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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Geometry of manifolds \\[15mm]
\small lecture 2: manifolds and ringed spaces}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf Math in Moscow and HSE
\\[2mm]  February 11, 2013
}
\end{center}

\newpage

{\bf \blue Sheaves and exact sequences}


\definition
{\bf\blue An open cover} of a topological space $X$ is a
family of open sets $\{U_i\}$ such that $\bigcup_iU_i=X$.


\remark
{\bf \purple A presheaf of functions} is a collection of subrings
of functions on open subsets, compatible with
restrictions. {\bf\purple A sheaf of fuctions is a presheaf
allowing ``gluing''} a function on a bigger open set
if its restrictions to smaller open sets are compatible.


\definition
A sequence $A_1 \arrow A_2 \arrow A_3 \arrow ...$
of homomorphisms of abelian groups or vector spaces
is called {\bf\blue exact} if the image of each map is
the kernel of the next one.


\claim
A presheaf
${\cal F}$ is a sheaf if and only if for every cover $\{U_i\}$ of
an open subset $U\subset M$, {\bf \red the sequence of restriction maps
$$0\to{\cal F}(U)\to\prod\limits_i{\cal F}(U_i)\to\prod\limits_{i\ne
j}{\cal F}(U_i\cap U_j)$$
is exact,} with $\eta\in {\cal F}(U_i)$ 
mapped to 
\[ \eta\restrict{U_i\cap U_j}-\eta\restrict{U_j\cap U_i}
\in  {\cal F}(U_i\cap U_j)\oplus {\cal F}(U_j\cap
U_i)\subset \prod\limits_{i\ne
j}{\cal F}(U_i\cap U_j)
\]

\newpage

{\bf \blue Ringed spaces}

A {\bf\blue ringed space} $(M,{\cal F})$ is a
topological space equipped with a sheaf of
functions. A~{\bf \blue morphism}
$(M,{\cal F})\stackrel\Psi\longrightarrow(N,{\cal F}')$ of ringed spaces
is a continuous map $M\stackrel\Psi\longrightarrow N$ such that, for
every open subset $U\subset N$ and every function $f\in{\cal F}'(U)$, the
function $\psi^* f:=f\circ\Psi$ belongs to the ring
${\cal F}\big(\Psi^{-1}(U)\big)$. An {\bf\blue  isomorphism} of ringed spaces
is a homeomorphism $\Psi$ such that $\Psi$ and $\Psi^{-1}$ 
are morphisms of ringed spaces.

\definition
Let $(M,{\cal F})$ be a topological manifold
equipped with a sheaf of functions. It is said to be a {\bf\blue smooth
manifold of class} $C^\infty$ or $C^i$ if every point in
$(M,{\cal F})$ has an open neighborhood isomorphic to the ringed space
$({\Bbb B}^n,{\cal F}')$, where ${\Bbb B}^n\subset \R^n$ 
is an open ball and ${\cal F}'$ is a ring of 
functions on an open ball ${\Bbb B}^n$ of this class.

\definition
{\bf\blue Diffeomorphism} of smooth manifolds
is a homeomorphism $\phi$ which induces an isomorphims
of ringed spaces, that is, $\phi$ and $\phi^{-1}$ map 
(locally defined) smooth functions to smooth functions.

{\bf \red Assume from now on that all manifolds are
  Hausdorff and of class $C^\infty$}.



\newpage

{\bf \blue Charts and coordinates}

\definition
{\bf \blue Coordinate system} on a manifold $M$
is an open subset $V\subset M$ equipped with 
an isomorphism of ringed spaces
$(V, C^\infty V)\cong({\Bbb B}^n, C^\infty{\Bbb B}^n)$ 
per definition of a manifold.

\definition
{\bf \blue A chart} on a smooth manifold 
$(M, C^\infty M)$ is an open subset $U\subset M$
together with an embedding $\psi:\; U \arrow \R^n$
given by smooth functions 
$\phi_1, ..., \phi_n\in C^\infty M$
inducing a diffeomorphism on any open 
subset $V\subset U$ equipped with a coordinate
system $(V, C^\infty V)\cong ({\Bbb B}^n, C^\infty{\Bbb B}^n)$.

\definition
{\bf \blue Transition map}
between two charts $\psi_1:\; U_1 \arrow \R^n$
and $\psi_2:\; U_2 \arrow \R^n$ is a map 
$\Psi_{ij}:\;\psi_1(U_1\cap U_2) \arrow \psi_2(U_1\cap U_2)$
defined as $\Psi_{ij}(x)= \psi_2(\psi_1^{-1}(x))$.

\claim
{\bf \red Transition maps are smooth.}

\proof In local coordinates all functions 
$\phi_1, ..., \phi_n$ used in the definition of the
transition map are smooth. \endproof

\newpage

{\bf \blue Atlases on manifolds}

\definition
{\bf \blue An atlas} on a smooth manifold
is a collection of charts $\{U_i, \psi_i:\; U_i \arrow
\R^n\}$ satisfying $\bigcup U_i =M$ together with their
transition maps.

\remark In such a situation, the {\bf \purple charts $U_i$ are
usually identified with their images $\psi(U_i)\subset \R^n$.}

\remark
{\bf \purple
The sheaf $C^\infty M$ can be reconstructed from an atlas}
as follows: a function $f$ on $U\subset M$ is smooth
if and only if its restrictions to $U\cap U_i$ are
smooth on all charts.

\newpage

{\bf \blue Embedded submanifolds}

\definition
 A {\bf\blue closed embedding} $\phi:\; N\hookrightarrow M$ of
topological spaces is an injective map from $N$
to a closed subset $\phi(N)$ inducing a homeomorphism
of $N$ and $\phi(N)$. 

\definition
$M\subset \R^n$ is called {\bf\blue
a submanifold} of dimension $m$ if for every point $x\in N$, there is a
neighborhood $U\subset \R^n$ diffeomorphic to an open ball, such that this
diffeomorphism maps $U\cap N$ onto a linear subspace of
dimension $m$.

\definition
Let $M\subset \R^n$ be a submanifold.
{\bf\blue A sheaf of smooth functions}  $C^\infty M$
is defined as the sheaf of all functions on $U\subset M$
which can be locally extended to smooth functions on an open
subset of $\R^n$ containing $U$.

\newpage

{\bf \blue Embedded submanifolds are smooth}

\example 
Let $M=\R^k\subset \R^n$ be a linear space, and $\pi:\;
\R^n\arrow M$ be a linear projection.
{\bf \purple A function $f$ on $U\subset M$ is smooth if and only if
$\pi^* f$ is smooth} (by definition,
$\pi^*f(z)=f(\pi(z))$). Therefore, {\bf \red any smooth
function on $U$ can be extended to $\pi^{-1}(U)\subset\R^n$.}
We obtain that $(M,C^\infty M)\cong (\R^k, C^\infty \R^k)$.

\claim
Let $M\subset \R^n$ be a submanifold,
and $C^\infty M$ the sheaf defined above.
{\bf \red Then $(M, C^\infty M)$ is a smooth manifold.}

\proof
Locally around each point of $M$, the pair
$(\R^n, M)$ is diffeomorphic to $(\R^n, \R^k)$.
Then the previous example can be applied.
\endproof


\theorem
{\bf \red Any manifold can be embedded to $\R^n$.}

Its proof takes some work and will be done in the next lecture.


\newpage

{\bf \blue Locally finite covers}

\definition
An open cover $\{U_\alpha\}$ of a topological space $M$
is called {\bf\blue locally finite} if every point in $M$ possesses a
neighborhood that intersects only a finite number of $U_\alpha$.

\claim
Let $\{U_\alpha\}$ be a locally finite atlas on a manifold $M$. 
{\bf \purple Then there exists 
a refinement $\{V_\beta\}$ of $\{U_\alpha\}$ such that
a closure of each $V_\beta$ is compact in $M$.}

{\bf \green Proof:}
Let $\{U_\alpha\}$ be a locally finite atlas on $M$, and
$U_\alpha\stackrel {\phi_\alpha}\arrow \R^n$ homeomorphisms.
Consider a cover $\{V_i\}$ of $\R^n$ given by open balls 
of radius 2 centered in integer points, and let $\{W_\beta\}$
be a cover of $M$ obtained as union of $\phi_\alpha^{-1}(V_i)$.
{\bf \purple Then $\{W_\beta\}$ is locally finite.}
\endproof


\definition
Let $U\subset V$ be two open subsets of $M$ such that the closure
of $U$ is contained in $V$. {\bf \blue In this case we write
$U\Subset V$.}

\newpage

{\bf \blue Locally finite covers and their subcovers}


\exercise
Let $K_1, K_2$ be non-intersecting closed subsets
of a Hausdorff topological space $M$, with $K_1$ compact. 
{\bf \red Show that there exist a pair of open subsets $U_1\supset K_1$,
$U_2\supset K_2$ satisfying $U_1 \cap U_2=\emptyset$.}


\claim
Let $U\subset M$ be an open subset with compact closure,
and $V\supset M\backslash U$ another open subset. {\bf
\purple Then
there exists $U'\subset U$ such that the closure of
$U'$ is contained in $U$, and $V\cup U'=M$.}

{\bf \green Proof. Step 1:}
The complement $M\backslash U$ does not intersect 
$M\backslash V$, these sets are closed, and
$M\backslash V$ is compact.

{\bf \green Step 2:}
Take open subsets $A, B \subset M$ separating
$M\backslash U$ and $M\backslash V$. Then 
the closure $\bar B$ does not intersect
$M\backslash U$, because $B$ does not intersect
an open neighbourhood of $M\backslash U$.
Therefore, $U':=B$ lies
in $U$ with its closure.

{\bf \green Step 3:} By construction, $\bar B$ contains
$M\backslash V$, hence $U'\cup V=M$.
\endproof



\newpage

{\bf \blue Locally finite covers and their subcovers}


\theorem
Let $\{U_\alpha\}$ be a countable locally finite
cover of a Hausdorff topological space,
such that a closure of each $U_\alpha$ is compact.
{\bf \red Then there exists another cover
$\{V_\alpha\}$ indexed by the same set,
such that $V_\alpha\Subset U_\alpha$.}

{\bf \green Proof. Step 1:}
Let $U_1, U_2, ...$ be all elements of the cover.
Suppose that $V_1, ..., V_{n-1}$ is already found. To take
an induction step it remains to find $V_n \Subset U_n$

{\bf \green Step 2:} Replacing $U_i$ by $V_i$ and
renumbering, we may assume that $n=1$. 
{\bf \purple Then the statement of Theorem follows
from the previous exercise applied to
$V=\bigcup_{i=2}^\infty U_i$ and $U=U_1$.}
\endproof

\newpage

{\bf \blue Construction of a partition of unity}


\remark
If all $U_\alpha$ are diffeomorphic to $\R^n$,
all $V_\alpha$ can be chosen diffeomorphic to an open ball.
Indeed, any compact set is contained in an open ball.

\corollary
Let $M$ be a manifold admitting a locally finite countable
cover $\{U_\alpha\}$, with $\phi_\alpha:\; U_\alpha
\arrow \R^n$ diffeomorphisms. {\bf \purple Then there exists 
another atlas $\{U_\alpha, \phi_\alpha':\; U_\alpha
\arrow \R^n\}$, such that $\phi'_\alpha({\Bbb B})$
is also a cover of $M$, and ${\Bbb B}\subset \R^n$ a
unit ball.}
\endproof

\exercise
{\bf \purple Find a smooth function $\nu:\; \R^n \arrow [0,1]$ which
vanishes outside of ${\Bbb B}\subset \R^n$ and is positive
on ${\Bbb B}$.}

\remark In assumptions of Corollary,
let $\nu_\alpha(z):= \nu(\phi_\alpha')$, and
$\mu_i:=\frac{\nu_i}{\sum_\alpha\nu_\alpha}$.
Then $\mu_\alpha:\; M \arrow [0,1]$ are smooth
functions with support in $U_\alpha$ satisfying $\sum_\alpha \mu_\alpha=1$.
Such a set of functions is called {\bf \blue a partition
  of unity}.


\newpage

{\bf \blue Partition of unity: a formal definition}

\definition
 Let $M$ be a smooth manifold and let $\{U_\alpha\}$
a locally finite cover of $M$. A {\bf \blue partition of unity}
subordinate to the cover $\{U_\alpha\}$ is a family of smooth functions
$f_i:M\to[0,1]$ with compact support indexed by the same indices as the
$U_i$'s and satisfying the following conditions.\\
(a) Every function $f_i$ vanishes outside $U_i$\\
(b) $\sum_if_i=1$

The argument of previous page proves the following
theorem.

\theorem
Let $\{U_\alpha\}$ be a countable, locally finite cover of a manifold
$M$, with all $U_\alpha$ diffeomorphic to $\R^n$. {\bf
  \red Then
there exists a  partition of unity subordinate to $\{U_\alpha\}$.}

\newpage

{\bf \blue Whitney's theorem for compact manifolds}

\theorem
Let $M$ be a compact smooth manifold. {\bf \red Then $M$ admits a closed
smooth embedding to $\R^N$.}

{\bf\green  Proof. Step 1:}
Choose a finite atlas 
$\{V_i, \phi_i:\; V_i \arrow \R^n, i=1, 2, ..., m\}$,
and subordinate partution of unity $\mu_i:\; M \arrow
[0,1]$. Let $\alpha:\; [0,1]\arrow [0,1]$
be a smooth, monotonous function mapping 0 to 0 and
$[1/2m,1]$ to 1, and $\nu_i:=\alpha(\mu_i)$.

{\bf\green Step 2:}
Denote by $W_i$ the set of interior points of
$\bar W_i:=\{z \ \ |\ \ \nu_i(z)=1\}$. Since
$\sum_{i=1}^m \mu_i =1$, the set
$\{W_i\}$ is a cover of $M$.

{\bf\green Step 3:}
For each $i$, the map  
$\Phi_i(z):= \left(\nu_i\phi_i(z),
\sqrt{1-|\phi_i(z)|^2\nu_i(z)^2}\right)$
is injective on $W_i$
and maps $M$ to a sphere $S^n \subset \R^{n+1}$.

{\bf\green Step 4:}
The product map
\[ \Psi:=\prod_{i=1}^m:\;\Phi_i:\; M \arrow \underbrace{S^n\times
S^n \times ...\times S^n}_{\text{$m$ times}}
\]
is an injective, continuous map from a compact,
hence it is a homeomorphism to its image {\bf \purple
  (prove it)}.


\newpage

{\bf \blue Whitney's theorem for compact manifolds (cont.)}

{\bf\green  Step 5:}
Any smooth function on 
$\bar W_i:=\{z \ \ |\ \ \nu_i(z)=1 \}$ can be obtained as a
restriction of a smooth function on $\Phi_i(\bar
W_i)\subset S^n$,
hence {\bf \purple this map induces an isomorphism of the
ring of smooth functions on $\bar W_i$ and its image.}

{\bf\green  Step 6:} Let $W_i$ be the
set of interior points of $\bar W_i$. 
We have constructed a cover $\{W_i\}$
of $M$ such that on each $W_i$ the map $\Psi$
induced an isomorphism of ringed spaces.
\endproof






\end{document}

