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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{2}{Symplectic handout 2: diffeomorphism flows}

%{\scriptsize
%  {\bf Rules:} You may choose to solve only 
%``hard'' exercises (marked with !, * and **) 
%or ``ordinary'' ones (marked with ! or unmarked),
%or both, if you want to have extra stuff to work.
%To have a perfect score, a student must obtain
%(in average) a score of 10 points per week.
%
%If you have got credit for 2/3 of ordinary problems
%or 2/3 of ``hard'' problems, you receive  
%$6t$ points, where $t$ is a number depending on the
%date when it is done. Passing all ``hard'' 
%or all ``ordinary'' problems brings you $10t$ points.
%Solving of ``**'' (extra hard) problems is not
%obligatory, but each such problem gives you a credit
%for 2 ``*'' or ``!'' problems in the ``hard'' set.
%
%The first 3 weeks after giving a handout, $t=1.5$,
%between 21 and 35 days, $t=1$, and afterwards, $t=0.7$.
%The scores are not cumulative, only the
%best score for each handout counts.
%}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Prove that each $\R$-linear derivation of the
ring $C^\infty \R^n$ is induced by a vector field
on $\R^n$.
\ez

\exercise[*]
Let $M$ be a manifold.
Prove that each $\R$-linear derivation of the
ring $C^\infty M$ is induced by a vector field
on $M$.
\ez

\exercise
Let $M$ be a compact manifold, and $\goth m$ a maximal ideal
in the ring $C^\infty M$ of all smooth real functions. 
Prove that $\goth m$ is the ideal
of all functions vanishing in a point of $M$.
\ez

\exercise
Let $M$ be a compact manifold and $A$ an $\R$-linear 
automorphism of $C^\infty M$. Prove that the corresponding map 
$A:\; \Spec_\R(C^\infty M)\arrow \Spec_\R(C^\infty M)$
is a homeomorphism with respect to the topology
induced from $M$.
\ez

\exercise
Let $M$ be a compact manifold, and $\Aut_\R(C^\infty M)$
the group of all $\R$-linear automorphisms of the ring
$C^\infty M$. Prove that the natural embedding
from $\Diff(M)$ to $\Aut_\R(C^\infty M)$
is an isomorphism.
\ez

\hint
Use the previous exercise.
\eh



\definition
Let $v_t\in TM$ be a vector field, depending on 
a parameter $t\in [0,a]$
and $\Psi_t\in \Diff(M)$ a flow of diffeomorphisms,
$t\in [0,a]$. We say that $\Psi_t$ is {\bf tangent to $v_t$},
or {\bf is obtained by integrating $v_t$}, or 
{\bf is equal to exponent of $v_t$} if for all $m\in M$, the
vector $\Psi_t^{-1} \frac {d\Psi_t}{dt} \in T_m M$
is equal to $v_t \restrict m$.
\ed

\exercise
Let $v_t=v\in TM$ be a time-independent 
vector field, and $\Psi_t\in \Diff(M)$ the
corresponding flow of automorphisms.
Prove that $v$ is $\Psi_t$-invariant.
\ez

\exercise
Find a vector field $v$ on $M=\R$ such that
there is no flow of diffeomorphisms
$\Psi_t\in \Diff(M)$, $t\in [0, \epsilon]$ tangent to $v$ 
for any $\epsilon >0$.
\ez

From now on, the manifold
$M$ is assumed to be equipped with a finite
covering $\{U_i\}$ by open balls,
with all successive intersection of balls diffeomorphic 
to open balls or empty. We start with proving Theorem 1
from Lecture 2.

\hfill

{\bf Theorem 1:}
Let $\alpha_t\in\Lambda^k(M)$, $t\in [0,1]$ be
a smooth family of exact forms on $M$. Then
there exists a smooth family of forms 
$\eta_t\in\Lambda^{k-1}(M)$, $t\in [0,1]$,
such that $d\eta_t = \alpha_t$.


\exercise
Prove Theorem 1 for an open ball.
\ez

\exercise
Prove Theorem 1 for a union of two open balls,
with the intersection diffeomorphic to an open ball.
\ez

\exercise
Prove Theorem 1 when $\alpha_t$ is a 1-form.
\ez

\exercise\label{_cohomo_smooth_choosing_Exercise_}
Let $[s_t]$ be a smooth family of cohomology classes
on a manifold $X$. Prove that it can be represented by
a smooth family $s_t\in \Lambda^*(X)$.
\ez

\exercise
Let $M = U \cup V$, where $U$ is a ball,
and $V$ a union of $n$ balls for which Theorem 1 
is already proven. Suppose that 
 $\alpha_t = du_t$ on $U$ and $\alpha_t = dv_t$ on $V$,
where $u_t, v_t$ are smooth families, $t\in [0,1]$.
\enum
\ite
Prove that $u_t-v_t$ is closed on $U\cap V$.
\ite Suppose that $u_t-v_t$ is exact on $U\cap V$.
Prove Theorem 1 in this situation.

\ite 
Consider the Mayer-Vietoris exact sequence
\[
H^{i-1}(U)\oplus H^{i-1}(V)\arrow
H^{i-1}(U\cap V)\stackrel \delta \arrow H^i(U\cup V)\arrow H^i(U)\oplus H^i(V).
\]
Let $\alpha$ be a closed form on $U\cup V$
such that $\alpha = du$ on $U$ and $\alpha = dv$ on $V$.
Let $[u-v]$ and $[\alpha]$ be cohomology classes represented
by these forms. Prove that $\delta([u-v]) = [\alpha]$.

\ite Prove that there exists a smooth family
of cohomology classes $[a_t]\in H^{k-1}(V)$ such that
$[a_t]\restrict {U\cap V}=[u_t-v_t]$.


\ite 
Prove that the antiderivatives
$u_t, v_t$ can be chosen in such a way that
$[u_t-v_t]$ is exact.

{\em Hint: Use Exercise \ref{_cohomo_smooth_choosing_Exercise_}.}

\ite Prove that there exists a smooth
family $\eta_t$ such that $d\eta_t=\alpha_t$.
\ee
\ez


\exercise
Let $A_0, A_1$ be volume forms on
a manifold $M$, with 
$\int_M A_0=\int_M A_1$. 
\enum
\ite Prove that $A_\lambda:= \lambda A_1 + (1-\lambda) A_0$
is always a volume form. Prove that 
$\frac{dA_\lambda}{d\lambda}= d(i_{X_\lambda} A_\lambda)$ for a
vector field $X_\lambda$ smoothly depending on $\lambda$.
\ite
Let $\Psi_\lambda$ be the exponent of the 
vector field $X_\lambda$ defined above.
Prove that $\Psi_\lambda(A_0)=A_\lambda$.
\ite
Let $A$ be the set of all volume
forms of constant volume on an
oriented compact manifold $M$.
Prove that the group $\Diff(M)$
of diffeomorphisms acts on 
$A$ transitively.
\ee
\ez

\definition
Let $(M,I)$ be an almost complex manifold,
and $\omega$ a symplectic form. We say that
$\omega$ is {\bf tamed by $I$} if
$\omega(Ix, x)>0$ for any non-zero
vector $x\in T_mM$.
\ed


\exercise
Let $(M,I)$ be an almost complex manifold,
and $\omega_1, \omega_2$ two symplectic forms
tamed by $I$. Assume that $\omega_1$ is
cohomologous to $\omega_2$. Prove that
there exists $h\in \Diff(M)$ such that
$h^*\omega_1=\omega_2$.
\ez

 
\end{document}


