\documentclass[10pt]{article} \usepackage{url} %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 % version 1.3, \ \ 15.09.2021, misprint in the last exercise % version 1.4, \ \ 18.09.2021, misprint in first, tilde{SL} in last \newcommand{\version}{version 1.4,\ \ 18.09.20201} \newcommand{\firstdate}{04.09.2021} %\addtolength{\topmargin}{-10mm} %\addtolength{\textheight}{20mm} %\addtolength{\oddsidemargin}{-10mm} %\addtolength{\textwidth}{20mm} \input{defs-listki-en.tex} \setlength{\headheight}{15pt} \pagestyle{fancy} \lhead{\tiny Symplectic geometry, HSE} \lfoot{\tiny Issued \firstdate} \cfoot{-- \thepage \ -- } \rfoot{\tiny \sc\version} \rhead{{\tiny Misha Verbitsky}} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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. %} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 \end{document}