\documentclass[10pt]{article} \usepackage{url} %version 1.0,\ \ 14.09.2021 \newcommand{\version}{version 1.0,\ \ 14.09.20201} \newcommand{\firstdate}{15.09.2021} %\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{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}