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%version 1.0,\ \   29.09.2021
%version 1.1,\ \   20.10.2021, ex. 4.9 had many errors
%version 1.2,\ \   23.10.2021, 4.1 and 4.4 (minor corrections)


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\lhead{\tiny Symplectic geometry, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{4}{Symplectic handout 4: Hamiltonian isotopies}

 

\exercise
Let $X, Y$ be symplectomorphic  vector fields on $(M, \omega)$. Prove that
$[X,Y]$ is Hamiltonian.
\ez


\exercise
Let $\omega$ be an exact symplectic form,
$\omega=d\eta$, and
and $v:= \omega^{-1}(\eta)$.
\enum
\ite Prove that $\Lie_v \omega=\omega$.
\ite Prove that $\Lie_v\eta=\eta$.
\ee
\ez



\exercise
Let $(M, \omega)$ be a symplectic manifold
such that $\omega=d\eta$, and $X\in TM$
a vector field. 
\enum
\ite Prove that $X$
is symplectomorphic  if and only if 
only if $\Lie_X\eta$ is closed.
\ite Prove that that $X$
 is Hamiltonian if and only if 
only if $\Lie_X\eta$ is exact.
\ee
\ez


\exercise
Let $\eta\in \Lambda^1(\R^{2k+1})$ be a
1-form such that the $(2k+1)$-form 
$\eta \wedge (d\eta)^k$ is non-degenerate,
and $v$ a vector field which satisfies
$\eta(v)=1$ and $\Lie_v(\eta)=0$.
Prove that $v$ is uniquely determined by these 
conditions.
\ez

\exercise
Let $\Theta$ be a volume form on $M$.
We call a vector field $v\in TM$ 
{\bf $\Theta$-hamiltonian} if  
$i_v\Theta$ is exact. Suppose that
$v\in TM$ is $\Theta$-Hamiltonian, and
$w\in TM$ satisfies $\Lie_w\Theta=0$.
Prove that $[v, w]$ is $\Theta$-Hamiltonian.
\ez


\definition
Let $H_t$ be a family of functions on $M$ smoothly
 depending on $t\in [0,1]$, and $X_t:= \omega^{-1}(dH_t)$
the corresponding Hamiltonian vector field. A {\bf Hamiltonian isotopy}
associated with $H_t$ is a diffeomorphism flow $\Psi_t$ which
is tangent to $X_t$ for all $t$, that is, satisfies
$\Psi_t^{-1}\frac{d\Psi_t}{dt} = X_t$, and satisfies
$\Psi_0=\Id$. 
%A {\bf Hamiltonian isotopy with compact support}
%is a Hamiltonian isotopy obtained from a hamily $H_t$ of
%Hamiltonians with compact support. 
A {\bf Hamiltonian
symplectomorphism} is the end point of a Hamiltonian
isotopy, that is, $\Psi_1$. 
\ed

\exercise
Prove that a non-trivial rotation of a 2-torus $T^2= S^1\times S^1$
along one of the two circles is never a Hamiltonian symplectomorphism.
\ez

\exercise
Prove that the group of Hamiltonian symplectomorphism
is a normal subgroup of the group of symplectomorphisms.
\ez


\definition
A submanifold $S\subset M$ of a symplectic manifold
$(M, \omega)$ is called {\bf Lagrangian} if $\dim S=\frac 1 2 \dim M$
and $\omega\restrict S=0$.
\ed

\exercise
Let $S \subset T^*M$ be a  submanifold
of a symplectic manifold $T^*M$ with the standard symplectic form.
\enum
\ite Suppose that $S=\Gamma_\zeta$ is a graph of a 1-form $\zeta\in \Lambda^1 M$
considered as a map $M \arrow T^*M$. Prove that $\Gamma_\zeta$ is Lagrangian
if and only if $\zeta$ is closed.

\ite
Let $\Gamma_\xi$ be a graph of an exact 1-form $\xi$.
Find a Hamiltonian symplectomorphism
which takes the zero section $\Gamma_0$ to
$\Gamma_\xi$.

\ite Let $M$ be compact.
Find an example of a compact, connected Lagrangian submanifold
$S\subset T^*M$ which is not obtained as a graph of a closed 1-form.
\ee
\ez




\exercise
Let $U(1)^{n+1}\subset U(n+1)$ be the subgroup of
rotations which are written diagonally in a certain basis in $\C^{n+1}$,
acting on $\C P^n$, and 
$\mu:\; \C P^n\arrow \goth (u(1)^{n+1})^*$ the corresponding
moment map. 
\enum
\ite
Prove that all fibers of $\mu$ are
Lagrangian subvarieties in $\C P^n$. Prove that all smooth
fibers are tori.
\ite
Prove that $\mu(\C P^n)$ vanishes on the diagonal 
$\1\Id_{\C^n}\subset u(1)^{n+1}$
\ee
\ez


\exercise\label{_Lie_commu_fixing_omega_Exercise_}
Let $\omega$ be a non-degenerate 2-form,
and $v_1, ..., v_{2n}$ a collection of 
commuting vector fields
such that $\Lie_{v_i}\omega=0$ for all $i$.
Suppose that at some point $x\in M$
the vectors $v_i\restrict x$ define
a basis in $T_x M$. Prove that
$\omega$ is closed in a certain 
neighbourhood of $x$.
\ez

\hint 
Find a coordinate system $z_1, ..., z_{2n}$ such that $v_i =\frac{d}{dz_i}$
and express $\omega$ in these coordinates.
\eh

\exercise[*]
Prove Exercise \ref{_Lie_commu_fixing_omega_Exercise_}
without assuming that $v_i$ commute, or find a counterexample.
\ez

\exercise[*]
Find a non-degenerate 2-form $\omega$ on $\R^6$ such that
the Lie algebra of vector fields $v$ such that $\Lie_v\omega=0$
is finite-dimensional.
\ez


\end{document}

\exercise
Let $\pi_1, \pi_2:\; M \times M\arrow M$ be the projections, 
$\omega$ a symplectic form, and
 $\Omega:= \pi_1^*\omega- \pi_2^*\omega$
the corresponding symplectic structure on $M\times M$.
\enum
\ite Let $\Psi:\; M \arrow M$ a diffeomorphism.
Prove that its graph $\Gamma_\Psi\subset M\times M$
is Lagrangian if and only if $\Psi$ is a symplectomorphism.
\ee
\ez