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

\listok{8}{Geometry 8: Vector bundles}

{\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{Tensor product}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\definition
Let $V,V'$ be $R$-modules, $W$ a free abelian group
generated by $v\otimes v'$, with $v\in V, v'\in V'$,
and $W_1\subset W$ a subgroup generated by combinations
$rv \otimes v'-v\otimes rv'$, $(v_1+ v_2)\otimes v'-
v_1 \otimes v' - v_2 \otimes v'$ and $v\otimes (v'_1+ v'_2)-
v\otimes v'_1 - v\otimes v'_2$.
Define {\bf the tensor product} $V \otimes_R V'$
as a quotient group $W/W_1$.
\ed

\exercise
Show that $r \cdot v\otimes v'\mapsto (rv)\otimes v'$
defines an $R$-module structure on $V \otimes_R V'$.
\ez

\exercise
Prove that $\Q \otimes_\Z (\Z/2\Z)=0$.
\ez

\exercise[*]
Find a non-zero $R$-module $V$ such that
 $V\otimes_R V=0$.
\ez

\exercise
Let $I_1, I_2$ be ideals in $R$.
Prove that $(R/I_1)\otimes_R (R/I_2)=R/(I_1 + I_2)$,
where $I_1 + I_2$ is an ideal generated by linear
combinations $I_1, I_2$.
\ez

\exercise
Prove that a tensor product of free $R$-modules is free.
\ez

\exercise
Let ${\cal F}$ be a sheaf of rings, and
${\cal B}_1$ and ${\cal B}_2$ be sheaves
of locally free $(M, {\cal F})$-modules.
Prove that 
\[ U \arrow {\cal B}_1(U)\otimes_{{\cal F}(U)} {\cal B}_2(U)
\]
is also a sheaf of modules.
\ez

\exercise[**]
Is the last statement true without the assumption
of local triviality?
\ez

\definition
{\bf Tensor product} of vector bundles
is a tensor product of the corresponding sheaves of modules.
\ed

\remark
In a similar way one defines exterior powers
and symmetric powers of a bundle.
\er

\exercise
Let  ${\cal B}_1$ and ${\cal B}_2$ 
be locally free sheaves of  $C^\infty M$-modules, and
${\cal B}_1\otimes_{C^\infty M}{\cal B}_2$
their tensor product. Show that the fiber 
${\cal B}_1\otimes_{C^\infty M}{\cal B}_2$
in $x$ is naturally identified with a tensor product
of the fibers:
\[\bigg({\cal B}_1\otimes_{C^\infty M}{\cal
  B}_2\bigg)\restrict x\cong 
{\cal B}_1\restrict x \otimes_\R{\cal B}_2\restrict x.
\]
\ez

\exercise
Let $V$ be an $R$-module, and  
$\Hom_R(V, R)$ the space of $R$-linear homomorphisms
from $V$ to $R$. Prove that the action
$r\cdot h(\dots) \mapsto rh(\dots)$ gives a structure of
$R$-module on $\Hom_R(V, R)$.
Prove that  $\Hom_R(R^n, R)$ 
with $R$-module structure defined this way
is isomorphic (non-canonically) to a free module
$R^n$.
\ez

\definition
Let $V$ be an $R$-module. {\bf A dual $R$-module}
$V^*$ is $\Hom_R(V, R)$ with the $R$-module structure
defined above.
\ed

\exercise
Consider $\Q/Z$ as a $Z$-module. Prove that
$(\Q/Z)^*=0$.
\ez

\exercise 
Prove that $\Hom_\Z(Q, \Z)=0$.
\ez

\exercise[*]
Let $R=C^\infty(\R)_0$ be a ring of germs of smooth functions at $0$,
and $K$ an ideal of functions vanishing in 0 with all
derivatives. Prove that 
$(R/K)^*:= \Hom_R(R/K, R)=0$, or disprove it.
\ez

\exercise[*]
Same question when $R=C^\infty(\R^n)_0$.
\ez

\exercise[!]
Let ${\cal B}$ be a vector bundle, that is,
a locally free sheaf of  $C^\infty M$-modules, and
$\Tot{\cal B}\stackrel \pi \arrow M$
its total space. Define ${\cal B}^*(U)$
as a space of smooth functions on $\pi^{-1}(U)$ 
linear in the fibers of $\pi$. 
\enum
\ite Show that the natural
restriction map
${\cal B}^*(U)\arrow {\cal B}^*(V)$
defines a sheaf ${\cal B}^*$.

\ite Show that this sheaf is locally trivial.

\ite[!] Show that
${\cal B}^*(U)$ is a dual $C^\infty(U)$-module
to ${\cal B}(U)$.
\ee
\ez

\definition
Let ${\cal B}$ be a vector bundle, and ${\cal B}^*$
a locally trivial sheaf of  $C^\infty M$-modules
defined above. It is called {\bf the dual bundle}
to ${\cal B}$. 
\ed

\exercise
Prove that the fiber ${\cal B}^*\restrict x$
is a vector space dual to
${\cal B}\restrict x$.
\ez

\exercise 
Let ${\cal B}$ be a non-trivial vector bundle.
Prove that ${\cal B}^*$ is also non-trivial.
\ez

\definition
{\bf Bilinear form} on a bundle ${\cal B}$ 
is a section of $({\cal B}\otimes {\cal B})^*$.
A symmetric bilinear form on ${\cal B}$
is called {\bf positive definite} if it gives
a positive definite form on all fibers of 
${\cal B}$. Symmetric positive definite
form is also called {\bf a metric}.
A skew-symmetric bilinear form on
${\cal B}$ is called {\bf non-degenerate}
if it is non-degenerate on all fibers of ${\cal B}$.
\ed

\exercise[!]
Let ${\cal B}$ be a vector bundle on a metrizable manifold
$M$. Prove that  ${\cal B}$ admits a metric.
\ez

\hint
Construct the metric locally, and use partition of unity.
\eh

\exercise
Construct a 2-dimensional vector bundle
which does not admit a non-degenerate skew-symmetric
bilinear form.
\ez

\exercise[**]
Let $M$ be a simply connected manifold,
and $B$ a $2n$-dimensional vector bundle.
Prove that $B$ admits a non-degenerate skew-symmetric
bilinear form, or find a counterexample.
\ez

\exercise[*]
Find a non-trivial 3-dimensional bundle
${\cal B}$ such that its exterior square
 $\Lambda^2 {\cal B}$ is trivial.
\ez

\exercise[*]
Find a 2-dimensional bundle 
which does not admit a non-degenerate
bilinear symmetric form of signature $(1,1)$.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Smooth morphisms of vector bundles and subbundles}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let ${\cal B}, {\cal B}'$ be sheaves on
$M$. {\bf A sheaf morphism} from ${\cal B}$ to ${\cal B}'$
is a collection of homomorphisms ${\cal B}(U)\arrow {\cal B}'(U)$,
defined for each open subset $U\subset M$,
and compatible with the restriction maps:
\[
\begin{CD}
{\cal B}(U) @>>> {\cal B}'(U)\\
@VVV@VVV\\
{\cal B}(U_1) @>>> {\cal B}'(U_1)
\end{CD}
\]
\ed

\remark
Morphisms of sheaves of modules are defined in the same
way, but in this case the maps ${\cal B}(U)\arrow {\cal
  B}'(U)$ should be compatible with the module structure.
\er

\definition
A sheaf morphism is called {\bf injective}
if it is injective on germs
and {\bf surjective}, if it is surjective on germs.
\ed

\exercise
Let ${\cal B} \stackrel \phi \arrow {\cal B}'$ 
be an injective morphism of sheaves on $M$. Prove that
$\phi$ induces an injective map ${\cal B}(M) \arrow {\cal B}'(M)$
on the spaces of global sections.
\ez

\exercise[*]
Find an example of a surjective sheaf morphism
which is not surjective on global sections.
\ez

\definition
Let ${\cal B} \stackrel \phi \arrow {\cal B}'$ be a
morphism of locally free sheaves of  $C^\infty M$-modules.
It is called {\bf a smooth morphism}, or
{\bf a morphism of vector bundles} if on each of the germ spaces
$\phi$ has free kernel and free cokernel.
\ed

\definition
Let ${\cal F}$ be a locally free sheaf of $C^\infty M$-modules,
and ${\cal F}_x$ its space of germs in $x$. Denote the quotient
${\cal F}_x/{\goth m}_x {\cal F}_x$
by ${\cal F}\restrict x$. This space is called
{\bf the fiber} of ${\cal F}$ in $x$.
A morphism of sheaves induces a linear
map on each of its fibers.
\ed


\exercise[!]
Find an example of an injective morphism of 
locally free $C^\infty M$-modules
which is not injective in some fiber.
\ez

\exercise[*]
Prove that a surjective morphism of locally
free sheaves of  $C^\infty M$-modules
is a smooth morphism of vector bundles,
in the sense of the above definition.
\ez

\exercise
Let ${\cal B} \arrow {\cal B}_1$ 
be a smooth morphism of vector bundles on $M$. 
\enum
\ite Prove that the corresponding map $\Psi$ of total spaces
is a homomorphism of relative vector spaces over $M$.
\ite Prove that $\Psi$ has no critical points.
\ee
\ez

\definition
{\bf A subbundle} ${\cal B}_1\subset {\cal B}$ 
is an image of an injective morphism of vector bundles.
\ed

\exercise
Let ${\cal B}_1\subset {\cal B}$  be a subbundle.
Prove that the quotient ${\cal B}/{\cal B}_1$ is also a
vector bundle.
\ez


\exercise[!]
Let ${\cal B}_1 \stackrel \phi\arrow {\cal B}_2$ 
be a morphism of vector bundles. Prove that the
image of $\phi$ is a subbundle in ${\cal B}_2$, 
and its kernel is a subbundle in ${\cal B}_1$.
\ez

\definition
{\bf Direct sum} of vector bundles is a direct
sum of corresponding sheaves.
\ed

\exercise
Prove that a total space of a direct sum of vector
bundles ${\cal B}\oplus {\cal B}'$ is homeomorphic to 
$\Tot {\cal B}\times_M \Tot {\cal B}'$.
\ez

\exercise
Let ${\cal B}$ be a vector bundle equipped with a metric
(that is, a positive definite symmetric form),
and ${\cal B}_1 \subset {\cal B}$ a subbundle.
Consider a subset $\Tot {\cal B}_1^\bot\subset \Tot {\cal B}$,
consisting of all $v\in {\cal B}\restrict x$
orthogonal to ${\cal B}_1\restrict x \subset {\cal
B}\restrict x$. Prove that $\Tot {\cal B}_1^\bot$ 
is a total space of a subbundle, denoted as ${\cal
B}_1^\bot\subset {\cal B}$.
\ez

\definition
A subbundle ${\cal B}_1^\bot\subset {\cal B}$
is called {\bf orthogonal complement} of
${\cal B}$ to ${\cal B}_1 \subset {\cal B}$.
\ed

\exercise
Let ${\cal B}_1 \subset {\cal B}$ be a sub-bundle.
Prove that ${\cal B}$ is isomorphic to a direct sum of
${\cal B}_1$ and another bundle.
\ez

\hint
Find a metric on ${\cal B}$ and use the previous exercise.
\eh

\remark
In this situation, it is said that ${\cal B}_1$
{\bf is a direct sum of ${\cal B}$}.
\er



\end{document}
