\documentclass[10pt]{article} % version 1.0 -- April 16, 2013, started the file (from pde-8.tex) % version 1.1 -- April 29, 2013, vector field tangent to diffeo flow \usepackage{diagrams} \usepackage{amscd} \newcommand{\version}{version 1.1,\ \ 29.04.2013} \newcommand{\firstdate}{29.04.2013} %\addtolength{\topmargin}{-5mm} %\addtolength{\textheight}{10mm} \addtolength{\oddsidemargin}{-10mm} \addtolength{\textwidth}{20mm} \input{defs-listki-en.tex} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \listok{11}{Geometry 11: Lie derivatives and Poincar\'e lemma} {\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 problems. To have a perfect score, a student must obtain (in average) a score of 10 points per week. It's up to you to ignore handouts entirely, because passing tests in class and having good scores at final exams could compensate (at least, partially) for the points obtained by grading handouts. Solutions for the problems are to be explained to the examiners orally in the class and marked in the score sheet. It's better to have a written version of your solution with you. It's OK to share your solutions with other students, and use books, Google search and Wikipedia, we encourage it. 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 (except at most 2) 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. Please keep your score sheets until the final evaluation is given. } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Lie derivative} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition An associative algebra $A^* = \oplus_{i\in \Z}A^i$ is called {\bf a graded algebra} if for all $a\in A^i$, $b\in A^j$, the product $ab$ lies in $A^{i+j}$. \ed \definition Let $A^* = \oplus_{i\in \Z}A^i$ be a graded algebra over a field. It is called {\bf graded commutative}, or {\bf supercommutative}, if $ab = (-1)^{ij} ba$ for all $a\in A^i, b \in A^j$. \ed \remark Grassmann algebra $\Lambda^* V$ is clearly supercommutative. \er \exercise Let $A^*, B^*$ be graded commutative algebras, and $A^* \otimes B^*$ their tensor product, with a grading $(A^* \otimes B^*)^p := \oplus_{i+j=p} A^i\otimes B^j$, and multiplication, defined as $a \otimes b \cdot a' \otimes b' = (-1)^{ij}aa' \otimes bb'$, where $a'\in A^i, b \in B^j$. Prove that it is supercommutative. \ez \exercise Let $V, W$ be vector spaces, and $A^*:= \Lambda^* V, B^*:=\Lambda^* W$ their Grassmann algebras. Prove that $\Lambda^*(V\oplus W)$ is isomorphic to a tensor product $A^* \otimes B^*$, defined as above. \ez \definition Let $A^*$ be a graded commutative algebra, and $D:\; A^* \arrow A^{*+i}$ be a map which shifts grading by $i$. It is called a {\bf graded derivation} if $D(ab) = D(a) b + (-1)^{ij} a D(b)$, for each $a \in A^j$. \ed \remark If $i$ is even, graded derivation is just a derivation. If it is even, it is called {\bf odd derivation}. \er \remark De Rham differential is an odd derivation, by definition. \er \definition Let $M$ be a smooth manifold, and $X\in TM$ a vector field. Consider an operation of {\bf convolution with a vector field} $i_X:\; \Lambda^i M \arrow \Lambda^{i-1}M$, mapping an $i$-form $\alpha$ to an $(i-1)$-form $v_1, ..., v_{i-1} \arrow \alpha(X, v_1, ..., v_{i-1})$ \ed \exercise Prove that $i_X$ is an odd derivation. \ez \exercise[*] Let $D:\; A^* \arrow A^{*+i}$ be a linear map such that for all $x\in A$ there exists $N$ such that $D^N(x)=0$. Prove that $e^D:= 1 + D + \frac {D^2}{2} + ... + \frac{D^i}{i!} + ... $ is an automorphism of $A^*$ if and only if $D$ is a derivation. \ez \definition Let $A^*$ be a graded vector space, and $E:\; A^*\arrow A^{*+i}$, $F:\; A^*\arrow A^{*+j}$ operators shifting the grading by $i, j$. Define {\bf the supercommutator} by the formula \[ \{E, F\}:= EF - (-1)^{ij} FE. \] \ed \remark An endomorphism which shifts a grading by $i$ is called {\bf even} if $i$ is even, and {\bf odd} otherwise. \er \exercise Prove that a supercommutator satisfies {\bf graded Jacobi identity,} \[ \{ E, \{F, G\}\} = \{\{ E, F\}, G\} + (-1)^{\tilde E \tilde F} \{ F, \{E, G\}\} \] where $\tilde E$ and $\tilde F$ are 0 if $E, F$ are even, and 1 otherwise. \ez \remark There is a simple mnemonic rule which allows one to remember a superidentity, if you know the commutative analogue. Each time when in commutative case two letters $A$, $F$ are exchanged, in supercommutative case one needs to multiply by $(-1)^{\tilde E\tilde F}$. \er \exercise Let $A^*$ be a graded commutative algebra and $a\in A$. Denote by $L_a:\; A\arrow A$ the operation of multiplication by $a$: $L_a(b) = ab$. Prove that $D$ is a superderivation if and only if $D(1)=0$ and for each $a\in A^i$, the supercommutator $\{D, L_a\}$ is equal to $L_b$ for some $b\in A^*$. \ez \exercise[!] Prove that a supercommutator of superderivations is again a superderivation. \ez \hint Use the Jacobi identity and apply the previous exercise. \eh \definition Let $B$ be a smooth manifold, and $v\in TM$ a vector field. An endomorphism $\Lie_v:\; \Lambda^* M \arrow \Lambda^* M$, preserving the grading is called {\bf a Lie derivative along $v$} if it satisfies the following conditions. \begin{description} \item[(i)] On functions $\Lie_v$ is equal to a derivative along $v$. \item[(ii)] $[\Lie_v, d]=0$ \item[(iii)] $\Lie_v$ is a derivation on the de Rham algebra. \end{description} \ed \exercise \label{_derivations_gene_Exercise_} Let $\nu_1, \nu_2:\; \Lambda^*(M) \arrow \Lambda^*(M)$ be derivations of the de Rham algebra. Suppose that $\nu_1$ is equal to $\nu_2$ on $C^\infty M=\Lambda^0(M)$ and on $d(C^\infty M)$. Prove that $\nu_1=\nu_2$. \ez \hint $\Lambda^*(M)$ is generated (multiplicatively) by $C^\infty M=\Lambda^0(M)$ and $d(C^\infty M)$. \eh \exercise Prove that the Lie derivative is uniquely determined by the properties (i)-(iii). \ez \hint Use the previous exercise. \eh \exercise Prove that $\{d, \{d, E\}\}=0$, for each $E\in \End(\Lambda^* M)$. \ez \hint Use the graded Jacobi identity. \eh \exercise Prove that $\{d, i_v\}$ commutes with $d$, where $i_v:\; \Lambda^* M \arrow \Lambda^{*-1}M$ is a convolution with $v$. \ez \hint Use the previous exercise. \eh \exercise[!] (Cartan formula) Prove that $\{d, i_v\}$ is a Lie derivative along $v$. \ez %\exercise[*] %Let $v, v'\in TM$ be two vector fields, and %$i_{v\otimes v'}:\; \Lambda^* M \arrow \Lambda^{*-2}M$ %is a substitution of $v, v'$ into a 2-form, %$i_{v\otimes v'}= i_v i_{v'}$. %Consider an $i$-form $\alpha \in \Lambda^* M$, %and let $L_\alpha$ be an operator of multiplication by $\alpha$. %Prove that an operator %\[ %x\arrow [i_{v\otimes v'}, L_\alpha](x) - L_{i_{v\otimes %v'}}(\alpha)\wedge x %\] %is a derivation. %\ez \exercise[*] Let $\tau:\; \Lambda^*(M) \arrow \Lambda^{*-1}(M)$ be a derivation shifting grading by $-1$. Prove that there exists a vector field $v\in TM$ such that $\tau=i_v$, or find a counterexample. \ez %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Poincar\'e lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \exercise Let $t$ be the coordinate function on a real line, $f(t)\in C^\infty \R$ a smooth function, and $v:= t \frac d {dt}$ a vector field. Define \[ R(f)(t):=\int^1_0 \frac{f(\lambda t)}{\lambda} d\lambda \] Prove that this integral converges whenever $f(0)=0$, and satisfies $\Lie_v R(f)=f$ in this case. \ez \exercise Let $t_1, ..., t_n$ be coordinate functions in $\R^n$, and $\vec{r}:=\sum_i t_i \frac d {dt_i}$ a radial vector field. Consider a function $f\in C^\infty \R^n$ satisfying $f(0)=0$, and let $x=(x_1, ..., x_n)$ be any point in $\R^n$. Prove that an integral \[ R(f)(x):=\int_0^1 \frac{f(\lambda x)}{\lambda} d\lambda \] converges, and satisfies $\Lie_{\vec{r}} R(f)=f$. \ez \hint Use the previous exercise. \eh \definition An open subset $U\subset \R^n$ is called {\bf starlike} if for any $x\in U$ the interval $[0,x]$ belongs to $U$. \ed \exercise[!] Let $U$ be a starlike subset in $\R^n$, and $i>0$. Construct an operator \[ R:\; \Lambda^i U \arrow \Lambda^i U\] which satisfies $\Lie_{\vec{r}} R\alpha=R\Lie_{\vec{r}} \alpha = \alpha$ for each $\alpha \in\Lambda^i U$. \ez \hint Define the integral $R(\alpha)$ as in the previous exercise, and check that it converges. Prove that $\Lie_{\vec{r}} R(\alpha)=\alpha$. \eh \exercise \label{_R_kernel_Zadacha_} Prive that any form $\alpha\in \Lambda^i U$ on a starlike set $U$ satisfying $\Lie_{\vec{r}} \alpha=0$ vanishes if $i>0$. \ez \hint Use the previous exercise. \eh \exercise[!] Prove that $\{R, d\}=0$ \ez \hint Check that \[ \{R, d\} \Lie_{\vec{r}} \alpha= Rd \Lie_{\vec{r}} \alpha + dR \Lie_{\vec{r}} \alpha = - R\Lie_{\vec{r}} d\alpha + d\alpha = 0. \] For any $\beta \in \ker \{R, d\}\cal \Lambda^iM$, satisfying $i>0$ or $\beta(0)=0$ for $i=0$, solve an equation $\Lie_{\vec{r}}\alpha=\beta$. \eh \exercise Prove that $\{ d, i_{\vec{r}}\} R(\alpha) =\alpha$, for any $i$-form $\alpha$ on a starlike set, $i>0$. \ez \definition Let $d$ be de Rham differential. A form in $\ker d$ is called {\bf closed}, a form in $\im d$ is called {\bf exact}. Since $d^2=0$, any exact form is closed. {\bf The group of $i$-th de Rham cohomology of $M$}, denoted $H^i(M)$, is a quotient of a space of closed $i$-forms by exact: $H^*(M)=\frac{\ker d}{\im d}$. \ed \exercise[!] Let $\alpha\in \Lambda^i U$ be a closed $i$-form on a starlike set $U$, with $i>0$. Prove that $\alpha = d i_{\vec{r}} R(\alpha)$. \ez \hint Use the previous exercise. \eh \exercise[!] (Poincar\'e lemma) Let $U$ be a starlike set. Prove that $H^i(U)=0$ for each $i>0$, and $H^0(M)=\R$. \ez \exercise Let $\theta$ be a closed form, and $d_\theta(x)=dx+\theta\wedge x$ the corresponding operator on $\Lambda^*M$. Its {\bf cohomology} are defined as $H^*(\Lambda^*(M), d_\theta):=\frac{\ker d_\theta}{\im d_\theta}$ \enum \ite Show that $d_\theta^2=0$. \ite[*] Let $\theta$ be an exact 1-form. Prove that $H^i(\Lambda^*(M), d_\theta)$ are isomorphic to $H^i(M)$. \ite[*] Let $\theta$ be a closed 1-form. Prove that $H^i(\Lambda^*(M), d_\theta)$ are isomorphic to $H^i(M)$, or find a counterexample. \ite[**] Let $\theta$ be a closed 3-form. Prove that $H^i(\Lambda^*(M), d_\theta)$ are isomorphic to $H^i(M)$, or find a counterexample. \ee \ez %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Pullback of a differential form} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $M \stackrel \phi\arrow N$ be a morphism of smooth manifolds, and $\Lambda^1 N \stackrel {\phi^*}\arrow \Lambda^1 M$ an induced morphism which maps $f dg$ to $\phi^* f d\phi^* g$. A form $\phi^* \alpha$ is called {\bf pullback of $\alpha$}. \ed \exercise Prove that $\phi^*$ can be extended from $\Lambda^1 N$ to a multiplicative homomorphism \[ \phi^*:\; \Lambda^* N \arrow \Lambda^* M. \] \ez \definition {\bf Pullback} of an $i$-form $\alpha$ is a form $\phi^* \alpha$ defined as above. If $M\stackrel \phi \arrow N$ is a closed embedding, the form $\phi^* \alpha$ is called {\bf restriction} of $\alpha$ to $M \hookrightarrow N$. \ed \exercise Let $x\in T_m M$ be a tangent vector, and $\alpha \in \Lambda^1 N$ a 1-form. Prove that $\phi^* \alpha (x)= \alpha\left(D_\phi(x)\right)$, where $D_\phi:\; T_m M \arrow T_{\phi(m)} N$ is a differential. \ez \exercise[!] Prove that $\phi^* d\alpha= d\phi^* \alpha$. \ez \hint Use exercise \ref{_derivations_gene_Exercise_}. \eh \definition Let $f:\; M \times [a,b]\arrow M$ be a smooth map such that for all $t\in [a,b]$ the restriction $f_t:= f\restrict{M\times\{t\}}:\; M \arrow M$ is a diffeomorphism. Then $f$ is called {\bf a flow of diffeomorphisms}. \ed \exercise Let $V_t$ be a flow of diffeomorphisms, $f\in C^\infty M$, and $V^*_t(f)(x):= f(V_t(x))$. Consider the map $\frac d{dt}V_t\restrict{t=c}:\; C^\infty M \arrow C^\infty M$, with $\frac d{dt}V_t\restrict{t=c}(f)= (V_c^{-1})^*\frac{dV_t}{dt}\restrict{t=c}f$. Show that $\frac d{dt}V_t\restrict{t=c}$ is a derivation (that is, a vector field). \ez \definition The vector field $\frac d{dt}V_t\restrict{t=c}$ is called {\bf a vector field tangent to a flow of diffeomorphisms $V_t$ at $t=c$.} \ed \definition Let $v$ be a vector field on $M$, and $V:\;M\times[a,b]\arrow M$ a flow of diffeomorphisms which satisfies $\frac d{dt}V_t\restrict{t=c}=v$ for each $c$, and $V_0=\Id$. Then $V_t$ is called {\bf an exponent of $v$}. \ed \exercise Prove that an exponent of a vector field is unique, if it exists. \ez \exercise[*] Let $v$ be a vector field, and $D_v:\; C^\infty M \arrow C^\infty M$ the corresponding derivation. Suppose that the serie $e^{tD_v}(f):=\sum_{i=0}^\infty \frac {t^i}{i!} (D_v)^i f$ converges for some $f\in C^\infty M$ and $t\in [0,a]$, and the exponent $V_t$ of $v$ exists for all $t\in[0,a]$. Prove that $e^{tD_v}(f)=V_t^*(f)$. \ez \exercise[!] Let $v$ be a vector field, and $V_t$ its exponent. For any $\alpha \in \Lambda^*M$, consider $V_t^*\alpha$ as a $\Lambda^*M$-valued function of $t$. Prove that $\Lie_v\alpha = \frac{d}{dt}(V_t^*\alpha)$. \ez \hint Use exercise \ref{_derivations_gene_Exercise_}. \eh %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Computation of cohomology} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \exercise[*] \label{_Fourier_exercise_} Let $M$ be a smooth manifold equipped with a smooth action $X_t$ of the group $S^1=U(1)$, $f\in C^\infty M$, and $f_t:= X_t^* f$. Prove that $f_t=\sum_{n\in \Z} e^{2\pi\1 nt} f_i$, where $f_i\in C^\infty M$ are $S^1$-invariant functions. \ez \definition Let $A\subset \Lambda^*M$ be a subspace which satisfies $d(A)\subset A$. {\bf Cohomology} of $A$ is the quotient $H^*(A):= \frac{\ker d\restrict A}{d(A)}$. If $A$ is graded, $A=\bigoplus A^i$, where $A_i:= \Lambda^i(M)\cap A$, one has $H^*(A)=\bigoplus_i H^i(A)$, where $H^i(A)=\frac{\ker d\restrict A^i}{d(A^{i-1})}$. \ed \exercise Let $M$ be a smooth manifold, and $v\in TM$ a vector field. Prove that the image $\Lie_v\Lambda^*M$ satisfies $d(\Lie_v\Lambda^*M)\subset \Lie_v\Lambda^*M$, and $H^*(\Lie_v\Lambda^*M)=0$. \ez \exercise Let $M$ be a smooth manifold, and $v\in TM$ a vector field. Consider the space $\Lambda^*_vM:= \ker\Lie_v\restrict{\Lambda^*M}$. Let $V_t$ be an exponent of $v$. Suppose that $V_t$ is periodic, $V_{t+2\pi}=V_t$. \enum \ite[!] Prove that $(\Lie_v\Lambda^*M)\cap \Lambda^*_vM=\emptyset$. \ite[*] Prove that $(\Lie_v\Lambda^*M)\oplus \Lambda^*_vM=\Lambda^*M$. \ee \ez \hint Use exercise \ref{_Fourier_exercise_}. \eh \exercise Let $M$ be a smooth manifold, and $v\in TM$ a vector field. Suppose that $(\Lie_v\Lambda^*M)\oplus \Lambda^*_vM=\Lambda^*M$. Prove that $H^*(M)=H^*(\Lambda^*_vM)$. \ez \exercise Let $M$ be a smooth manifold, $K\subset \R^n$ a starlike set, and $v$ the radial vector field on $K$ lifted to $M\times K$. \enum \ite[!] Prove that $\Lambda^*_v(M\times K)=\Lambda^*(M)$. \ite Prove that $H^*(M\times K)=H^*(M)$. \ee \ez \exercise Let $v_1, v_2$ be vector fields which satisfy $\Lie_{v_i}\Lambda^*M\oplus \Lambda^*M_{v_i}=\Lambda^*M$. \enum \ite Prove that \[ \Lambda^*M= (\ker\Lie_{v_1}\cap \ker \Lie_{v_2})\oplus (\im\Lie_{v_1}\cup \im \Lie_{v_2}). \] \ite[!] Prove that $H^*(\im\Lie_{v_1}\cup \im \Lie_{v_2})=0$. \ee \ez \exercise[*] Let $G$ be a Lie group (that is, a smooth manifold equipped with a smooth group structure) generated by circles $S^1\subset G$, $M$ a manifold with $G$-action, and $\Lambda^*_GM$ the space of $G$-invariant forms. Prove that $H^*(\Lambda^*_GM)=H^*M$. \ez \hint Use the previous exercise. \eh \exercise[*] Compute cohomology of a sphere $S^n$ and a real projective space $\R P^n$. \ez \hint Use the previous exercise. \eh %\definition %Дифференциальная форма $\alpha$ называется %{\bf формой с компактным носителем}, если %$\alpha=0$ вне какого-то компактного множества. %\ed % %\definition %Пусть $M$ -- $n$-мерное многообразие, %а $\Lambda^n_c(M)$ -- пространство $n$-форм %с компактным носителем. Определим {\bf интеграл} %$\int_M:\; \Lambda^n_c(M)\arrow \R$ как отображение, %которое удовлетворяет следующим свойствам. %\begin{description} %\item[(i)] $\int_M (\alpha + \beta) = \int_M \alpha +\int_M \beta$. %\item[(ii)] $\int_M \phi^*\alpha= \int_N\alpha$, %для любого собственного отображения %$M \stackrel \phi \arrow N$ и $\alpha \in \Lambda^n_c(N)$. %\item[(iii)] Для любого гладкого, собственного отображения %$\R^n \stackrel \phi\arrow M$, и формы $\alpha \in \Lambda^n_c(M)$, %имеем $\int_M \alpha = \int_{\R^n} f d\mu$, %где $d\mu$ есть обычная мера Лебега на $\R^n$, %$f= \frac{\phi^* \alpha}{\Vol \R^n}$, а $\Vol \R^n$ -- %стандартная форма объема, $\Vol \R^n= dt_1 \wedge dt_2 %\wedge ... \wedge dt_n$. %\end{description} %\ed % %\exercise[!] Докажите, что интеграл, определенный %вышеприведенными аксиомами, существует, и единственен. %\ez % %\definition %Пусть $N \stackrel \pi \arrow M$ -- неособый %морфизм многообразий, со слоями размерности %$k$, а $\alpha \in \Lambda^{i+k}_c N$ -- форма %с компактным носителем. {\bf Прямой образ %$\pi_* \alpha \in \Lambda^i_c M$} определяется %как форма, которая удовлетворяет %\[ %\int_M \pi_*\alpha\wedge \beta= \int_N \alpha \wedge \pi^* \beta, %\] %для любой формы $\beta \in \Lambda^{\dim M -i} M$. %\ed % %\exercise[!] %Докажите, что $\pi_*\alpha$ определяется %этой формулой однозначно. %\ez % %\exercise[*] Докажите, что $\pi_* \alpha\in \Lambda^i_c M$ определен %для любой $\alpha \in \Lambda^{i+k}_c N$ %\ez % %\hint %Это "интеграл вдоль слоев" $\pi$. %\eh % %\exercise[*] %Докажите, что $\pi_* d\alpha = d \pi_* \alpha$. %\ez % % %\exercise[*] %Пусть $\alpha$ -- $(n-1)$-форма с компактным %носителем на многообразии $M$ с краем $\partial M$. %Определим $\int_{\partial M} \alpha$ как интеграл %от ограничения (обратного образа) $\alpha$ на $\partial M$. %Докажите {\bf формулу Стокса}: %\[ %\int_M d\alpha = \int_{\partial M} \alpha. %\] %\ez % % %\hint %Воспользуйтесь предыдущей задачей, и примените индукцию. %\eh \end{document}