\def\li{\item{$\bullet$}}
\def\pr#1.{\bigskip\noindent{\bf#1.}\enskip}
\def\supp{\mathop{\rm supp}}
\def\sm{C^\infty}
\def\smsh{\sm_{\rm sheaf}}
\def\Diff{\mathop{\rm Diff}}
\def\Symb{\mathop{\rm Symb}}
\def\Hom{\mathop{\rm Hom}}
\def\End{\mathop{\rm End}}
\def\Sym{\mathop{\rm Sym}}
\def\colim{\mathop{\rm colim}}
\def\R{{\bf R}}
\def\C{{\bf C}}
\def\id{{\rm id}}

\centerline{\bf Spring 2011, Math 276: Index Theory, Homework~1}
\bigskip
\leftline{Please submit by February~1 and contact Dmitri Pavlov (pavlov@math) for all questions about homework.}

\pr Problem 1: Differential operators.
Suppose that $V$~and~$W$ are smooth vector bundles over a smooth manifold~$M$.
Define the set of all differential operators $\Diff^{\le k}(V,W)$ of order at most~$k$
from~$V$ to~$W$ as follows: For $k<0$ we have $\Diff^{\le k}(V,W)=\{0\}$
and otherwise it consists of $\C$-linear maps $D\colon\sm(V)\to\sm(W)$
such that $Dm_V(a)-m_W(a)D\in\Diff^{\le k-1}(V,W)$ for all $a\in\sm(M)$.
Here $m_V(a)$ denotes the operator $\sm(V)\to\sm(V)$ given by the multiplication by~$a$
and likewise for~$m_W(a)$.
In particular, $\Diff^{\le0}(V,W)$ consists of all $\sm(M)$-linear morphisms $\sm(V)\to\sm(W)$,
i.e., morphisms of vector bundles $V\to W$.

\item{(a)} Prove that the coordinate definition of a differential operator
is equivalent to the one above, i.e.,
for any open set $U\subset\R^n$ and for any family of smooth functions $f_\alpha\in\sm(U)$
prove that every operator of the form $\sum_{|\alpha|\le k}f_\alpha\partial_\alpha$
is an element of $\Diff^{\le k}(U)$ and every element of $\Diff^{\le k}(U)$ can be represented
in this form.
Here $\alpha$ denotes multiindices and $\partial_\alpha$ denotes the corresponding
composition of partial derivatives.

\item{*(b)} A {\it differential operator\/} is
a morphism of sheaves of real vector spaces $\smsh(V)\to\smsh(W)$.
Alternatively, a differential operator is a morphism of real vector spaces $D\colon\sm(V)\to\sm(W)$
that preserves supports of sections: $\supp(D(f))\subset\supp(f)$ for all $f\in\sm(V)$.
Denote by $\Diff(V,W)$ the set of all differential operators from~$V$ to~$W$
and prove that $\Diff^{\le k}(V,W)\subset\Diff(V,W)$ for all~$k$.
Show that the canonical morphism $\colim_k\Diff^{\le k}(V,W)\to\Diff(V,W)$
given by the universal property of the colimit is an isomorphism.
Here the colimit is taken in the category of sheaves (think of fiberwise colimit
followed by sheafification).

\pr Problem 2: Symbols of differential operators.
Recall that in Problem~1 we constructed for an arbitrary smooth manifold~$M$
a category~$\Diff$ of vector bundles and differential operators
together with the filtration $\Diff^{\le k}(U,V)$.
In this problem we study the associated graded category of this filtered category.

\item{(a)} In the notation of Problem~1 prove the following relations:
$\Diff^{\le i}(V,W)\Diff^{\le j}(U,V)\subset\Diff^{\le i+j}(U,W)$
and $[\Diff^{\le i}(V,V),\Diff^{\le j}(V,V)]\subset\Diff^{\le i+j-1}(V,V)$
for all $i$~and~$j$.
In particular, the category~$\Diff$ is filtered by~$\Diff^{\le k}$.

\item{(b)} Consider the category of symbols~$\Symb$ over~$M$,
whose objects are vector bundles over~$M$ and morphisms from~$E$ to~$F$
are $\Symb(E,F):=\Sym(TM)\otimes_{\sm(M)}\Hom(E,F)$.
Composition and identity morphisms are defined in a natural way.
The category $\Symb$ admits a natural filtration $\Symb^{\le k}$
(polynomials of degree at most~$k$).
Construct a natural map of the associated graded categories of $\Symb$ and $\Diff$:
$\Symb^{\le k}(E,F)/\Symb^{\le k-1}(E,F)\to\Diff^{\le k}(E,F)/\Diff^{\le k-1}(E,F)$.
(Hint: Use the fact that sections of~$TM$ are derivations of~$\sm(M)$
and combine it with part~(a).)

\item{(c)} Construct a natural map
$\Diff^{\le k}(E,F)/\Diff^{\le k-1}(E,F)\to\Symb^{\le k}(E,F)/\Symb^{\le k-1}(E,F)$.
(Hint: An element of $\Symb^k(E,F):=\Symb^{\le k}(E,F)/\Symb^{\le k-1}(E,F)$
can be constructed fiberwise.
The fiber of~$\Symb^k(E,F)$ at point~$x$ can be identified
with $\Hom(\sm(E)m_x^k/\sm(E)m_x^{k+1},\sm(F)/\sm(F)m_x)$,
where $m_x$ is the ideal of functions in~$\sm(M)$ that vanish at~$x$.
Elements of $\Diff^{\le k}(E,F)$ act on $\sm(E)$.)

\item{(d)} Prove that the two natural maps constructed in parts (b)~and~(c)
are the mutual inverses of each other.
In particular, they give an equivalence of the associated graded categories
of $\Symb$~and~$\Diff$.

\pr Problem 3: Connections and parallel transport.
A connection on a smooth vector bundle~$V$ over a smooth manifold~$M$
is an $\R$-linear map $\nabla\colon\sm(V)\to\Omega^1(M)\otimes_{\sm(M)}\sm(V)=\sm(T^*M\otimes V)$
such that for all $f\in\sm(M)$ and $s\in\sm(V)$ we have
$\nabla(fs)=f\nabla(s)+df\otimes s$.

\item{(a)} Prove that any connection is a differential operator of order~1.
Prove that if $\nabla$ is a connection and
$A\in\Hom_{\sm(M)}(\sm(V),\Omega^1(M)\otimes_{\sm(M)}\sm(V))
=\Omega^1(M,\End(V))$,
then $\nabla+A$ is also a connection.
Prove that the difference of any two connections is such a $\sm(M)$-linear map.

\item{(b)} Consider the trivial vector bundle $V=\R^n\times\nobreak M\to M$.
Prove that the de Rham differential
induces a connection $d\otimes\id\colon\sm(M)\otimes\R^n\to\Omega^1(M)\otimes\R^n$ on~$V$.
Use part~(a) to find an explicit expression for all connections on~$V$
and conclude that connections exist on any vector bundle.

\item{(c)} A section $s\in\sm(V)$ is called {\it parallel\/} if $\nabla(s)=0$.
For $M=[0,1]$, prove that the restriction map~$r_x$ from the set of all parallel sections of~$V$
to the fiber over a point $x\in M$ is an isomorphism of vector spaces.
The morphism $r_1r_0^{-1}$ is called the {\it parallel transport\/} from~0 to~1.
Write down an explicit formula for the parallel transport from~0 to~1
in terms of the explicit description of connections over~$M$ obtained in part~(b).
Hint: $\Omega^1(M,\End(E))$ can be identified with $\sm(M,\End(V))$.

\item{(d)} Use part~(c) to define parallel transport for an arbitrary vector bundle~$V$
equipped with a connection~$\nabla$
over a smooth manifold~$M$ along an arbitrary path~$p\colon[0,1]\to M$.
Hint: Parallel transport commutes with pull-backs of connections.

\bye