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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Geometry of manifolds \\[15mm]
\small lecture 1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf Math in Moscow and HSE
\\[2mm]  February 04, 2013
}
\end{center}

\newpage

{\bf \blue The Plan.}

{\bf \green Preliminaries:} 
I assume knowledge of {\bf \blue topological spaces}
and {\bf \blue continuous maps}, {\bf \blue
  homeomorphisms}, {\bf \blue connected spaces},
{\bf \blue path connected spaces},
{\bf \blue metric spaces}, {\bf compact spaces},
{\bf \blue groups}, {\bf \blue abelian groups}, {\bf \blue homomorphisms}
and {\bf \blue vector spaces}.

{\bf \green Plan of today's talk:}

1. Topological manifolds.

2. Smooth manifolds.

3. Sheaves of functions.

4. 3 different definitions of a smooth manifold

5. Partition of unity.

\newpage

{\bf \blue Topological manifolds}

\remark
Manifolds can be smooth (of a given ``differentiability
class''), real analytic, or topological (continuous). 

\definition 
{\bf\blue Topological manifold} is
a topological space which is locally homeomorphic to
an open ball in $\R^n$.

{\bf \green PROBLEM:}
{\bf \purple Show that a group of homeomorphisms acts on 
a connected manifold transitively. }

\definition Such a topological
space is called {\bf \blue homogeneous}.

\newpage

{\bf \blue Topological manifolds: some unsolved problems}


\definition
{\bf \blue Geodesic} in a metric space is an
isometry $[0,1]\arrow M$.

\definition
A {\bf \blue Busemann space} is a metric space $M$ such that
any two points can be connected by a geodesic, any 
closed, bounded subset of $M$ is compact, and a geodesic connecting
$x$ to $y$ is unique when $d(x,y)$ is sufficiently small.

\remark
A Busemann space is homogeneous.

\conjecture (Busemann, 1955)\\
{\bf \red Any Busemann space is a topological manifold.}

{\it\green    ...Although
this (the Busemann Conjecture) is probably true for any
G-space, the proof, if the
conjecture is correct, seems quite inaccessible in the
present state of topology... (Herbert Busemann)}

There are many other conjectures about path connected,
homogeneous topological spaces (Bing-Borsuk, Moore, de
Groot...), implying that they are
manifolds, none of them proven, except in low dimension.


\newpage

{\bf \blue Conflict}


\begin{center}

{\small Herbert Busemann\\ (Berlin, 1905 - Santa Ynez,
  1994)}

\epsfig{file=Herbert-Busemann-Conflict-2772-noho.jpg,width=0.72\linewidth}

 {\em Conflict,} 1972

\end{center}


\newpage

{\bf \blue Atlases on manifolds}


\definition
{\bf\blue An open cover} of a topological space $X$ is a
family of open sets $\{U_i\}$ such that $\bigcup_iU_i=X$. A cover
$\{V_i\}$ is a {\bf \blue refinement} of a cover $\{U_i\}$ if every $V_i$ is
contained in some $U_i$.

\remark
{\bf \purple Any two covers  $\{U_i\}$,  $\{V_i\}$ of a topological space
admit a common refinement $\{U_i\cap V_j\}$.}

\definition
Let $M$ be a topological manifold.
 A cover $\{U_i\}$ of $M$ is an {\bf\blue atlas} if for every
$U_i$, we have a map $\varphi_i:U_i\to\R^n$ giving a
homeomorphism of $U_i$ with an open subset in $\R^n$. 
In this case, one defines
the {\bf
\blue transition maps}
\[ 
\Phi_{ij}:\varphi_i(U_i\cap U_j)\to\varphi_j(U_i\cap U_j)
\]

\definition
A function $\R \arrow \R$ is {\bf \blue of
differentiability class $C^i$} if it is $i$ times
differentiable, and its $i$-th derivative is continuous.
A map $\R^n \arrow \R^m$ is {\bf \blue of
differentiability class $C^i$} if all its 
coordinate components are. A {\bf\blue smooth function}
(map) is a function (map) of class $C^\infty =\bigcap C^i$.

\definition
 An atlas is {\bf \blue smooth} if
all transition maps are smooth (of~class $C^\infty$, i.e., infinitely
differentiable), {\bf \blue smooth of class} $C^i$ if all transition
functions are of differentiability class $C^i$, 
and~{\bf \blue real analytic} if all transition maps 
admit a Taylor expansion at each point.


\newpage

{\bf \blue Smooth structures}

\definition
A {\bf\blue refinement} of an {\bf\blue
atlas} is a refinement of the corresponding cover $V_i\subset U_i$
equipped with the maps $\varphi_i:V_i\to\Bbb R^n$ that are the
restrictions of $\varphi_i:U_i\to\Bbb R^n$. Two atlases
$(U_i,\varphi_i)$ and $(U_i,\psi_i)$ of class $C^\infty$ or $C^i$
(with the same cover) are {\bf\blue equivalent} in this class if, for
all $i$, the map $\psi_i\circ\varphi_i^{-1}$ defined on the
corresponding open subset in $\Bbb R^n$ belongs to the mentioned
class. Two~arbitrary atlases are {\bf\blue equivalent} if the
corresponding covers possess a common refinement giving
equivalent atlases.


\definition
 A {\bf\blue smooth structure} on a manifold (of class
$C^\infty$ or $C^i$) is an atlas of class $C^\infty$ or $C^i$
considered up to the above equivalence. A {\bf\blue smooth manifold} is a
topological manifold equipped with a smooth structure.

\definition
A {\bf\blue smooth function} on a manifold
$M$ is a function $f$ whose restriction to the chart $(U_i,\varphi_i)$
gives a smooth function 
$f\circ \varphi_i^{-1}:\; \varphi_i(U_i)\arrow \R$ for each open
subset $\varphi_i(U_i)\subset\Bbb R^n$.

\newpage

{\bf \blue Smooth maps and isomorphisms}

From now on, {\bf \red I shall identify the charts $U_i$
with the corresponding subsets of $\R^n$,} and forget
the differentiability class.

\definition
{\bf \blue A smooth map} of $U\subset \R^n$ to a
manifold $N$ is a map $f:\; U \arrow N$ such that for each chart 
$U_i\subset N$, the restriction
$f\restrict{f^{-1}(U_i)}:\;f^{-1}(U_i)\arrow U_i$
is smooth with respect to coordinates on $U_i$.
{\bf \blue A map of manifolds}  
$f:\; M \arrow N$ is {\bf \blue smooth}
if for any chart $V_i$ on $M$, the restriction
$f\restrict{V_i}:\; V_i \arrow N$ is smooth
as a map of $V_i\subset \R^n$ to $N$.

\definition
{\bf \blue An isomorphism} of smooth manifolds
is a bijective smooth map $f:\; M \arrow N$ such that
$f^{-1}$ is also smooth.

\newpage

{\bf \blue Smooth structures, smooth finctions and sheaves}

\remark
For two equivalent atlases of a given differentiability
class $C^i$,  {\bf \purple the spaces $C^iM$ of $C^i$-functions 
coincide.}

Converse is also true.

{\bf \green PROBLEM:}
Let $f:\; M \arrow N$ be a map of smooth manifolds
such that $f^*\mu$ is smooth for any smooth function
$\mu:\; N \arrow \R$. {\bf \red Show that $f$ is a smooth map.}

\remark
It's better to define smooth structures in terms
of smooth functions, but for practical work
{\bf \purple it's most convenient to use sheaves}.

\newpage

{\bf \blue Sheaves}

\definition
A {\bf\blue presheaf of functions} on a
topological space $M$ is a collection of subrings 
${\cal F}(U)\subset C(U)$ in
the ring $C(U)$ of all functions on $U$, for each open 
subset $U\subset M$,  such that the restriction of
every $\gamma\in{\cal F}(U)$ to an open subset $U_1\subset U$ belongs to
${\cal F}(U_1)$. 


\definition
A presheaf of functions
${\cal F}$ is called {\bf\blue a sheaf of functions} 
if these subrings
satisfy the following condition. Let $\{U_i\}$ be a cover of an open
subset $U\subset M$ (possibly infinite)
and $f_i\in{\cal F}(U_i)$ a family of
functions defined on the open sets of the cover and 
compatible on the pairwise intersections:
$$f_i|_{U_i\cap U_j}=f_j|_{U_i\cap U_j}$$
for every pair of members of the cover. {\bf \purple Then there exists
$f\in{\cal F}(U)$ such that $f_i$ is the restriction of $f$ to $U_i$ for
all $i$.}


\newpage

{\bf \blue Sheaves and exact sequences}


\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 $\eta\restrict{U_i\cap U_j}$
and $-\eta\restrict{U_j\cap U_i}$. 


\newpage

{\bf \blue Sheaves and presheaves: examples}

{\bf \green Examples of sheaves:}

* Space of continuous functions

* Space of smooth functions, any differentiability class

* Space of real analytic functions


{\bf \green Examples of presheaves which are not sheaves:}

* Space of constant functions {\bf \purple (why?)}

* Space of bounded functions {\bf \purple (why?)}

\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.

\example
Let $M$ be a manifold of class $C^i$ and let
$C^i(U)$ be the space of functions of this class. 
{\bf \purple Then $C^i$ is a sheaf of functions, and 
$(M, C^i)$ is a ringed space.}


\remark
Let $f:\; X \arrow Y$ be a smooth map of smooth manifolds.
Since a pullback $f^*\mu$ of a smooth function $\mu\in C^\infty(M)$
is smooth, {\bf \purple a smooth map of 
smooth manifolds defines a morphism of ringed spaces.}

{\bf \red Converse is also true:}

\newpage

{\bf \blue Ringed spaces and smooth maps}

\claim
Let $(M, C^i)$ and $(N,C^i)$ be manifolds of class $C^i$.
Then {\bf \red there is a bijection between smooth maps $f:\; M \arrow N$
and the morphisms of corresponding ringed spaces.}

{\bf \green Proof:}
Any smooth map induces a morphism of ringed spaces.
Indeed, {\bf \purple a composition of smooth functions is smooth,
hence a pullback is also smooth.}

Conversely, let $U_i \arrow V_i$ be a restriction
of $f$ to some charts; to show that $f$ is smooth,
it would suffice to show that $U_i \arrow V_i$ is smooth.
However, we know that a pullback of any smooth function
is smooth. {\bf \purple Therefore, Claim is implied by the following
lemma.}

\lemma
Let $M,N$ be open subsets in $\Bbb R^n$ and let
$f:\; M\to N$ map such that a pullback of any
function of class $C^i$ belongs to $C^i$. {\bf \red Then 
$f$ is of class $C^i$.}

{\bf \green Proof:} Apply $f$ to coordinate functions. \endproof

\newpage

{\bf \blue A new definition of a manifold}

As we have just shown, this definition is equivalent to
the previous one.

\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
$(\Bbb R^n,{\cal F}')$, where ${\cal F}'$ is a ring of 
functions on $\Bbb R^n$ of this class.

\definition
{\bf \blue A chart}, or 
{\bf\blue a coordinate system} on an open subset $U$
of a manifold $(M,{\cal F})$ is an isomorphism between $(U,{\cal F})$ and
an open subset in $(\Bbb R^n,{\cal F}')$, where ${\cal F}'$ are functions
of the same class on $\Bbb R^n$.

\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 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
{\bf \blue A morphism} of embedded submanifolds
$M_1 \subset \R^n$ to $M_2 \subset \R^n$
is a map $f:\; M_1 \arrow M_2$ such that
any point $x\in M_1$ has a neighbourhood 
$U$ such that $f\restrict{M_1 \cap U}$
can be extended to a smooth map $U \arrow \R^n$.

\remark
The third definition of a smooth manifold:
{\bf \red a smooth manifold can be defined as a smooth submanifold
in $\R^n$.}

This definition becomes equivalent to the usual
one if one proves the Whitney's theorem.

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

Its proof takes some work.


\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}



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

\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 {\bf \red (why it exists?)}
$\{V_i, \phi_i:\; V_i \arrow \R^n, i=1, 2, ..., m\}$,
and subordinate partution of unity $\nu_i:\; M \arrow
[0,1]$. For each $i$, the map 
\[ \Phi_i(z):= \left(\nu_i\phi_i(z),
\sqrt{1-\nu_i(z)^2}\right)
\]
is injective on the set $\{z \ \ |\ \ \nu_i(z)> 0\}$,
and maps $M$ to a sphere $S^n \subset \R^{n+1}$.

{\bf\green Step 2:}
The product map
\[ \prod_{i=1}^m:\;\Phi_i:\; M \arrow \underbrace{S^n\times
S^n \times ...\times S^n}_{\text{$m$ times}}
\]
is an injective, continuous map from a compact,
hence it is a homeomorphism to its image {\bf \purple (prove it)}.

{\bf\green  Step 3:}
Any smooth function on $W_i:=\{z \ \ |\ \ \nu_i(z)> \frac
1 {2m}\}$ can be obtained as a
restriction of a smooth function on $\R^{n+1}$,
hence {\bf \purple this map induces an isomorphism of the
corresponding sheaves of smooth functions.}
\endproof






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