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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Symplectic geometry\\[15mm]
\small lecture 4: Hamiltonian symplectomorphisms}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf HSE, room 306, 16:20,
\\[2mm] 
September 15, 2021
}
\end{center}

\newpage

{\bf \blue Symplectic structure on the total space of 
cotangent bundle}

From now on, {\bf \blue the total space of the cotangent bundle}
to $M$ is denoted as $T^*M$. The {\bf \blue diffeomorphism group} of $M$
is denoted $\Diff(M)$.

\theorem Let $M$ be a smooth manifold. {\bf \red Then $T^*M$
is equipped with a natural, $\Diff(M)$-invariant
symplectic form $\omega$.}

\proof
Let $\pi:\; TM \arrow M$ denote the projection.
Consider a point $(x, \xi)\in T^*M$, where
$x\in M$ and $\xi \in T^*_x M$.
Let $\theta\in \Lambda^1 (TM)$ be a 1-form which takes
a tangent vector $v \in T_{(x, \xi)}T^*M$ and maps
it to $\langle D\pi(v), \xi\rangle$.
Here, $D\pi:\; T(T^*M) \arrow TM$ is the differential,
and $\langle \cdot, \cdot\rangle$ the pairing between
$T_xM$ and $T^*_xM$.

If we introduce the coordinates $p_1, ..., p_n$ on $M$
and $q_1, ..., q_n$ on the fibers of $T^*M\arrow M$
dual to $p_1, ..., p_n$, the form $\theta$
at a point $p_1,..., p_n, q_1, ... q_n$
takes a vector $\sum_i f_i \frac d{dp_i} + \sum_i g_i \frac d{dpq_i}$
to $\sum_i f_i q_i$. Therefore, $\theta=\sum_i q_i dp_i$.
{\bf \purple In the same coordinates $d\theta = \sum_i dq_i \wedge dp_i$;
this form is symplectic and by construction $\Diff(M)$-invariant.}
\endproof


\newpage

{\bf \blue Lagrangian subspaces}

\remark
Let $(V, \omega)$ be a symplectic vector space,
$\dim_\R V=2n$,
and $W\subset V$ be a subspace of dimension $m$.
Denote by $W^\bot$ the symplectic orthogonal to $W$.
Then $\frac{W}{W^\bot \cap W}$ is a symplectic
vector space, hence $\rk (\omega\restrict W)=  \dim W - \dim (W^\bot \cap W)$. 
Since $\dim W^{\bot}= 2n - \dim W$, the intersection
$W^\bot \cap W$ has dimension at most $n$.

\definition
A subspace $W\subset V$ is called\\
(i) {\bf \blue isotropic} if $\omega\restrict W=0$;
in this case $W\subset W^\bot$, hence $\dim W \leq n$.\\
(ii) {\bf \blue coisotropic} if $\omega\restrict W^\bot=0$,
or, equivalently, $W^\bot \subset W$. 
In this case $W\supset W^\bot$, hence $\dim W \geq n$.\\
(iii) {\bf \blue Lagrangian} if it is isotropic and
coisotropic, that is, $W^\bot=W$.

\remark
Isotropic and coisotropic subspaces {\bf \purple have minimal possible
rank of $\omega$ for a given dimension.}

\remark
{\bf \purple An $\omega$-orthogonal complement to an 
isotropic space is coisotropic,}
and vice versa.


\newpage

{\bf \blue Lagrangian submanifolds}

\definition
Let $X\subset M$ be a submanifold in a symplectic manifold.
It is called {\bf \blue Lagrangian} if $T_x X \subset T_xM$
is Lagrangian for all $x\in X$.

\theorem
Let $\xi\in \Lambda^1 M$ be a 1-form,
and $\Gamma_\xi\subset T^* M$ its graph, considered
as a submanifold in the total space of the cotangent
bundle. {\bf \red Then $\Gamma_\xi$ is Lagrangian
if and only if $d\xi=0$.}

\proof
Let $\sigma:\; x \mapsto (x, \xi(x))$ be the standard
diffeomorphism from $M$ to $\Gamma_\xi$.
Consider the restriction of $\theta$ to $\Gamma_\xi$.
For each  $u\in T_{(x, \xi(x))}\Gamma_\xi$,
the form $\theta$ takes $u$ to $\xi(d\pi(u))$.
This implies that $\sigma^* \theta\restrict{\Gamma_\xi} = \xi$,
hence $\sigma^* \omega\restrict{\Gamma_\xi} = d\xi$.
\endproof

\newpage

{\bf \blue Hamiltonian vector fields}

\definition
Let $v\in TM$ be a vector field on a symplectic
manifold $(M, \omega)$. We say that
$v$ is {\bf \blue symplectomorphic} if $\Lie_v\omega=0$,
that is, if $\omega$ is invariant under the corresponding
diffeomorphism flow.

\remark From Cartan's formula, we have
$\Lie_v\omega=d (i_v\omega)$,
hence {\bf \purple $v$ is symplectomorphic if and only
if the $\omega$-dual 1-form is closed.}

\definition
Let $v\in TM$ be a symplectomorphic 
vector field on a symplectic manifold $(M, \omega)$,
and $\eta:= i_v\omega$ the corresponding 1-form.
We say that $v$ is {\bf \blue a Hamiltonian vector field}
if $i_v\omega$ is exact. Its {\bf\blue Hamiltonian}
is a function $f$ such that $df= i_v\omega$.
The group of {\bf \blue Hamiltonian symplectomorphisms} is
generated by diffeomorphisms obtained by exponents
of a time-dependent vector field $v_t$, which is Hamiltonian for all
$t\in [0,1]$.

\remark We have an exact sequence 
\[
0 \arrow \R \arrow C^\infty(M) \stackrel \delta \arrow
\Ham(M) \arrow 0
\]
If we identify  $\Ham(M)$ with exact 1-forms,
the differential $\delta:\; C^\infty(M)\arrow \Ham(M)$
{\bf \purple is identified with the de Rham differential.}

\newpage

{\bf \blue The flux}

\definition
Let $\Psi_t$ be a flow of symplectomorphisms of $(M,\omega)$,
with $t\in [0, a]$ and $\Psi_0=\Id_M$.
Given a homology class $[u]\in H_1(M, \Z)$,
consider its representative as a smooth circle
$u:\; S^1\arrow M$. Then $\Psi_t(u)$ takes
$(s\in S^1, t\in [0,1])$ to $\Psi_t(u(s))$,
giving a smooth map 
$\Psi_t(u):\; S^1 \times [0, 1]\arrow M$.
Denote its image by $\Psi_{[0,1]}(u)$.
Define {\bf \blue the flux}
of $\Psi_t$ on $u$ as $\int_{\Psi_{[0,1]}(u)}\omega$.

\claim The flux of $\Psi_t$ on $u$ {\bf \red is independent
from the choice of $u$ in $[u]$.}

\proof 
Suppose that $u_\tau$, $\tau\in [0,1]$ 
is a homotopy of $u_0$ to $u_1$. Consider the map
$\tilde\Psi:\; S^1\times [0,1] \times [0,1]$
mapping $(s, t, \tau)$ to $\Psi_t(u_\tau)$.
The surfaces $\Psi_{[0,1]}(u_0)$, $\Psi_{[0,1]}(u_1)$,
$\cup_{\tau\in [0,1]} u_\tau$
and $\cup_{\tau\in [0,1]} \Psi_1(u_\tau)$
bound a solid torus $\im \tilde \Psi$,
hence {\bf \purple to prove that 
$\int_{\Psi_{[0,1]}(u_1)}\omega = \int_{\Psi_{[0,1]}(u_2)}\omega$,
it would suffice to show that
$\int_{\cup_{\tau\in [0,1]} u_\tau}\omega=
\int_{\cup_{\tau\in [0,1]} \Psi_a(u_\tau)}\omega$,}
which is clear because $\Psi_1$ is a 
symplectomorphism.

To pass from homotopy to homology, we notice
that flux is additive on unions of curves,
hence {\bf \purple vanishes on a commutator of $\pi_1(M)$, }
and defines a map from $\frac{\pi_1(M)}{[\pi_1(M), \pi_1(M)]}=H_1(M)$.
\endproof

\newpage

{\bf \blue The flux conjecture}


\claim {\bf \red Flux of a Hamiltonian diffeomorphism vanishes.}

\proof
Let $\Psi_t$ be a Hamiltonian symplectomorphism flow on 
$(M,\omega)$, with $t\in [0, 1]$ and $\Psi_t^{-1}\frac {d\Psi_t}{dt}=v_t$,
where $v_t$ is a Hamiltonian vector field with Hamiltonian $H_t$,
and $u:\; S^1 \arrow M$ a circle. To prove that the flux
vanishes it would suffice to show that
$\frac{d}{dt} \int_{\Psi_{[0,t]}(u)}\omega=0$.
This derivative can be computed
as $\int_{\Psi_t(u))} \omega(\frac {d\Psi_t}{dt}, \cdot)$.
However, the 1-form $\omega(\frac {d\Psi_t}{dt}, \cdot)= \Psi_t^*(dH_t)$
is exact, because $v_t$ is Hamiltonian, hence 
$\frac{d}{dt} \int_{\Psi_{[0,t]}(u)}\omega=0$.
\endproof

\remark The famous ``flux conjecture'' states
that vanishing of the flux is a necessary and sufficient condition:
{\bf \blue if $\Psi_t$, $t\in [0,1]$ is a flow of symplectomorphisms, 
and the flux of $\Psi_1$ vanishes, then $\Psi_1$
is a Hamiltonian symplectomorphism. }
Flux conjecture was proven by Kaoru Ono using Floer theory.

\remark Flux conjecture implies that {\bf \purple Hamiltonian
symplectomorphisms is closed in the group  of symplectomorphisms.}


\newpage

{\bf \blue Commutator of Hamiltonian vector fields}

\example 
Let $u_\epsilon$ be a rotation of a 2-dimensional torus $S^1\times S^1$
with the product metric which comes from unit circles 
and the symplectic structure given
by the volume form. Consider a rotation along the first circle
with angle $\epsilon$. Then its flux through the second circle
is the area of the segment bounded by the first
circle and its image, that, is $2\pi\epsilon$.
In particular, {\bf \red rotation of a 
torus is not a Hamiltonian symplectomorphism}.

\theorem
Let $X$, $Y$ be a Hamiltonian vector fields
on $(M, \omega)$. {\bf \red Then the commutator $[X,Y]$ is Hamiltonian.}

\proof
Let $f, g\in C^\infty M$, and let $X_f, X_g$ be the
corresponding Hamiltonian vector fields.
Define {\bf \blue the Poisson bracket}
as $\{f, g\}:= \omega(X_f, X_g)$.
By definition, $\omega(X_f, X_g)= \langle dg, X_f\rangle = \Lie_{X_f}(g)$,
which gives $[X_f, X_g] =  \Lie_{X_f}(X_g)= X_{\Lie_{X_f}(g)}= X_h$,
where $h= \omega(X_f, X_g)$.
Therefore, {\bf \purple commutator of Hamiltonian vector fields
corresponds to the Poisson bracket on their Hamiltonian functions.}
\endproof


\newpage

{\bf \blue Hamiltonian vector fields on Lagrangian fibrations}

\definition
Let $(M, \omega)$ be a symplectic manifold,
and $\pi:\; M \arrow X$ a smooth submersion.
It is called {\bf \blue a Lagrangian fibration}
if all its fibers are Lagrangian.

\claim Let $\pi:\; M \arrow X$ 
be a Lagrangian fibration, and $H$ a function on $X$.
Then {\bf \red the corresponding Hamiltonian vector field $v$
is tangent to the fibers of $\pi$.} Moreover, 
{\bf \red $v$ is non-degenerate
everywhere on a fiber $\pi^{-1}(x)$ 
if and only if $dH\neq 0$ in $x$;}
otherwise $v\restrict \pi^{-1}(x)=0$.

\proof Let $L:= \pi^{-1}(x)$.
Consider $\omega$ as a map from $T_m M$
to $T_m^* M$. Then $\omega^{-1}$ takes 1-forms vanishing
in $T_m L$ to vectors $v\in T_m M$ such that
$\omega(v, \cdot)$ vanishes on $T_mL$.
However, $T_mL^\bot= T_mL$ because $L$
is Lagrangian. {\bf \purple Therefore, $\omega^{-1}$
takes 1-forms vanishing on $T L$
to the vector fields tangent to $L$.}

The forms vanishing on $TL$ are generated by $\pi^* \Lambda^1 X$,
hence the corresponding Hamiltonian vector fields are tangent to
the fibers of $\pi$. 
\endproof


\remark 
Let $L= \pi^{-1}(x)$ be a fiber of a 
Lagrangian fibration. Then $\omega$ defines
a non-degenerate pairing between $T_mL$
and $T_x X$. This implies that {\bf\red the bundle
$TL$ is trivial.}


\newpage

{\bf \blue Collections of commuting Hamiltonians}

\remark 
Let $\pi:\; M \arrow X$ be a Lagrangian fibration,
and $v_1, v_2$ two Hamiltonian vector fields obtained
from $H_1, H_2\in C^\infty X$ as above. 
{\bf \purple Then $H_1$, $H_2$ commute.} Indeed, the Poisson commutator 
$\{H_1, H_2\}$ is expressed as $\omega(v_t, u_t)$, vanishing for the
vector fields tangent to fibers of $\pi$.

\proposition
Let $\pi:\; M \arrow X$ be a Lagrangian fibration,
and $L$ its fiber. Then {\bf \red the bundle $TL$ has a basis of 
globally defined commuting vector fields.}
Moreover, {\bf \red if $L$ is compact, it is a torus.}

\proof
Let $L=\pi^{-1}(x)$ be a fiber, and
$H_1, ..., H_n\in C^\infty X$ be a collection of
functions such that $dH_1, ..., dH_n\restrict T_x X$ is a basis
in $T^*_x X$. Then {\bf \purple the corresponding vector fields
commute and define a basis in $TL$}. If these vector fields
are {\bf \blue complete} (that is, can be integrated to a flow
for all $t\in \R$), the manifold $L$ is 
equipped with a transitive $\R^n$-action,
which gives $L= \R^n/\Lambda$, where $\Lambda$
is a discrete subgroup. If $L$ is compact,
$\Lambda$ is a cocompact lattice, hence $L$ is a torus.
\endproof



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