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Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \setlength\paperheight {10in}% \setlength\paperwidth {13.5in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \newcommand{\exercise}{{\bf \green EXERCISE:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 \end{document}