\documentclass[10pt]{article}
\usepackage{url}

%version 1.0,\ \   25.06.2020


\newcommand{\version}{version 1.0,\ \   01.10.2020}
\newcommand{\firstdate}{03.10.2020}

\addtolength{\topmargin}{-5mm}
\addtolength{\textheight}{10mm}
\addtolength{\oddsidemargin}{-5mm}
\addtolength{\textwidth}{10mm}

\input{defs-listki-en.tex}


\setlength{\headheight}{15pt}
\pagestyle{fancy} 
\lhead{\tiny Complex geometry, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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

\listok{2}{Complex geometry handout 2: Distributions and 
almost complex structures}

%{\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
Find a vector field $v$ on a 2-dimensional torus $T^2$
such that all orbits of the corresponding diffeomorphism
flow are dense.
\ez

\exercise
Construct a 4-manifold $M$ and a rank 2 distribution
$B\subset TM$ such that $[B, B]$ has rank 3 and
$[[B, B], B]$ has rank 4.
\ez

\definition
{\bf A contact structure} on an $2n+1$-dimensional manifold $M$
is a rank $2n$ distribution $B\subset TM$ such that
$TM/B$ is a trivial rank one bundle and
the Frobenius form $\Lambda^2 B \arrow TM/B$
is non-degenerate.
\ed


\exercise
Let $\theta$ be a 1-form on an $2n+1$-dimensional  manifold $M$
such that $\theta\wedge (d\theta)^{n}$ is non-degenerate,
$f\in C^\infty M$ a nowhere vanishing function, and $\theta'=f\theta$.
Prove that $\theta'\wedge (d\theta')^{n}$ is
non-degenerate.
\ez

\exercise
Let $\theta$ be a 1-form on an $2n+1$-dimensional  manifold $M$
such that $\theta\wedge (d\theta)^{n}$ is non-degenerate.
Prove that $\ker \theta\subset TM$
is a contact distribution.
\ez

\exercise
Let $M$ be an odd-dimensional manifold,
and $B\subset TM$ a contact distribution.
Prove that there exists a 1-form $\theta$
such that $B=\ker\theta$ and $\theta\wedge (d\theta)^{n}$
is non-degenerate.
\ez


\exercise
Construct a contact structure on a sphere $S^{2n+1}$
for any $n=1, 2, 3, ...$
\ez

\hint 
Use the previous exercise.
\eh

\exercise
Let $M$ be a contact manifold. Prove that $M$
admits a pseudo-Riemannian structure of
signature $(1,2n)$.
\ez

\exercise[*]
Let $M$ be a compact almost complex manifold, 
and $f$ a holomorphic function on $M$. Prove that $f$
is constant.
\ez

\exercise
Let $\eta, \eta'$ be non-vanishing closed $(p,0)$-forms
on an almost complex manifold, satisfying
$\eta=f\eta'$ for some $f\in C^\infty M$. Prove that $f$
is holomorphic.
\ez




\definition
Let $M$ be an almost complex manifold,
and $A:\; \Lambda^* M\arrow \Lambda^*M$
a linear map. {\bf Hodge components} of $A$
are operators $A^{p,q}$ such that
$A=\sum_{p,q}A^{p,q}$ and 
$A^{p,q}(\Lambda^{i,j}(M))\subset \Lambda^{i+p,j+q}(M)$.
\ed

\exercise
Prove that the de Rham differential on an almost
complex manifold has at most 4 non-zero Hodge 
components: $d=d^{2,-1}+d^{1,0}+d^{0,1}+d^{-1,2}$. 
\ez




 
\end{document}