsorted/Talbot%20Talk%202.md)
	- [Lie group](Unsorted/Lie%20group.md)

---

# symplectic

![](attachments/2023-03-06-sympbas.png)

**Definition (Symplectic Manifold):**
Recall that $M^{2n}$ is a **symplectic manifold** iff $W$ is smooth of even dimension and admits a  ^395eb1

- **closed**: $d\omega = 0$
	Motivation: the [Lie derivative](Lie%20derivative) of $\omega$ along $V_H$ is 0, i.e. $\mathcal{L}_{V_H}(\omega) = 0$.
- **nondegenerate** $\omega_p: T_p M \cross T_p M \to \RR$; $\omega_p(\vector v,\vector w) = 0~~\forall \vector w \implies \vector v= 0$.
	- Motivation: for every $dH$ there exists a vector field $V_H$ such that $dH = \omega(V_H, \wait)$.
- **skew-symmetric**: $\omega_p(\vector v, \vector w) = \omega_p(\vector w, \vector v)$. 
	- Motivation: $H$ should be constant along flow lines, i.e. $dH(V_H) = \omega(V_H, V_H) = 0$
- **bilinear**: Lifts to a map $T_pM\tensor T_P M \to \RR$
	- Motivation: send any form $\inner{\wait}{\wait}: V\times V \to k$ to the linear map $f: V \to V\dual$ where $v\mapsto f(v) \da \inner{v}{\wait}$. If the pairing is nondegenerate, $\ker f = 0$, and we get an identification $V\cong V\dual$.
	- Yields $TM \cong T\dual M$, which can be combined with $\iota$ to obtain an isomorphism $\mathfrak{X}(M) \cong \Omega^1(M)$ between vector fields and 1-forms.
- **2-form** 
$$
\omega \in \Omega^2(M) = \globsec{\Extalg_\RR^2 \T\dual M}
.$$ 
An important consequence:  to any $f\in C^\infty(M \to \RR)$, we can associate to it a vector field $X_f$.
So there is a map $C^\infty(M\to \RR) \to \globsec{\T M}$?

![](attachments/Pasted%20image%2020220425180053.png)

## Motivations


Relation to [string theory](Unsorted/string%20theory.md):
![](attachments/Pasted%20image%2020220424165246.png)

See [open string](open%20string), [closed string](closed%20string), [Floer homology](Floer%20homology), [quantum cohomology](Unsorted/quantum%20cohomology.md), [symplectic cohomology](symplectic%20cohomology), [Lagrangian Floer cohomology](Unsorted/Lagrangian%20Floer%20homology.md), [wrapped Floer cohomology](wrapped%20Floer%20cohomology), [Fukaya category](Unsorted/Fukaya%20category.md), [homological mirror symmetry](Unsorted/homological%20mirror%20symmetry.md).

# Results

**Proposition:**
$(M, \omega \in \Omega^2(M))$ is symplectic iff $\omega^n \neq 0$ everywhere (c.f. Mike Hutchings).

**Corollary:**
Every symplectic manifold is orientable (since it has a nonvanishing volume form).

> **Important Remark:**
Symplectic structures on smooth manifolds give us a way to generate *flows* on a manifold (by defining a [Hamiltonian](Hamiltonian.md) or a symplectic vector field).

**Definition (Exact Symplectic Manifold):**
$W$ is an **exact** symplectic manifold iff there exists a 1-form $\lambda \in \Omega^1(W)$ such that $d\lambda \in \Omega^2(W)$ is non-degenerate. ^9a87d0

> **Remark**:
> If $(W, \lambda)$ is exact symplectic then $(W, d\lambda)$ is symplectic.
> $\lambda$ is sometimes referred to as a *Liouville form*.

> **Important Remark:**
> If $(W, \lambda)$ is exact and $H: \RR \cross M \to \RR$ is smooth, then the [Hamiltonian flow](Hamiltonian%20flow) $\phi_H^t: M \to M$ is defined for all time and is an *exact symplectomorphism*.

**Theorem:**
There are no closed (compact and boundaryless) exact symplectic manifolds.

*Proof:*
\[
\int_{\bd M} \lambda \wedge \omega^{n-1} 
&= \int_M d(\lambda \wedge \omega^{n-1}) \\ 
&= \int_M d\lambda \wedge \omega^{n-1} + (-1)^{\abs \lambda}\lambda\wedge d\omega^{n-1} \\
&= \int_M \omega \wedge \omega^{n-1} + (-1)^{\abs\lambda} \lambda \wedge 0 \\
&= \int_M \omega^n \\
&= \mathrm{Vol}_{\text{Sp}}(M) \\
&> 0
,\]

so $\bd M \neq 0$, and thus $M$ can not be closed.


**Definition ([isotropic](isotropic)):**
Let $\Lambda$ be the image of an embedded sphere $S^k \to W$.
Then $\Lambda$ is *isotropic* iff $\restrictionof{\lambda}{\Lambda} = 0$.

Given a [almost complex structure](almost%20complex%20structure.md),...?

> Reference: p 68-70 of Cannas da Silva


# Symplectic vector spaces

![](attachments/Pasted%20image%2020220806162036.png)
![](attachments/Pasted%20image%2020220806162114.png)