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\lhead{\tiny Hogde theory, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{6}{Hodge theory 6: Connections}

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

{\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.
}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Connections}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $B$ be a vector bundle on a smooth manifold $M$,
and 
 \[ \nabla:\; B \arrow B \otimes \Lambda^1 M \] 
a differential operator which satisfies 
\[
\nabla(fb) = b \otimes df + f \nabla b,
\]
for any $f\in C^\infty(M)$ and any $b\in B$.
Then $\nabla$ is called {\bf connection} on $B$.
Given a vector field $X$, consider an operator
$\nabla_X:\; B \arrow B$ obtained by the convolution of $\nabla(b)$ with $X$.
This operator is called {\bf covariant derivative} along $X$. 
\ed



\exercise
Prove that the covariant derivative $\nabla_X$ satisfies 
the Leibniz rule $\nabla_X(fb)= \langle df, X\rangle b + f \nabla_X b$.
Here $\langle df, X= \Lie_X f = df\cntrct X$ denotes the derivative
of $f$ along $X$.
\ez

\exercise
Let $B$ be a vector bundle on $M$. Suppose that for any 
vector field $X\in TM$ we are provided with a covariant derivative operator
$\nabla_X:\; B \arrow B$ satisfying the Leibniz rule and $C^\infty M$-linear
on $X$. Prove that $\nabla_X$ is obtained from a connection.
\ez

\exercise
 Prove that the symbol $\Symb(\nabla)\in TM \otimes \Hom(B,
\Lambda^1 M \otimes B)$ of a connection is given by an identity
map $\Id_{TM}\otimes \Id_B:\; TM \otimes T^* M \otimes
\Hom(B,B)$.
\ez

\exercise[!]
Let $D$ be a first order differential operator on 
a bundle $B$ with symbol $\Symb(D)\in TM \otimes \Hom(B,
\Lambda^1 M \otimes B)$  given by an identity
map $\Id_{TM}\otimes \Id_B:\; TM \otimes T^* M \otimes
\Hom(B,B)$. Prove that it is a connection.
\ez


\exercise[!]
Let $\nabla$ be a connection on $B$, $B^*$ the dual bundle. 
Prove that there exists a unique operator $\nabla^*:\; B^* \arrow B^* \otimes \Lambda^1 M$
such that
\[
d\langle b , b'\rangle = \langle \nabla b, b'\rangle + \langle b, \nabla^* b'\rangle
\]
for any $b\in B, b' \in B^*$.
Prove that $\nabla^*$ is a connection on $B^*$.
\ez

\exercise
Let $B_1, ..., B_n$ be vector bundles with connections, denoted by $\nabla$
(people often use the same letter $\nabla$ to denote different connections if they
are defined on different bundles). Consider the following differential
operator 
\[ \nabla:\; B_1 \otimes ... \otimes B_n \arrow 
B_1 \otimes ... \otimes B_n \otimes \Lambda^1 M,
\]
$\nabla(b_1 \otimes b_2 \otimes ... \otimes b_n)=
\nabla(b_1) \otimes b_2 \otimes ... \otimes b_n +
b_1 \otimes \nabla(b_2) \otimes ... \otimes b_n + ... +
b_1 \otimes b_2 \otimes ... \otimes \nabla(b_n)$.
Prove that $\nabla$ defines a connection on the vector bundle
$B_1 \otimes ... \otimes B_n$.
\ez

\remark
Previous two exercises show that a connection on a bundle
$B$ defines a connection on any tensor power $B^{\otimes n} \otimes (B^*)^{\otimes m}$.
This connection is almost always denoted by the same letter.
\er

\exercise[!]
Let $B$ be a vector bundle over a manifold admitting
partition of unity. Prove that $B$ admits a connection.
\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Holonomy}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $(B, \nabla)$ be a bundle with connection.
A tensor $\Psi\in B^{\otimes i} \otimes (B^*)^{\otimes j}$
is called {\bf parallel} if $\nabla(\Psi)=0$. 
In this case we also say that $\Psi$ {\bf is preserved}
by $\nabla$.
\ed

\exercise
Let $g$ be a tensor on a bundle $B$ over $M$.
Construct a connection $\nabla$ such that $\nabla(g)=0$
if
\enum
\ite[!] $g$ is a non-degenerate bilinear symmetric form on
$B$.
\ite[*] $g$ is a non-degenerate bilinear antisymmetric
form on $B$.
\ite[*] $g$ is a bilinear symmetric form of constant rank
on $B$.
\ee
\ez


\exercise
Let $B$ be a trivial vector bundle with connection over $\R$.
Prove that for each $x\in \R$ and each vector $b_x \in B\restrict x$
there exists a unique section $b\in B$ such that
$\nabla b=0$, $b\restrict x= b_x$.
\ez


\definition
Let $\gamma:\; [0, 1] \arrow M$ be a smooth path
in $M$ connecting $x$ and $y$, and $(B, \nabla)$ 
a vector bundle with connection. Restricting 
$(B, \nabla)$ to $\gamma[0,1]$, we obtain a 
bundle with connection on an interval.
Solve an equation  $\nabla(b)=0$
for $b\in B\restrict{\gamma([0,1])}$
and initial condition $b\restrict x= b_x$.
This process is called {\bf parallel transport}
along the path via the connection.
The vector $b_y:= b\restrict y$
is called {\bf vector obtained by parallel
transport of $b_x$ along $\gamma$}.
{\bf Holonomy group} of $\gamma$ is the group
of endomorphisms of the fiber $B_x$ obtained
from parallel transports along all paths 
starting and ending in $x\in M$
\ed

\exercise
Let $(B, \nabla)$ be a vector bundle over
a connected manifold $M$, and $x, y \in M$.
Construct an isomorphism of the corresponding
holonomy groups $\Hol_x(\nabla)\arrow \Hol_y(\nabla)$.
\ez

\exercise
Find a bundle with connection over $S^1$
which has non-trivial holonomy. 
\ez



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Iterated connection}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $M$ be a smooth manifold. A connection on $TM$ or on $\Lambda^1M$
is called {\bf connection on $M$}. This connection defines a connection
on all tensor powers of $TM$ and $\Lambda^1 M$. A tensor product of several
copies of $TM$ and $\Lambda^1 M$ is called {\bf a tensor bundle} on $M$,
and its section {\bf a tensor}. Similarly, a section of
a tensor product of several copies of $B$ and $B^*$ is
called {\bf a tensor over a bundle $B$}.
\ed

\definition
Let $B$ be a vector bundle with connection $\nabla_0$ over
a manifold $M$, and $\nabla$ a connection on $\Lambda^1 M$.
Define a connection 
\begin{equation}\label{_nabla_i_Equation_}
\nabla_i:\; 
B\otimes  \underbrace{\Lambda^1M\otimes ...}_\text{{$i$ times}}\arrow
B\otimes  \underbrace{\Lambda^1M\otimes ...}_\text{{$i+1$ times}}
\end{equation}
using the Leibniz formula
\[
\nabla_i(b \otimes \xi_1 \otimes ... \otimes \xi_i) =
\nabla_{i-1}(b \otimes \xi_1 \otimes ... \otimes \xi_{i-1}) \otimes \xi_i+
b \otimes \xi_1 \otimes ... \otimes \xi_{i-1} \otimes \nabla \xi_i.
\]
Denote by 
\[ \nabla^i:\; B \arrow 
B\otimes \underbrace{\Lambda^1M\otimes ...}_\text{$i$ times}
\]
the composition
$\nabla_0 \circ \nabla_1 \circ ... \circ \nabla_{i}$.
This operator is called {\bf an $i$-th power of the connection $\nabla$}.
\ed

\exercise
\enum
\ite Prove that the symbol of $\nabla^2$, considered as an element
of 
\[
\Sym^2 TM \otimes \Hom (B, B\otimes \Lambda^1M\otimes\Lambda^1M)
\]
is symmetric under the permutation of the tensor multipliers
$\Lambda^1M\otimes\Lambda^1M$.
\ite[*]
Let $S$ be the symbol of $\nabla^i$,
\[
S\in \Sym^i TM \otimes \Hom \bigg (B, B\otimes 
\underbrace{\Lambda^1M\otimes ...}_\text{$i$ times}\bigg)
\]
Prove that $S$ is symmetric under the permutations of the
tensor multipliers  $\Lambda^1M\otimes\Lambda^1M\otimes ... \otimes \Lambda^1 M$.
\ee
\ez

\exercise\label{_diff_op_via_conne_Exercise_}
Let $D\in \Diff^s(B,C)$ be a differential operator 
on vector bundles $B$, $C$. 
\enum
\ite[!] Prove that there exists a $C^\infty$-linear 
map 
\[ \Psi:\; B \otimes \bigoplus_{i=0}^s(\Lambda^1 M)^{\otimes i}\arrow C\]
such that $D(b)=\Psi\left(\bigoplus_{i=0}^s \nabla^i b\right)$.
\ite[**] Prove that there exists a $C^\infty$-linear 
map and a connection $\nabla$ such that 
$\Phi:\;  B \otimes (\Lambda^1 M)^{\otimes s}\arrow C$
such that $D(b)=\Phi(\nabla^s b)$.
\ee
\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Adjoint operators}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise[!]\label{_first_order_via_nabla_Exercise_}
Let $D:\; B \arrow C$ be a first order differential operator 
on a vector bundle with connection $\nabla$. 
Prove that $D$ is obtained as a linear combination
of $b \arrow A(\nabla_{X_i})$, where $X_i$ are vector fields
and $A:\; B \arrow C$ is a linear map.
\ez

\definition
Let $\Vol$ be a volume form on a manifold $M$.
{\bf Divergence} of a vector field $X$ is the form 
$\Lie_X \Vol = d(\Vol\cntrct X)$.
\ed

\exercise[!]\label{_div_free_generate_Exercise_}
Let $\Vol$ be a volume form on a manifold $M$,
and ${\cal D}\subset TM$ the sheaf of vector fields
with zero divergence. Prove that the global sections of 
the sheaf ${\cal D}$ generate
all vector fields on <A HREF="$TM$"> over $C^\infty M$.
\ez

\exercise
Let $B$ be a vector bundle over a manifold $M$, $\Vol$ a volume form
and $\nabla$ a connection on $B$.
Consider the pairing between sections of $B$ and the dual
bundle $B^*$, with 
\begin{equation}\label{_section_pairing_via_Vol_Equation_}
\langle b, \lambda\rangle = \int_M (b,
  \lambda) \Vol
\end{equation}
for each $b\in B, \lambda\in B^*$.
\enum 
\ite
Let $X\in TM$ a vector field with zero divergence.
Prove that the operators 
$\nabla_X:\; B \arrow B$ and $\nabla_X^*:\; B^* \arrow B^*$
are adjoint with respect to the pairing
\eqref{_section_pairing_via_Vol_Equation_}.

\ite
Let $X$ be an arbitrary vector field, and
$\nabla_X:\; B \arrow B$ the corresponding covariant derivative.
Consider the adjoint $(\nabla_X)^*:\; B^* \arrow B^*$
associated with the pairing \eqref{_section_pairing_via_Vol_Equation_}.
Prove that 
\[ \int_M (b, \nabla_X^* b')\Vol + \int_M (\nabla_X b,
b')\Vol = -\int_M (b,b') \Lie_X\Vol
\]
\ite[!]
Prove that $(\nabla_X)^*$ is a differential operator.
\ee
\ez

\exercise
Prove that
an adjoint map of a first order differential
operator on vector bundles
$D:\; B \arrow B_1$ is a first order differential
operator from $B_1^*$ to $B^*$.
\ez

\hint Use Exercise \ref{_first_order_via_nabla_Exercise_}.
\eh



\end{document}
