\input amstex
\documentstyle{amsppt}\nologo\footline={}\subjclassyear{2000}

\def\GL{\mathop{\text{\rm GL}}}

\hsize450pt\vsize590pt\topmatter\title Analysis 1: Smooth
manifolds.\endtitle\author Misha
Verbitsky\endauthor\address\endaddress\email\endemail
%\subjclass\endsubjclass\abstract\endabstract
\endtopmatter\document

In order to be approved, you should solve in every sheet either all
problems with asterisks or all problems without asterisks. The problems
with two asterisks are optional: $k$ problems with two asterisks
substitute $2k$ problems with one asterisk. The problems with (!)
are obligatory for everybody.

\bigskip

\noindent
{\bf1.1.~Topological manifolds}

\medskip

\noindent
{\bf Remark.} Manifolds can be smooth (of a given ``class of
smoothness''), real analytic, or topological (continuous). These
types of manifolds admit different definitions. One may specify a
type if necessary, but usually it is clear from the context.

\smallskip

\noindent
{\bf Definition 1.1.} A {\bf topological manifold} of dimension $n$ is
a topological space where every point has a neighborhood homeomorphic
to $\Bbb R^n$.

\smallskip

\noindent
{\bf Remark.} Let $G$ be a group acting on a set $M$. The {\bf
stabilizer} of $x\in M$ is the subgroup of all elements in $G$ that
fix $x$. An action is {\bf free} if the stabilizer of every point is
trivial.

\smallskip

\noindent
{\bf Remark.} Any action of a group on a topological space is
continuous by default.

\smallskip

\noindent
{\bf Problem 1.1.} Suppose that a finite group $G$ acts freely on a
Hausdorff manifold $M$. Show that the quotient space $M/G$ is a
manifold.

\smallskip

\noindent
{\bf Problem 1.2 (!).} Construct an example of a finite group $G$
acting non-freely on a manifold $M$ such that $M/G$ is not a manifold.

\smallskip

\noindent
{\bf Problem 1.3.} Consider the quotient of $\Bbb R^2$ by the action
of $\{\pm1\}$ that maps $x$ to $-x$. Is the quotient space a manifold?

\smallskip

\noindent
{\bf Problem 1.4 (*).} Show that the $n$-dimensional sphere $\Bbb S^n$,
the $n$-dimensional real projective space $\Bbb P_\Bbb R^n$, and the
$n$-dimensional complex projective space $\Bbb P_\Bbb C^n$ are
manifolds.

\smallskip

\noindent
{\bf Remark.} In the above definition of a manifold, we do not require
it to be Hausdorff. Nevertheless, in~many cases, manifolds are assumed
to be Hausdorff by default.

\smallskip

\noindent
{\bf Problem 1.5.} Construct an example of a non-Hausdorff manifold.

\smallskip

\noindent
{\bf Problem 1.6.} Show that $\Bbb R^2/\Bbb Z^2$ is a manifold.

\smallskip

\noindent
{\bf Problem 1.7.} Let $\alpha$ be an irrational number. The group
$\Bbb Z^2$ acts on $\Bbb R$ by the formula $t\mapsto t+m+n\alpha$.
Show that this action is free, but the quotient $\Bbb R/\Bbb Z^2$ is
not a manifold.

\smallskip

\noindent
{\bf Problem 1.8 (**).} Construct an example of a manifold of positive
dimension such that the closures of two arbitrary nonempty sets always
intersect or show that such a manifold cannot exist.

\smallskip

\noindent
{\bf Problem 1.9 (**).} Let $G\subset\GL(n,\Bbb R)$ be a compact
subgroup. Show that $G$ is a manifold and that the quotient space
$\GL(n,\Bbb R)/G$ is also a manifold.

\bigskip

\noindent
{\bf1.2.~Smooth manifolds}

\medskip

\noindent
{\bf Definition 1.2.} A {\bf 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 refinement} of a cover $\{U_i\}$ if every $V_i$ is
included in some $U_i$.

\smallskip

\noindent
{\bf Problem 1.10.} Show that any two cover of a topological space
admit a common refinement.

\smallskip

\noindent
{\bf Definition 1.3.} A cover $\{U_i\}$ is an {\bf atlas} if, for every
$U_i$, we have a map $\varphi_i:U_i\to\Bbb R^n$ that provides a
homeomorphism of $U_i$ with an open subset in $\Bbb R^n$. The {\bf
transition maps}
$$\Phi_{ij}:\varphi_i(U_i\cap U_j)\to\varphi_j(U_i\cap U_j)$$
are induced by the above homeomorphisms. An atlas is {\bf smooth} if
all transition maps are smooth (of~class $C^\infty$, i.e., infinitely
differentiable), {\bf smooth of class} $C^i$ if they are $i$ times
differentiable, and~{\it real analytic} if all transition maps are
developable in Taylor's series at every point.

\smallskip

\noindent {\bf Definition 1.4.} A {\bf refinement} of an {\bf
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 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 equivalent} if the
corresponding cover possess a common refinement and the
corresponding refinements of the atlases are equivalent.

\smallskip

\noindent
{\bf Definition 1.5.} A {\bf 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 smooth manifold} is a
topological manifold equipped with a smooth structure.

\smallskip

\noindent
{\bf Remark.} Terrible, is not it?

\smallskip

\noindent
{\bf Problem 1.11 (*).} Construct an example of two nonequivalent
smooth structures on $\Bbb R^n$.

\smallskip

\noindent {\bf Definition 1.6.} A {\bf smooth function} on a manifold
$M$ is a function $f$ whose restriction to the chart $(U_i,\varphi_i)$
provides a smooth map $f\circ \varphi_i^{-1}$ defined on the open
subset $\varphi_i(U_i)\subset\Bbb R^n$.

\smallskip

\noindent
{\bf Remark.} There are several ways to define a smooth manifold. The
above way is canonical. It is not the most convenient one but you
should know it. Two other ways (via sheaves of functions and via
Whitney's theorem) are presented in these sheets.

\smallskip

\noindent
{\bf Definition 1.7.} A {\bf pre-sheaf} of {\bf functions} on a
topological space $M$ is given by the following data. For every open
subset $U\subset M$, it is given a subring $\Cal F(U)\subset F(U)$ in
the ring $F(U)$ of all functions on $U$ such that the restriction of
every $\gamma\in\Cal F(U)$ to an open subset $U_1\subset U$ belongs to
$\Cal F(U_1)$. A pre-sheaf is called a {\bf sheaf} if such subrings
satisfy the following conditions. Let $\{U_i\}$ be a cover of an open
subset $U\subset M$ and let $f_i\in\Cal F(U_i)$ be a family of
functions defined on the members of the cover and satisfying the
condition
$$f_i|_{U_i\cap U_j}=f_j|_{U_i\cap U_j}$$
for every pair of members of the cover. Then there exists
$f\in\Cal F(U)$ such that $f_i$ is the restriction of $f$ to $U_i$ for
all $i$.

\smallskip

\noindent
{\bf Problem 1.12 (!).} Let $\Cal F$ be a pre-sheaf of functions. Show
that $\Cal F$ is a sheaf if and only if, for every cover $\{U_i\}$ of
an open subset $U\subset M$, the sequence
$$0\to\Cal F(U)\to\prod\limits_i\Cal F(U_i)\to\prod\limits_{i\ne
j}\Cal F(U_i\cap U_j)$$
is exact.

\smallskip

\noindent
{\bf Remark.} An {\bf exact sequence} is a sequence of abelian groups
and homomorphisms
$$\dots\to A_1\to A_2\to A_3\to\dots$$
such that the kernel of every arrow coincides with the image of the
previous one.

\smallskip

\noindent
{\bf Problem 1.13.} Show that the following spaces of functions on
$\Bbb R^n$ are rings and define sheaves of functions.

a. The space of continuous functions

b. The space of infinitely smooth functions

c. The space of $i$ times differentiable functions

d(*). The space of functions that are pointwise limits of sequences of
continuous functions

e. The space of functions vanishing outside a set of measure $0$

\smallskip

\noindent
{\bf Problem 1.14.} Show that the following spaces of functions on
$\Bbb R^n$ are pre-sheaves but are not sheaves.

a. The space of constant functions

b. The space of limited functions

c. The space of functions vanishing outside a limited subset

d(*). The space of Lebesgue measurable functions with finite measure % ???

\smallskip

\noindent
{\bf Definition 1.8.} A {\bf ringed space} $(M,\Cal F)$ is a
topological space equipped with a sheaf of functions. A~{\bf morphism}
$(M,\Cal F)\overset\Psi\to\longrightarrow(N,\Cal F')$ of ringed spaces
is a continuous map $M\overset\Psi\to\longrightarrow N$ such that, for
every open subset $U\subset N$ and every function $f\in\Cal F'(U)$, the
function $f\circ\Psi$ belongs to the ring
$\Cal F\big(\Psi^{-1}(U)\big)$. An {\bf isomorphism} of ringed spaces
is a homeomorphism $\Psi$ such that $\Psi$ and $\Psi^{-1}$ satisfy the
above condition (i.e., are morphisms of ringed spaces).

\smallskip

\noindent
{\bf Remark.} Frequently, the term ``ringed space'' stands for a more
general concept, where the ``sheaf of functions'' is an abstract
``sheaf of rings,'' not necessarily a subsheaf in the sheaf of all
functions on $M$. The above definition is simpler although not quite
standard.

\smallskip

\noindent
{\bf Problem 1.15.} Let $M,N$ be open subsets in $\Bbb R^n$ and let
$\Psi:M\to N$ be a smooth map. Show that $\Psi$ defines a morphism of
spaces ringed by smooth functions.

\smallskip

\noindent
{\bf Problem 1.16.} Let $M$ be a smooth manifold of some class and let
$\Cal F$ be the space of functions of this class. Show that $\Cal F$ is
a sheaf.

\smallskip

\noindent
{\bf Problem 1.17 (!).} Let $M$ be a topological manifold and let
$(U_i,\varphi_i)$ and $(V_j,\psi_j)$ be smooth structures on $M$. Show
that these structures are equivalent if and only if the corresponding
sheaves of smooth functions coincide.

\smallskip

\noindent
{\bf Remark.} The above problem implies that the following definition
is equivalent to Definition 1.5.

\smallskip

\noindent
{\bf Definition 1.9.} Let $(M,\Cal F)$ be a topological manifold
equipped with a sheaf of functions. It is said to be a {\bf smooth
manifold} of {\bf 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 $\Cal F'$ are functions on $\Bbb R^n$ of
the mentioned class.

\smallskip

\noindent
{\bf Definition 1.10.} A {\bf 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 $\Bbb R^n$.

\smallskip

\noindent
{\bf Remark.} In order to avoid a complicated notation, from now on,
we assume all manifolds to be Hausdorff and infinitely smooth. The case
of other classes can be considered in the same manner.

\smallskip

\noindent
{\bf Problem 1.18.} Let $(M,\Cal F)$ and $(N,\Cal F')$ be manifolds
and let $\Psi:M\to N$ be a continuous map. Show that the following
conditions are equivalent.

(a) In local coordinates, $\Psi$ is given by a smooth map

(b) $\Psi$ is a morphism of ringed spaces.

\smallskip

\noindent
{\bf Remark.} An isomorphism of smooth manifolds is called a {\bf
diffeomorphism}. A diffeomorphism is a homeomorphism that maps smooth
functions onto smooth ones.

\smallskip

\noindent
{\bf Problem 1.19 (*).} Let $\Cal F$ be a pre-sheaf of functions on
$\Bbb R^n$. Figure out a minimal sheaf that contains $\Cal F$ in the
following cases.

(a) Constant functions

(b) Functions vanishing outside a limited subset

(c) Limited functions

\smallskip

\noindent
{\bf Problem 1.20 (*).} Consider the ringed space $(\Bbb R^n,C^i)$ with
$i$ times differentiable functions. Describe all morphisms from
$(\Bbb R^n,C^{i+1})$ to $(\Bbb R^n,C^i)$.

\bigskip

\noindent
{\bf1.2.~Embedded manifolds}

\medskip

\noindent
{\bf Definition 1.11.} A {\bf closed embedding} $N\hookrightarrow M$ of
topological spaces is a homeomorphism of $N$ with its image that is
closed in $M$.

\smallskip

\noindent
{\bf Definition 1.12.} Let $M$ be a smooth manifold of dimension $m$
and let $N\subset M$ be a subset. Then $N$ is called an {\bf embedded
manifold} of dimension $n$ and the map $N\hookrightarrow M$ is called
a {\bf smooth embedding} if, for every point $x\in N$, there is a
neighborhood $U\subset M$ diffeomorphic to $\Bbb R^m$ such that the
diffeomorphism maps $U\cap N$ onto a linear subspace of dimension $n$.
If the image of $N$ is closed in $M$, the map $N\hookrightarrow M$ is
called a {\bf closed embedding}.

\smallskip

\noindent
{\bf Problem 1.21 (!).} Let $(M,\Cal F)$ be a smooth manifold and let
$N\subset M$ be an embedded submanifold. Consider the space
$\Cal F'(U)$ of functions on $U\subset N$ that are extendable to
functions on $M$ defined on some neighborhood of $U$.

a. Show that $\Cal F'$ is a sheaf.

b. Show that this sheaf defines a smooth structure on $N$.

c. Show that the natural embedding $(N,\Cal F')\to(M,\Cal F)$ is a
morphism of manifolds.

\smallskip

\noindent
{\bf Problem 1.22.} Let $N_1,N_2$ be two manifolds and let
$\varphi_i:N_i\to M$ be smooth embeddings. Suppose that the image of
$N_1$ coincides with that of $N_2$. Show that $N_1$ and $N_2$ are
isomorphic.

\smallskip

\noindent
{\bf Remark.} By the above problem, in order to define a smooth
structure on $N$, it suffices to embed $N$ into $\Bbb R^n$. As it will
be clear in the next sheet, every manifold is embeddable into
$\Bbb R^n$ (assuming reasonable conditions). Therefore, in place of a
smooth manifold, we can use ``manifolds that are smoothly embedded into
$\Bbb R^n$.''

\smallskip

\noindent
{\bf Problem 1.23.} Construct a smooth embedding of
$\Bbb R^2/\Bbb Z^2$ into $\Bbb R^3$.

\smallskip

\noindent
{\bf Problem 1.24 (*).} Show that $\Bbb P_\Bbb R^n$ does not admit
a smooth embedding into $\Bbb R^{n+1}$ for $n>1$.

\bigskip

\noindent
{\bf1.4.~Partition of unity}

\medskip

\noindent
{\bf Definition 1.13.} A cover $\{U_i\}$ of a topological space $M$
is called {\bf locally finite} if every point in $M$ possesses a
neighborhood that intersects only a finite number of members of the
cover.

\smallskip

\noindent
{\bf Problem 1.25.} Let $\{U_i\}$ be a locally finite cover of $M$
such that every $U_i$ is homeomorphic to $\Bbb R^n$. Show that the
cover possesses a locally finite refinement $\{V_i\}$ such that
the closure of every $V_i$ is compact in $M$.

\smallskip

\noindent
{\bf Hint.} Cover every $U_i=\Bbb R^n$ by the balls of radius $1$
centered at integer points.

\smallskip

\noindent
{\bf Problem 1.26 (!).} Given a locally finite cover $\{U_i\}$ of a
manifold $M$ such that every $U_i$ is equipped with a homeomorphism
$U_i\overset\varphi_i\to\longrightarrow\Bbb R^n$ and has compact
closure in $M$, show that there exists a collection of numbers $r_i>0$
such that the $\varphi_i^{-1}(B_{r_i})$ form a cover of $M$, where
$B_{r_i}$ stands for the open ball of radius $r_i$ centered at $0$.

\smallskip

\noindent
{\bf Problem 1.27 (!).} Let $M$ be a manifold admitting a locally
finite cover by open subsets homeomorphic to $\Bbb R^n$. Show that the
cover has a locally finite refinement $\{U_i\}$ such that every $U_i$
can be equipped with a homeomorphism
$U_i\overset\varphi_i\to\longrightarrow\Bbb R^n$ and the inverse images
$\varphi_i^{-1}(B_1)$ of unit balls cover $M$ as well. Verify that we
can take smooth $\varphi_i$'s if $M$ is equipped with a smooth
structure.

\smallskip

\noindent
{\bf Definition 1.14.} Let $M$ be a smooth manifold and let $\{U_i\}$
be a locally finite cover of $M$. A {\bf partition} of {\bf unity}
subordinate to the cover $\{U_i\}$ 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) $\sum_if_i=1$

\smallskip

\noindent
{\bf Remark.} Note that the sum $\sum_if_i$ is well defined because the
cover $U_i$ is locally finite.

\smallskip

\noindent
{\bf Problem 1.28.} Show that all derivatives of $e^{-\frac1{x^2}}$ at
$0$ vanish.

\smallskip

\noindent
{\bf Problem 1.29.} Define the following function $\lambda$ on
$\Bbb R^n$
$$\lambda(x):=\left\{\matrix e^{\frac1{|x|^2-1}}&\text{ if
}|x|<1\\0\hfill&\text{ if }|x|\ge1\endmatrix\right.$$
Show that $\lambda$ is smooth and that all its derivatives vanish at
the points of the unit sphere.

\smallskip

\noindent
{\bf Problem 1.30.} Let $\{U_i,\ \varphi_i:U_i\to\Bbb R^n\}$ be an
atlas on a smooth manifold $M$. Consider the following function
$\lambda_i:M\to[0,1]$
$$\lambda_i(m):=\left\{\matrix\lambda\big(\varphi_i(m)\big)&\text{ if
}m\in U_i\\0\hfill&\text{ if }m\notin U_i\endmatrix\right.$$
Show that $\lambda_i$ is smooth.

\smallskip

\noindent
{\bf Problem 1.31 (!).} Let $\{U_i,\ \varphi_i:U_i\to\Bbb R^n\}$ be a
locally finite atlas on a manifold $M$ such that $\varphi_i^{-1}(B_1)$
cover $M$ as well (such an atlas was constructed in Problem 1.27).
Consider the functions $\lambda_i$'s constructed in the above problem.
Show that $\sum_j\lambda_j$ vanishes nowhere and that the family of
functions $\Big\{f_i:=\frac{\lambda_i}{\sum_j\lambda_j}\Big\}$ provides
a partition of unity on $M$.

\smallskip

\noindent
{\bf Problem 1.32 (!).} Show that every manifold with enumerable base
of topology admits a partition of unity.

\bigskip

\noindent
{\bf1.5.~Whitney's theorem for compact manifolds}

\medskip

\noindent
{\bf Definition 1.15.} Define $\Bbb R^\infty$ as the union of all
$\Bbb R^i$ embedded one into the other by the maps
$(x_1,\dots,x_n)\allowmathbreak\hookrightarrow(x_1,\dots,x_n,0)$

\smallskip

\noindent
{\bf Problem 1.33 (*).} Show that $\Bbb R^\infty$ is not locally
compact.

\smallskip

\noindent
{\bf Problem 1.34.} Show that $\Bbb R^\infty$ is a topological abelian
group (i.e., equipped with a continuous commutative group operation,
namely, the addition).

\smallskip

\noindent
{\bf Problem 1.35 (*).} Consider the unit sphere
$\Bbb S^\infty\subset\Bbb R^\infty$. Show that it is contractible.

\smallskip

\noindent
{\bf Problem 1.36 (*).} Is the corresponding projective space
$\Bbb P_\Bbb R^\infty:=\Bbb S^\infty/\{\pm1\}$ contractible?

\smallskip

\noindent
{\bf Problem 1.37.} Let $M$ be a smooth manifold, let
$\{U_i,\ \varphi_i:U_i\to\Bbb R^n\}$ be a locally finite atlas, and let
$\{f_i\}$ be a partition of unity subordinated to the atlas and such
that $f_i=0$ outside some compact subset in $U_i$. Consider the
following map $\Psi_i:M\to\Bbb R^{n+1}$
$$\Psi_i(m):=\left\{\matrix\big(f_i(m)\varphi_i(m),f_i(m)\big)&\text{
if }m\in U_i\\(0,\dots,0)\hfill&\text{ if }m\notin
U_i\endmatrix\right.$$

a. Show that $\Psi_i$ is injective on the set where $f_i\ne0$.

b. Suppose the the atlas $\{U_i\}$ is finite and contains $m$ charts.
Show that $\bigoplus_i\Psi_i$ defines a closed embedding of $M$ into
$\Bbb R^{(n+1)m}$.

c(*). Show that $\bigoplus_i\Psi_i$ defines a closed embedding of $M$
into $\Bbb R^\infty$ if the number of charts in the atlas $\{U_i\}$ is
infinite.

\smallskip

\noindent
{\bf Problem 1.38 (!).} Prove Whitney's theorem (for compact
manifolds) : every compact manifold admits a closed smooth embedding
into $\Bbb R^n$.

\smallskip

\noindent
{\bf Problem 1.39 (*).} Let $U\subset M$ be an open subset in a smooth
manifold. Suppose that $U$ is homeomorphic to $\Bbb R^n$ and let
$V\subset U$ correspond to the unit ball. Construct a smooth map from
$M$ onto the unit sphere $\Bbb S^n$ in $\Bbb R^{n+1}$ which is
injective on $V$ and maps into $(0,\dots,0,1)$ the complement to $U_i$.

\smallskip

\noindent
{\bf Problem 1.40 (**).} Show that the map constructed in the above
problem is necessarily surjective.

\enddocument
