\documentclass{slides} \usepackage{amssymb, amsmath, amscd, color, epsfig} %\usepackage[matrix,arrow]{xy} \newcommand{\green}{\color[rgb]{0,0.4,0}} \newcommand{\purple}{\color[rgb]{0.4,0,0.4}} \newcommand{\red}{\color[rgb]{0.7,0,0}} \newcommand{\blue}{\color{blue}} \def\eqref#1{(\ref{#1})} \newcommand{\goth}{\mathfrak} \newcommand{\g}{{\frak g}} \newcommand{\arrow}{{\:\longrightarrow\:}} \newcommand{\Z}{{\Bbb Z}} \newcommand{\C}{{\Bbb C}} \newcommand{\R}{{\Bbb R}} \newcommand{\Q}{{\Bbb Q}} \renewcommand{\H}{{\Bbb H}} \newcommand{\6}{\partial} \def\1{\sqrt{-1}\:} \newcommand{\restrict}[1]{{\left|_{{#1}}\right.}} \newcommand{\cntrct} % contraction with a vector field {\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}} \def\Bbb#1{\mathbb #1} \newcommand{\calo}{{\cal O}} \newcommand{\cac}{{\cal C}} % Correcting TeX... %\let\oldtilde=\tilde %\renewcommand{\tilde}{\widetilde} \renewcommand{\bar}{\overline} \renewcommand{\phi}{\varphi} \renewcommand{\epsilon}{\varepsilon} \renewcommand{\geq}{\geqslant} \renewcommand{\leq}{\leqslant} % Operatornames \newcommand{\Der}{\operatorname{Der}} \newcommand{\even}{{\rm even}} \newcommand{\ev}{{\rm even}} \newcommand{\odd}{{\rm odd}} \newcommand{\const}{{\it const}} \newcommand{\diam}{{\sf diam}} \newcommand{\fl}{{\rm fl}} \newcommand{\im}{\operatorname{im}} \newcommand{\End}{\operatorname{End}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Hol}{\operatorname{{\cal H}ol}} \newcommand{\Tot}{\operatorname{Tot}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\id}{\operatorname{\text{\sf id}}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Alt}{\operatorname{Alt}} \newcommand{\Iso}{\operatorname{Iso}} \newcommand{\Sec}{\operatorname{Sec}} \newcommand{\Can}{\operatorname{Can}} \newcommand{\Sing}{\operatorname{Sing}} \newcommand{\Spin}{\operatorname{Spin}} \newcommand{\codim}{\operatorname{codim}} \newcommand{\coim}{\operatorname{coim}} \newcommand{\coker}{\operatorname{coker}} \newcommand{\slope}{\operatorname{slope}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\Def}{\operatorname{Def}} \newcommand{\Lie}{\operatorname{Lie}} \newcommand{\Tw}{\operatorname{Tw}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\Sup}{\operatorname{Sup}} \newcommand{\Diff}{\operatorname{Diff}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\inbfpare}[1]{{% \mbox{\tt (}\hspace{-5pt}\mbox{\tt (} #1 % \mbox{\tt )}\hspace{-5pt}\mbox{\tt )}% }} \newcommand{\comment}[1]{{}} \def\blacksquare{\hbox{\vrule width 10pt height 10pt depth 0pt}} \def\endproof{\blacksquare} \def\shortdash{\mbox{\vrule width 4.5pt height 0.55ex depth -0.5ex}} \makeatletter %\@ifundefined{Bbb} % {\newcommand{\Bbb}[1]{{\mathbb #1}}}% %{}% {\edef\Bbb#1{{\Bbb #1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Pagestyle % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\ps@verbit}{% \renewcommand{\@oddhead}{% \scriptsize {\it \small Geometry of manifolds, lecture 4 \hfil \tiny M. 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 5: Derivations in rings} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf Math in Moscow and HSE \\[2mm] March 4, 2013 } \end{center} \newpage {\bf \blue Rings and derivations} \remark All rings in these handouts are assumed to be commutative and with unit. Algebras are associative, but not necessarily commutative (such as the matrix algebra). {\bf\blue Rings over a field $k$} are rings containing a field $k$. We assume that $k$ has characteristic 0. \definition Let $R$ be a ring over a field $k$. A $k$-linear map $D\; R \arrow R$ is called {\bf\blue a derivation} if it satisfies {\bf\blue the Leibnitz equation} $D(fg) = D(f) g + gD(f)$. The space of derivations is denoted as $\Der_k(R)$. \example $\frac{d}{dt}:\; \C[t]\arrow \C[t]$. \ \ $\frac{d}{dt}:\; C^\infty \R\arrow C^\infty\R$. \remark Any derivation $\delta\in\Der_k(R)$ vanishes on $k\subset R$. Indeed, $\delta(1) = \delta(1\cdot 1) = 2\delta(1)$. \claim Let $K$ be {\bf \blue a finite extension} of a field $k$, that is, a field containing $k$ and finite-dimensional as a $K$-linear space. {\bf \purple Then $\Der_k(K)=0$.} \proof Indeed, any $x\in K$ satisfies a non-trivial polynomial equation $P(x)=0$ with coefficients in $k$. Chose $P(t)$ of smallest degree possible. {\bf \purple For any $\delta\in\Der_k(R)$, we have $0=\delta(P(x))= P'(x) \delta(x)$,} and unless $\delta(x)=0$, one has $P'(x)=0$, giving a contradiction. \endproof \newpage {\bf \blue Modules over a ring} \definition Let $R$ be a ring over a field $k$. {\bf \blue An $R$-module} is a vector space $V$ over $k$, equipped with an algebra homomorphism $R\arrow \End(V)$, where $\End(V)$ denotes the endomorphism algebra of $V$, that is, the matrix algebra. \remark Let $R$ be a field. Then $R$-modules are the same as vector spaces over $R$. \definition Homomorphisms, isomorphisms, submodules, quotient modules, direct sums of modules are defined in the same way as for the vector spaces. A ring $R$ is itself an $R$-module. A direct sum of $n$ copies of $R$ is denoted $R^n$. Such $R$-module is called {\bf\blue a free $R$-module}. \example {\bf \purple $R$-submodules in $R$ are the same as ideals in $R$.} \definition {\bf\blue Finitely generated} $R$-module is a quotient module of $R^n$. \newpage {\bf \blue Noetherian rings} \definition {\bf\blue A Noetherian ring} is a ring $R$ with all ideals finitely generated as $R$-modules. \theorem Let $R$ be a Noetherian ring. {\bf \red Then any submodule of a finitely generated $R$-module is finitely generated.} {\bf \green Proof. Step 1:} Consider an exact sequence of $R$-modules\\ $0\arrow M_1 \arrow M \arrow M_2\arrow 0$. Then $M$ is called {\bf\blue an extension} of $M_1$ and $M_2$. An extension of finitely-generated modules is finitely generated. Indeed, take a finite set of generators in $M_2$, and let $\{\xi_i\}$ be preimages of these generators in $M$. Let $\{\zeta_j\}$ be a finite set of generators in $M_1\subset M$. {\bf \purple Then $\{\zeta_j +\xi_i\}$ generate $M$.} {\bf \green Step 2:} {\bf \blue A filtration} on a module $M$ as a sequence of submodules \\ $0=M_0\subset M_1 \subset ... \subset M_n =M$. From Step 1 and induction it follows that any {\bf \purple $M$ admitting a filtration with finitely-generated $M_i/M_{i-1}$ is also finitely-generated.} {\bf \green Step 3:} Let $M\subset R^n=W$. Consider a filtration $W_0\subset W_1 \subset ... \subset W_{n}=W$, with $W_i=R^i$, and let $M_i =M\cap W_i$. {\bf \purple Then $M_i/M_{i-1}$ is a submodule of $W_i/W_{i-1}=R$, hence finitely-generated.} \endproof \newpage {\bf \blue Ring of smooth functions} \theorem Let $R$ be a ring of smooth real functions on $\R^n$. {\bf \red Then $R$ is not Noetherian.} Moreover, the ideal $I$ of all functions with all derivatives vanishing at 0 is not finitely generated. {\bf \green Proof. Step 1:} If $I$ is generated by $f_1,..., f_n$, then for each $g\in I$, one can express $g$ as $g=\sum g_i f_i$. {\bf \purple Then \[ \lim\limits_{x\rightarrow 0}\sup \frac{g}{\sum f_i^2}\leq \sum |g_i|\frac{f_1}{\sum f_i^2}< \infty.\]} {\bf \green Step 2:} Let $x_i$ be coordinate functions. {\bf \purple The function $F:=\frac{\sum f_i^2}{\sum x_i^2}$ is smooth}, and all its derivatives vanish at 0, however, \[ \lim\limits_{x\rightarrow 0}\sup \frac{F}{\sum f_i^2}= \lim\limits_{x\rightarrow 0}\sup\frac{1}{\sum x_i^2}= \infty. \] \endproof \newpage {\bf \blue Derivations as an $R$-module} \remark Let $R$ be a ring over $k$. {\bf \blue The space $\Der_k(R)$ of derivations is also an $R$-module,} with multiplicative action of $R$ given by $rD(f) = r D(f)$. \claim Let $R= k[t_1, .., t_k]$ be a polynomial ring. {\bf \red Then $\Der_k(R)$ is a free $R$-module isomorphic to $R^n$,} with generators $\frac d {dt_1}, \frac d{dt_2}, ...,\frac d{dt_n}$. \proof Consider a map $\Der_k(R) \arrow R^n$, \[ D \arrow (D(t_1), D(t_2), ..., D(t_n)) \] It is surjective, because it maps each $\frac d {dt_i}$ to $(0, ..., 0, 1, 0, ..., 0)$, and injective, because each derivation which vanishes on $t_i$, vanishes on the whole polynomial ring. \endproof {\bf \green Now we prove a similar result for $C^\infty \R^n$.} \newpage {\bf \blue Hadamard's Lemma} \lemma {\bf \blue (Hadamard's Lemma)}\\ Let $f$ be a smooth function $f$ on $\R^n$, and $x_i$ the coordinate functions. {\bf \red Then $f(x)=f(0)+\sum_{i=1}^n x_i g_i(x)$, for some smooth $g_i\in C^\infty \R^n$.} {\bf \green Proof:} Let $t\in \R^n$. Consider a function $h(t)\in C^\infty \R^n$, $h(t)=f(tx)$. Then $\frac{dh}{dt}=\sum \frac{df(tx)}{dx_i}(tx)x_i$, giving \[ f(x)-f(0)= \int_0^1 \frac{dh}{dt} dt=\sum_i x_i \int_0^1\frac{df(tx)}{dx_i}(tx)dt. \] \endproof \corollary Let $\goth m_0$ be an ideal of all smooth functions on $\R^n$ vanishing in $0$. {\bf \purple Then $\goth m_0$ is generated by coordinate functions.} \endproof \corollary Let $f$ be a smooth function on $\R^n$ satisfying $f(x)=0$ and $f'(x)=0$. {\bf \red Then $f\in {\goth m}_x^2$.} \proof $f(x)=\sum_{i=1}^n x_i g_i(x)$, where all $g_i$ vanish in 0. \endproof \newpage {\bf \blue Derivations of $C^\infty \R^n$} \theorem Let $x_1, ..., x_n$ be coordinates on $\R^n$, $R=C^\infty\R^n$, and $\Der(R) \stackrel \Pi \arrow(C^\infty\R^n)^n$ map $D$ to $(D(x_1), D(x_2), ..., D(x_n))$. {\bf \red Then $D:\; \Der(C^\infty\R^n) \arrow R^n$ is an isomorphism.} {\bf \green Proof. Step 1:} Since $\Pi$ maps each $\frac d {dt_i}$ to $(0, ..., 0, 1, 0, ..., 0)$, it is {\bf \purple surjective.} {\bf \green Step 2:} Let ${\goth m}_0$ be an ideal of 0, and $D\subset \ker \Pi$. Then $\Pi(x_i)=0$, where $x_i$ are coordinate functions. By Hadamard's Lemma, $f(x)=f(0)+\sum_{i=1}^n x_i g_i(x)$, hence $D(f)=\sum_{i=1}^n x_i D(g_i)$. {\bf \purple Therefore, $D(f)$ lies in ${\goth m}_0$.} {\bf \green Step 3:} {\bf \purple Same argument proves that $D(f)$ vanishes everywhere,} for all $f\in C^\infty M$. \endproof \newpage {\bf \blue Sheaves} \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 The definition of a sheaf below is a more abstract version of the notion of ``sheaf of functions'' defined previously. \definition A {\bf\blue presheaf} on a topological space $M$ is a collection of vector spaces ${\cal F}(U)$, for each open subset $U\subset M$, together with {\bf \blue restriction maps} $R_{UW}{\cal F}(U)\arrow {\cal F}(W)$ defined for each $W\subset U$, such that for any three open sets $W\subset V\subset U$, $\Psi_{UW}=\Psi_{UV}\circ \Psi_{VW}$. Elements of ${\cal F}(U)$ are called {\bf \blue sections of ${\cal F}$ over $U$}, and restriction map often denoted $f\restrict W$ \definition A presheaf ${\cal F}$ is called {\bf\blue a sheaf} if for any open set $U$ and any cover $U=\bigcup U_I$ the following two conditions are satisfied.\\ \phantom{xu} 1. Let $f\in {\cal F}(U)$ be a section of ${\cal F}$ on $U$ such that its restriction to each $U_i$ vanishes. {\bf \purple Then $f=0$.} \\ \phantom{xu} 2. Let $f_i\in{\cal F}(U_i)$ be a family of sections 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} \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 Ringed spaces (reminder)} \definition {\bf\blue A sheaf of rings} is a sheaf ${\cal F}$ such that all the spaces ${\cal F}(U)$ are rings, and all restriction maps are ring homomorphisms. \definition {\bf \blue A sheaf of functions} is a subsheaf in a sheaf of all functions, closed under multiplication. {\bf \red For simplicity, I assume now that a sheaf of rings is a subsheaf in a sheaf of all functions}. \definition A {\bf\blue ringed space} $(M,{\cal F})$ is a topological space equipped with a sheaf of rings. 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. \newpage {\bf \blue Smooth manifold (reminder)} \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 Partition of unity (reminder)} \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) $\sum_if_i=1$ \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 $\{U_\alpha\}$.} \definition Let $U\subset V$ be open subsets in $M$. We write $U\Subset V$ if the closure of $U$ is contained in $V$. \definition Let $f\in {\cal F}(M)$ be a section of a sheaf ${\cal F}$ on $M$. A point $x\in M$ does not lie in the {\bf\blue support} $\Sup(f)$ of $f$ if $f\restrict U=0$ for some neighbourhood $U\ni x$. \remark {\bf \purple Support of a section is obviously closed.} \newpage {\bf \blue Smooth functions with prescribed support } \exercise Let $X,Y\subset M$ be non-intersecting closed subsets in a metric space. {\bf \purple Find non-intersecting open neighbourhoods $U_1\supset X$ and $U_2\supset U$.} \claim Let $U\Subset V$ -- open subsets in a smooth metrizable manifold. {\bf \red Then there exists a smooth function $\Phi_{U,V}\in C^\infty M$, supported on $V$, and equal to 1 on $U$.} {\bf \green Proof. Step 1:} Find non-intersecting open neighbourhoods $U_1$ and $U_2$ of $\bar U$ and $M \backslash V$, and choose a partition of unity $\{V_i, \phi_i\}$ subordinate to the cover $U_1, U_2, U_3=V\backslash \bar U$. {\bf \purple Then for each $i$, either $\Sup(\phi_i)\cap U_1=\emptyset$, or $\Sup(\phi_i)\cap U_2=\emptyset$.} {\bf \green Step 2:} Let $\Phi_{U,V}:= \sum_S \phi_i$, where the sum is taken over the set $S$ all $\phi_i$ satisfying $\Sup(\phi_i)\cap U_1\neq \emptyset$. {\bf \purple Since support of all such $\phi_i$ lies in $M \backslash U_2\subset V$, one has $\Sup(\Phi_{U,V})\subset V$.} Also, {\bf \purple for each $x\in U_1$, one has $\sum_{i\in S} \phi_i(x)=1$, hence $\Phi_{U,V}=1$ on $U_1\supset U$.} \endproof \newpage {\bf \blue Vector fields as derivations } \definition Let $M$ be a smooth manifold. A {\bf\blue vector field} on $M$ is an element in $\Der(C^\infty M)$. \example For $M=\R^n$, {\bf \red the space $\Der(C^\infty M)$ is a free module generated by $\frac{d}{dx_i}$, $i=1, ..., n$.} \remark We want to prove that vector fields form a sheaf. {\bf \purple However, it is not immediately clear how to restrict a vector field from $U$ to $W\subset U$.} \theorem Let $U\Subset V$ be open subset of a smooth metrizable manifold, and $D\in (C^\infty M)$ a derivation. Consider a smooth function $\Phi_{U,V}\in C^\infty M$ supported on $V$, and equal to 1 on $U$. Given $f\in C^\infty V$, define $D(f)\restrict U:=D(\Phi_{U,V}f)$. Choosing a cover of such $U_i$, we can glue together a section $D(f)$ of $C^\infty V$ from such $D(f)\restrict {U_i}$ {\bf \red This operation is independent of all choices we made and gives an element $D\restrict V\in \Der(V)$. Moreover, this restriction maps define a structure of a sheaf on $\Der(M)$.} {\bf \green Proof: next lecture.} The proof uses germs. %\newpage %{\bf \blue A sheaf of vector fields } \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 $W$ is mapped surjectively 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}