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%version 1.0,\ \   04.09.2021
%version 1.1, \ \ 06.09.2021, a misprint fixed (Qianyi Shu)
%version 1.2, \ \ 11.09.2021, more misprints (1.6, 1.8) Vadim, Ivan Frolov
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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{1}{Symplectic handout 1: Non-degenerate 2-forms}

%{\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.
%}

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\definition
Let $V$ be a vector space. A {\bf complex structure}
on $V$ is an operator $I\in \End(V)$ which satisfies
$I^2=-\Id$.
\ed


\exercise
Let $V$ be a  real vector space,
and $X$ the set of complex subspaces $W \subset V\otimes_\R \C$ 
which satisfy $W \cap \bar W=0$,
$W+ \bar W= V\otimes_\R \C$. Construct a $GL(V)$-invariant
bijective correspondence between $X$
and the space of  complex structures on $V$.
\ez

\definition
Let $M$ be a manifold. An endomorphism $I\in \End(TM)$, $I^2=-\Id_{TM}$
is called {\bf an almost complex structure}. An $I$-invariant
Riemannian metric is called {\bf an Hermitian metric}.
\ed

\exercise
Let $(M,I)$ be an almost complex manifold, $\dim_\C M=n$.
Prove that $(M,I)$ always admits a Hermitian 
metric $g$. Consider the orientation form $\omega^n$, obtained as
the top exterior power of the corresponding Hermitian form $\omega$.
Prove that the orientation defined by $\omega^n$
is independent from the choice of $g$.
\ez

\exercise
Let $g$ be a positive definite scalar product on
a vector space $V$. 
\enum
\ite Construct a bijection between bilinear symmetric
forms on $V$ and operators $A\in \End(V)$ satisfying
$g(A(x), y)= g(x, A(y))$ (such operators are called
{\bf symmetric}, or {\bf self-adjoint}).
\ite Construct a bijection between bilinear anti-symmetric
forms on $V$ and operators $A\in \End(V)$ satisfying
$g(A(x), y)= -g(x, A(y))$ (such operators are called
{\bf anti-symmetric}).
\ite A symmetric matrix $A$ is called {\bf positive}
if the bilinear symmetric form $x, y \arrow g(A(x), y)$
is positive definite. Construct a bijection
between positive symmetric matrices and positive
definite bilinear symmetric forms.
\ite Let $A$ be a non-degenerate anti-symmetric operator.
Prove that $-A^2$ is positive symmetric.
\ee
\ez

\exercise
Let $g, g_1$ be bilinear symmetric forms on $V$, with $g$
positive definite and $S_g\subset V$ the sphere $\{v\in V\ \ |\ \ g(v,v)=1\}$.
\enum
\ite Denote by $x\in S_g$ an extremum of the function $x\arrow g_1(x,x)$ on $S_g$.
Prove that the 1-forms $g(x,\cdot)$ and $g_1(x, \cdot)$
are proportional.
\ite (``Simultaneous diagonalization theorem'')
Prove that there exists an orthonormal basis
$x_1, ..., x_n$ in $V$ such that $g_1(x_i, x_j)=0$
for $i\neq j$.
\ee
\ez


\hint Take for $x_1$ the point of $S_g$ 
where $x\arrow g_1(x,x)$ reaches maximum, pass to $x^\bot$ 
and apply induction on $\dim V$.
\eh


\exercise
Let $A$ be a positive symmetric operator,
and $A_1$ a positive symmetric operator 
satisfying $A_1^2=A$. Prove that
$A_1$ is unique.
\ez

\hint Use the simultaneous diagonalization theorem.
\eh


\exercise
Let $\omega$ be an antisymmetric 2-form
on a vector space $V=\R^{2n}$, and $g$
a positive definite scalar product.
Prove that there exists a basis $x_1, ..., x_{2n}$,
orthonormal with respect to $g$, 
such that $\omega$ is written in this basis as 
\[
{\small \begin{pmatrix} 
0 & a_1&0 & 0& ... & 0& 0 \\ -a_1 & 0&0 & 0& ... & 0& 0\\
0 & 0 & 0 & a_2 & ... & 0 & 0\\
0 & 0 & -a_2 & 0 & ... &0 & 0\\
...\\
0 & 0 & 0 & 0 & ... &0 & a_n\\
0 & 0 & 0 & 0 & ... &-a_n & 0\\
 \end{pmatrix}}
\]
where $a_1, ..., a_n$ are non-negative real numbers.
\ez

\hint Apply the simultaneous diagonalization theorem
to $g, -A^2$, where $A$ is the anti-symmetric operator
corresponding to $\omega$.
\eh


\exercise
Let $(M,g)$ be a Riemannian manifold,
and $A\in \End(TM)$ a positive symmetric
operator, smooth in $M$. Prove that there exists
a positive symmetric operator
$A_1\in \End(TM)$ such that $A_1^2=A$.
Prove that $A_1$ is smooth.
\ez

\hint Use the uniqueness of $A_1$.
\eh


\exercise
Let $g$ be a positive definite form on a manifold,
$\omega$ a non-degenerate 2-form, and $A\in \End(TM)$
the corresponding anti-symmetric operator.
\enum
\item Prove that there exists a positive symmetric operator
$A_1\in \End(TM)$, smooth in $M$, such that $A_1^2=-A^2$.
\item 
Consider the symmetric form $g_1(x, y):= g(A_1(x), y)$,
and let $I$ be an operator which satisfies
$g_1(I(x), y)=\omega(x, y)$. Prove that $I^2=-\Id$.
\item
Let $M$ be a manifold admitting a non-degenerate 2-form.
Prove that $M$ admits an almost complex structure.
\ee
\ez

\exercise[*]
Prove that the space of almost complex structures
on $M$ is homotopy equivalent to the space of 
non-degenerate 2-forms on $M$.
\ez

\exercise[*]
Let $\Omega$ be a non-degenerate complex linear
3-form on $\C^3$, and $\rho:= \Re(\Omega)$
the corresponding form on $\R^{6}$.
Let $\widetilde{SL}(3, \C)$ be the group
generated by $SL(2, \C)$ and the complex conjugation.
Prove that the group of all $A\in GL(6, \R)$
such that $A(\rho)=\rho$ is isomorphic to 
$\widetilde{SL}(3, \C)$.
\ez



 
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