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\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
%���������������� ����� $\alpha$ ����������
%{\bf ������ � ���������� ���������}, ����
%$\alpha=0$ ��� ������-�� ����������� ���������.
%\ed
%
%\definition
%����� $M$ -- $n$-������ ������������,
%� $\Lambda^n_c(M)$ -- ������������ $n$-����
%� ���������� ���������. ��������� {\bf ��������}
%$\int_M:\; \Lambda^n_c(M)\arrow \R$ ��� �����������,
%������� ������������� ��������� ���������.
%\begin{description}
%\item[(i)] $\int_M (\alpha + \beta) = \int_M \alpha +\int_M \beta$.
%\item[(ii)] $\int_M \phi^*\alpha= \int_N\alpha$,
%��� ������ ������������ ����������� 
%$M \stackrel \phi \arrow N$ � $\alpha \in \Lambda^n_c(N)$.
%\item[(iii)] ��� ������ ��������, ������������ �����������
%$\R^n \stackrel \phi\arrow M$, � ����� $\alpha \in \Lambda^n_c(M)$,
%����� $\int_M \alpha = \int_{\R^n} f d\mu$,
%��� $d\mu$ ���� ������� ���� ������ �� $\R^n$,
%$f= \frac{\phi^* \alpha}{\Vol \R^n}$, � $\Vol \R^n$ --
%����������� ����� ������, $\Vol \R^n= dt_1 \wedge dt_2
%\wedge ... \wedge dt_n$.
%\end{description}
%\ed
%
%\exercise[!] ��������, ��� ��������, ������������ 
%���������������� ���������, ����������, � �����������. 
%\ez
%
%\definition
%����� $N \stackrel \pi \arrow M$ -- ��������
%������� ������������, �� ������ �����������
%$k$, � $\alpha \in \Lambda^{i+k}_c N$ -- �����
%� ���������� ���������. {\bf ������ ����� 
%$\pi_* \alpha \in \Lambda^i_c M$} ������������
%��� �����, ������� �������������
%\[
%\int_M \pi_*\alpha\wedge \beta= \int_N \alpha \wedge \pi^* \beta,
%\]
%��� ����� ����� $\beta \in \Lambda^{\dim M -i} M$.
%\ed
%
%\exercise[!]
%��������, ��� $\pi_*\alpha$ ������������
%���� �������� ����������.
%\ez
%
%\exercise[*] ��������, ��� $\pi_* \alpha\in \Lambda^i_c M$ ���������
%��� ����� $\alpha \in \Lambda^{i+k}_c N$
%\ez
%
%\hint
%��� "�������� ����� �����" $\pi$.
%\eh
%
%\exercise[*]
%��������, ��� $\pi_* d\alpha = d \pi_* \alpha$.
%\ez
%
%
%\exercise[*]
%����� $\alpha$ -- $(n-1)$-����� � ���������� 
%��������� �� ������������ $M$ � ����� $\partial M$.
%��������� $\int_{\partial M} \alpha$ ��� ��������
%�� ����������� (��������� ������) $\alpha$ �� $\partial M$.
%�������� {\bf ������� ������}:
%\[
%\int_M d\alpha = \int_{\partial M} \alpha.
%\]
%\ez
%
%
%\hint
%�������������� ���������� �������, � ��������� ��������.
%\eh




\end{document}

