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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:\ }} \newcommand{\proof}{{\bf \green Proof:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Geometry of manifolds \\[15mm] \small lecture 3: partition of unity} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf Math in Moscow and HSE \\[2mm] February 18, 2013 } \end{center} \newpage {\bf \blue Sheaves of functions (reminder)} \definition {\bf\blue An open cover} of a topological space $X$ is a family of open sets $\{U_i\}$ such that $\bigcup_iU_i=X$. \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 $i$.} \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. \newpage {\bf \blue Ringed spaces (reminder)} 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 (reminder)} \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 (reminder)} \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 Locally finite covers (reminder)} \definition An open cover $\{U_\alpha\}$ of a topological space $M$ is called {\bf\blue locally finite} if every point in $M$ has a neighborhood intersecting 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 $M$.} \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 $V_\alpha\Subset U_\alpha$.} \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 $\{V_\alpha\}$, with $\phi_\alpha:\; V_\alpha \arrow \R^n$ 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 \newpage {\bf \blue Construction of a partition of unity} \claim {\bf \red There exists a smooth function $\nu:\; \R^n \arrow [0,1]$ which vanishes outside of a unit ball ${\Bbb B}\subset \R^n$ and is positive on ${\Bbb B}$.} {\bf \green Proof. Step 1:} There exists a {\bf \purple smooth function $a:\; \R \arrow \R$ which is positive on $\R^{>0}$ and 0 on $\R^{\leq 0}$.} Take $a=e^{-x^{-2}}$ on $\R^{>0}$ and $a=0$ on $\R^{\leq 0}$. {\bf \green Step 2:} There exists a {\bf \purple smooth function $b:\; \R \arrow \R$ vanishing outside of $[0,1]$ and positive on the open interval $]0,1[$.} Take $b(x)=a(x) a(1-x)$. {\bf \green Step 3:} There exists a {\bf \purple smooth function $c:\; \R \arrow \R$ equal to 0 on $]1,\infty[$ and equal to 1 on $]-\infty,0[$.} Take $c(x)= 1- \lambda^{-1}\int_{-\infty}^x b(x) dx$, where $\lambda:=\int_{-\infty}^\infty b(x) dx$. {\bf \green Step 4:} Now, let $\nu(z):= c(|z|^2)$. This function is smooth, vanishes on $|z|\geq1$, and positive on $|z|<1$. \endproof \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 $\{U_\alpha\}$.} \endproof \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 $\bar W_i:=\{z \ \ |\ \ \nu_i(z)=1\}= \{z \ \ |\ \ \mu_i(z)\geq \frac{1}{2m}\}.$ {\bf \purple 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 $\Phi_i(z):= \frac{(\nu_i\phi_i(z),1-\nu_i(z))}{|(\nu_i\phi_i(z),1-\nu_i(z))|}$ {\bf \purple is smooth and induces a diffeomorphism of $W_i$ and an open subset of $S^{n}\subset \R^{n+1}$.} {\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 {\bf \purple it is a homeomorphism to its image.} It is a smooth embedding, because its differential is injective. \endproof \newpage {\bf \blue Embedding to $\R^\infty$} \question {\bf \purple What if $M$ is non-compact?} \definition Define $\R^I_f$ as a direct sum of several copies of $\R$ indexed by a set $I$, that is, the set of points in a product where only finitely meny of coordinates can be non-zero. {\bf \blue The set $\R^I_f$ has metric \[ d((x_1, ..., x_n, ...), (y_1, ..., y_n, ...)):= \sqrt{|x_1-y_1|^2+|x_2-y_2|^2+...+|x_n-y_n|+...}. \]} {\bf \purple It is well-defined, because only finitely many of $x_i, y_i$ are non-zero.} \theorem Let $M$ be a compact smooth manifold, $\{V_i, \phi_i:\; V_i \arrow \R^n, i\in I\}$ be a locally finite atlas, and $\mu_i:\; M \arrow [0,1]$ a subordinate partition of unity. Define $\nu_i:=\alpha(\mu_i)$ and $\Phi_i$ as above, and let \[ \Psi:=\prod_I:\;\Phi_i:\; M \arrow \underbrace{S^n\times S^n \times ...\times S^n}_{\text{$I$ times}} \subset (\R^{n+1})^I \] be the corresponding product map. Then {\bf \red $\Psi$ is a homeomorphism to its image.} \newpage {\bf \blue Embedding to $\R^\infty$ (cont.)} \theorem Let $M$ be a compact smooth manifold, $\{V_i, \phi_i:\; V_i \arrow \R^n, i\in I\}$ be a locally finite atlas, and $\mu_i:\; M \arrow [0,1]$ a subordinate partition of unity. Define $\nu_i:=\alpha(\mu_i)$ and $\Phi_i$ as above, and let \[ \Psi:=\prod_I:\;\Phi_i:\; M \arrow \underbrace{S^n\times S^n \times ...\times S^n}_{\text{$I$ times}} \subset (\R^{n+1})^I \] be the corresponding product map. Then {\bf \red $\Psi$ is a homeomorphism to its image.} {\bf\green Proof. Step 1:} $\Psi$ is injective by construction. To prove that it is a homeomorphism, it suffices to check that an image of an open set $U$ is open in $\Psi(M)$, for each $U\subset W_i$, for some open cover $\{W_i\}$ {\bf\green Step 2:} However, the set $\Psi(W_i)$ is determined by $\nu_i(z)=1$, that is, by $\Phi_i(z)_{n+1}=1$, where $\Phi_i(z)_{n+1}$ is the last coordinate of $\Phi_i(z)$. Therefore, {\bf \purple $\Psi$ maps $W_i$ to an open subset of $\Psi(M)$.} {\bf\green Step 3:} Since $\Phi_i \restrict{\bar W_i}$ (restriction to a closure) is a continuous, bijective map from a compact, it's a homeomorphism. Therefore, {\bf \purple an image of any open subset $U\subset W_i$ is open in $\Psi(W_i)$, which is open in $\Psi(M)$ as follows from Step 2.} \endproof \newpage {\bf \blue Paracompactness} \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 $U_i$. \definition A cover of $M$ is called {\bf \blue locally finite} if any point $x\in M$ has a neighbourhood intersecting only finitely many of the elements of a cover. \definition A topological space is called {\bf \blue paracompact} if any cover admits a locally finite refinement. \exercise Let $M$ be a paracompact topological space, and $Z\subset M$ a closed supset. {\bf \purple Prove that $Z$ is paracompact.} \newpage {\bf \blue Paracompactness and partitions of unity} \theorem Let $M$ be a manifold. {\bf \red Then the following conditions are equivalent: \\ \phantom{huyak} (i) $M$ is metrizable \\ \phantom{huyak} (ii) $M$ admits a partition of unity \\ \phantom{huyak} (iii) $M$ is paracompact.} \proof Implication (iii) $\Rightarrow$ (ii) is proven above. Metrizability of $M$ follows from existence of partition of unity, because $M$ admits a homeomorphism to a subset of a metric space $\R^I$. This proves (ii) $\Rightarrow$ (i). It remains to prove that metrizability implies paracompactness. We don't need it, but I will give a short sketch of a proof. \newpage {\bf \blue Paracompactness and partitions of unity (cont.)} {\bf \green Step 1:} Consider the function $\rho:\; M \arrow \R^{\geq 0}$ mapping $x\in M$ to a supremum of all $r$ such that the open ball $B_r(x)$ is contained in one of $U_i$, and its closure is compact. It is easy to check that {\bf \purple $\rho$ is continuous,} and, indeed, 1-Lipschitz {\bf \red (prove it).} {\bf \green Step 2:} Now we can replace $\{U_i\}$ by a cover $\{B_{\rho(x)}(x)\ \ |\ \ x\in M\}$, which is its refinement. {\bf \green Step 3:} Take a maximal subset $Z\subset M$ such that for each distinct $x,y\in Z$, one has $d(x,y) \geq 1/8\rho(x)$. Such a subset exists by Zorn's Lemma. Since $\rho$ is 1-Lipschitz, $\{W_i\} := \{B_{1/2\rho(x)}(x)\ \ |\ \ x\in Z\}$ is also a cover of $M$. It is a refinement of $\{U_i\}$, as follows from Step 2. {\bf \green Step 4:} Now, each $W_i=B_{1/2\rho(z_i)}{(z_i)}$ intersects only those $W_j=B_{1/2\rho(z_j)}{(z_j)}$ for which $d(z_i, z_j)\geq 1/8\rho(z_i)$, and there are only finitely many of them, by compactness of $\bar W_i$. \endproof \newpage {\bf \blue Measure 0 subsets and Sard's theorem } \definition A subset $Z\subset\R^n$ has {\bf \blue measure zero} if, for every $\varepsilon>0$, there exists a countable cover of $Z$ by open balls $U_i$ such that $\sum_i\Vol U_i<\varepsilon$. \definition A subset $Z\subset M$ of a manifold $M$ has {\bf \blue measure 0} if intersection of $M$ with each chart $U_i\hookrightarrow \R^n$ has measure 0. {\bf \green Properties of measure 0 subsets}.\\ \phantom{huy} {\bf \red A countable union of measure 0 subsets has measure 0.} \phantom{huy} A measure 0 subset $Z\subset M$ is {\bf \blue nowhere dense}, that is, $(M \backslash Z)\cap U \neq \emptyset$ for any non-empty open subset $U\subset M$. \theorem {\bf \blue (a special case of Sard's Lemma)} Let $f:\; M\arrow N$ be a smooth map of manifolds, $\dim M < \dim N$. {\bf \red Then $f(M)$ has measure zero in $N$.} {\bf \green Its proof will be given in the next lecture} (if needed). \newpage {\bf \blue Whitney's theorem (with a bound on dimension): strategy of the proof } \theorem Let $M$ be a smooth $n$-manifold. {\bf \red Then $M$ admits a closed embedding to $\R^{2n+2}$.} {\bf \green Strategy of the proof:}\\ \phantom{huy} 1. $M$ is embedded to $\R^\infty$.\\ \phantom{huy} 2. We find a linear projection $\R^\infty\stackrel \pi \arrow \R^{2n+2}$ such that $\pi\restrict M$ is a closed embedding of manifolds. \lemma Let $M\subset \R^I$ be a subset, and $\pi:\; \R^I \arrow \R^J$ a linear projection. Consider the set $W$ of all vectors $\R(x-y)$, where $x,y \in M$ are distinct points. {\bf\purple Then $\pi\restrict M$ is injective if and only if $\ker \pi\cap W=0$.} \proof $\pi\restrict M$ is not injective if and only if $\pi(x)=\pi(y)$, which is equivalent to $\pi(x-y)=0$. \endproof \newpage {\bf \blue Whitney's theorem: injectivity of projections } \remark Let $M\subset \R^I$ be a submanifold, and $W\subset \R^I$ the set of all vectors $\R(x-y)$, where $x,y \in M$ are distinct points. {\bf \red Then $W$ is an image of a $2m+1$-dimensional manifold}, hence (by Sard's Lemma) {\bf \purple for any projection of $\R^I$ to a $(2m+2)$-dimensional space, image of $W$ has measure 0.} \corollary Let $M\subset \R^I$ be an $m$-dimensional submanifold, and $S\subset \R^I$ a maximal linear subspace not intersecting $W$. {\bf \purple Then the projection of $W$ to $\R^I/S$ is surjective.} \proof Suppose it's not surjective: $v\notin S$. Then $S \oplus \R v$ satisfies assumptions of lemma, hence $M \arrow \R^I/(S+ \R v)$ is also injective. \endproof \theorem Let $M$ be a smooth $n$-manifold, $M\hookrightarrow \R^I$ an embedding constructed earlier. {\bf \red Then there exists a projection $\pi:\; \R^I\arrow \R^{2n+2}$ which is injective on $M$.} \proof Let $S$ be the maximal linear subspace such that the restriction of $\pi:\; \R^I\arrow \R^I/S$ to $M$ is injective. Then the $2m+1$-dimensional manifold $W$ surjects to $\R^I/S$, hence $\dim \R^i/S\leq 2m+1$ by Sard's lemma. \endproof \newpage {\bf \blue Tangent space to an embedded manifold } \definition Let $M\hookrightarrow\Bbb R^n$ be a smooth $m$-submanifold. The {\bf\blue tangent plane} at $p\in M$ is the plane in $\Bbb R^n$ tangent to $M$ (i.e, the plane lying in the image of the differential given in local coordinates). A {\bf\blue tangent vector} is an arbitrary vector in this plane with the origin at $p$. The space of all tangent vectors at $p$ is denoted by $T_pM$. Given a metric on $\Bbb R^n$, we can define the space of {\bf\blue unit tangent vectors} $\Bbb S^{m-1}M$ as the set of all pairs $(p,v)$, where $p\in M$, $v\in T_pM$, and $|v|=1$. \remark $\Bbb S^{m-1}M$ is a smooth manifold, projected to $M$ with fibers isomorphic to $m-1$-spheres, hence {\bf \purple $\Bbb S^{m-1}M$ is $(2m-1)$-dimensional.} \lemma Let $M\subset \R^I$ be a subset, and $\pi:\; \R^I \arrow \R^J$ a linear projection. Consider the set $W'$ of all vectors $\R t$, where $t\in T_xM$ {\bf\purple Then the differential $D\pi\restrict M$ is injective if and only if $\ker \pi\cap W'=0$.} \endproof Now the above argument is repeated: we take a maximal space $S\supset \R^I$ such that the restriction of $\pi:\; \R^I\arrow \R^I/S$ to $M$ is injective and has injective differential, and the projection of $W\cup W'$ to $\R^I/S$ has to be surjective. However, $W'$ is an image of an $2m$-dimensional manifold ${\Bbb S}^{m-1}M\times \R$, hence {\bf \red the projection of $W\cup W'$ to $\R^I/S$ can be surjective only if $\dim \R^I/S\leq 2m +2$.} This proves Whitney's theorem. \end{document}