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\lhead{\tiny Math in Moscow, HSE, Spring 2013} 
\lfoot{\tiny Issued \firstdate} 
\rhead{\tiny Differential geometry 
exam (May 20, 2013)}
\cfoot{-- \thepage \ -- } \rfoot{\tiny \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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

\listok{1}{Differential geometry exam (May 20, 2013)}

{\scriptsize
{\bf Rules:} Each student should count his or her 
handout score (as explained in the handouts rules page) and
give me the score sheets with the scores marked. 

Each examinee
receives a set of 12 problems to solve. 
The solutions should be explained to 
examiners orally.

The points for the exam are computed by
summing up points for the problems. Results will be
announced at \url{http://bogomolov-lab.ru/KURSY/GEOM-2013/}

The final score is determined (for HSE purposes) as
$2 + [0.1 b]$, where $b$ is total of points for handouts, exam and
two test assignments.
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Hausdorff dimension}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise[5]
Let $f:\; \R \arrow \R$ be any function,
and $\Gamma_f\subset \R \times \R$ its graph.
Prove that the Hausdorff dimension of $\Gamma_f$ is $\geq 1$.
\ez

\exercise[10]
Let $f:\; \R \arrow \R$ be a convex function,
and $\Gamma_f\subset \R \times \R$ its graph.
Find the Hausdorff dimension of $\Gamma_f$.
\ez

\exercise[15]
Construct a continuous function 
$f:\; \R \arrow \R$ such that the Hausdorff dimension of 
its graph $\Gamma_f\subset \R^2$
is bigger than 1.
\ez

\exercise[10]
Construct a function
$f:\; [0,1] \arrow [0,1]$ which is continuous, but not Lipschitz
on any interval $[a,b]\subset [0,1]$.
\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Geometry of manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise[5]
Prove that any compact manifold is metrizable.
\ez

\exercise[10]
Let $M=\bigcup_iM_i$ be a union of countably many compact manifolds
$M_i$ with boundary, $M_1\subset M_2 \subset M_3 \subset ...$.
Prove that $M$ is metrizable.
\ez

\exercise[5]
Let $V \subset U \subset M$ be open subsets of a metric space
$M$, and the closure of $V$ is contained in $U$.
Prove that there exists a continuous function
$M\arrow [0,1]$ which is equal to 1 on $V$ and
0 outside of $U$.
\ez

\exercise[10]
Let $M$ be a metric space.
Suppose that $M$ is equipped with locally finite covers
$U_\alpha$ and $V_\alpha$, indexed by the same set, and
the closure of $V_\alpha$ is contained in $U_\alpha$ for each $\alpha$.
Construct a set of continuous functions
$\phi_\alpha:\; í \arrow [0,1]$ such that $\phi_\alpha=0$ 
outside of $U_\alpha$, $\phi_\alpha=1$ on $V_\alpha$,
and $\sum_\alpha \phi_\alpha \geq 1$.
\ez


\exercise[10]
Let $M$ and $M'$ be smooth manifolds, such
that the ring $C^\infty M$ is isomorphic to $C^\infty M'$.
Prove that $M$ is diffeomorphic to $M'$.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Germs of functions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\exercise[5]
Let $R$ be a ring of germs of continuous
functions on $\R$ in 0,
and $I\subset R$ an ideal. Consider the subset $I_+\subset I$
of all non-negative functions in $I$. Prove that $I$ is generated
by $I_+$, or find a counterexample.
\ez


\exercise[5]
Let $R$ be a ring of germs of continuous
functions on $\R$ in 0,
and $I\subset R$ an ideal. Consider the subset $I_s\subset I$
of all smooth functions in $I$. Prove that $I$ is generated
by $I_s$, or find a counterexample.
\ez


\exercise[5]
Let $R$ be a be a ring of germs of smooth
functions on $\R^n$ in 0,
and $K\subset R$ -- intersection of all powers of 
maximal ideal. Prove that $K$ is generated by all
non-negative $\phi\in K$.
\ez


\exercise[15]
Let $R$ be a ring of continuous $\R$-valued functions
on a topological space $M$, and $I\subset R$ a prime ideal.
Prove that $I^2=I$.
\ez

\exercise[5]
Find all prime ideals in a ring of germs of continuous functions
on $\R$ in $0$.
\ez



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Topology of manifolds and vector bundles}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise[5]
Let $M$ be a smooth manifold,
and $TM$ the total space of its tangent bundle.
Prove that $TM$ is orientable, or find a counterexample.
\ez

\exercise[5]
Let $B$ be a 1-dimensional real vector bundle
on a smooth manifold $M$. Show that $B\otimes B$ is 
a trivial bundle.
\ez

\exercise[10]
Let $B$ be a non-trivial vector bundle on a smooth manifold $M$,
and $B_1:= B\oplus C^\infty M$ a direct sum of $B$ and a trivial
1-dimensional bundle. Find a non-trivial bundle 
$B$ such that $B_1$ is trivial.
\ez

\exercise[15]
Let $M$ be a compact, even-dimensional manifold. Prove that
$M$ admits a non-trivial vector bundle.
\ez

\exercise[10]
Let $M=S^{2n}$ be an even-dimensional sphere.
Prove that $TM$ does not admit a metric of signature
$(1,2n-1)$.
\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Differential forms and differential operators}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise[10]
Let $M$ be a smooth manifold. Denote by $L_a\in \End(C^\infty M)$
an operator of multiplication by $a\in C^\infty M$.
{\bf A differential operator of first order}
is a map $D:\; C^\infty M \arrow C^\infty M$
which satisfies $[D, L_a](z)= z [D, L_a](1)$
for each $a\in C^\infty M$. Prove that 
the differential operators of first order 
are {\bf local}, that is, for each $f\in C^\infty M$ 
and an open set $U\subset M$, the functions
$D(f)\restrict U$ is determined by $f\restrict U$.
\ez

\exercise[5]
Let $\alpha$ be a closed differential form with compact support on
$\R^n$. Prove that there exists a $k$-form $\beta$ with support
in unit ball such that $\alpha-\beta=d\gamma$, where $\gamma$
is a form with compact support.
\ez

\exercise[10]
Let $\delta:\; \Lambda^*M \arrow \Lambda^* M$ be a derivation
of de Rham algebra, and $\alpha$ a differential form with support 
in a closed set $S\subset M$.
Prove that $\delta(\alpha)$ has support in $S$.
\ez

\exercise[15] Let $t\in \Lambda^2 TM$ be a 
skew-symmetric bivector on a manifold $M$, and 
$i_t:\;\Lambda^iM\arrow \Lambda^{i-2}M$
an operation of substitution a bivector into an $i$-form.
Consider a k-form $\eta$, and let $e_\eta(v) = \eta\wedge v$
be the corresponding map on de Rham algebra, 
$e_\eta:\; \Lambda^i M \arrow \Lambda^{i+k}(M)$.
Prove that the commutator $[e_\eta, i_t]$ is a derivation.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Fibered spaces and fibrations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Recall that $U(n)$ is a group of all unitary automorphisms
of $\C^n$, $SO(n)$ a group of all orthogonal, oriented
automorphisms of $\R^n$, and $Sp(n)$ a group of all 
quaternionic linear, unitary automorphisms of ${\Bbb H}^n$.
These groups are called {\bf classical Lie groups}.
\ed

\exercise[10]
Let $G$ be a classical Lie group. Find a fibration
with a total space $G$, base $S^n$ (a sphere),
and fiber $G_0$, which is also a Lie group.
Find $G_0$ in each of these cases. 
\ez

\exercise[10]
Compute the real dimension of each classical Lie group.
\ez

\exercise[5]
Construct a fibration with total space $SU(3)$, base $S^5$
and fiber $S^3$.
\ez


\exercise[5]
Construct a non-trivial fibration with 
fiber $S^3$.
\ez



\end{document}

