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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:\ }} \newcommand{\pstep}{% {\bf \green Proof. Step 1:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Complex geometry\\[15mm] \small lecture 3: Frobenius theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE, room 306, 16:20, \\[2mm] September 30, 2020 } \end{center} \newpage {\bf \blue Distributions (reminder)} \definition {\bf \blue Distribution} on a manifold is a sub-bundle $B\subset TM$ \remark Let $\Pi:\; TM \arrow TM/ B$ be the projection, and $x, y \in B$ some vector fields. Then $[fx, y]= f[x,y] - D_y (f) x$. This implies that {\bf \purple $\Pi([x,y])$ is $C^\infty(M)$-linear as a function of $x$ and $y$.} \definition The map $[B,B]\arrow TM/B$ we have constructed is called {\bf \blue Frobenius bracket} (or {\bf \blue Frobenius form}); it is a skew-symmetric $C^\infty(M)$-linear form on $B$ with values in $TM/B$. \definition A distribution is called {\bf \blue integrable}, or {\bf \blue holonomic}, or {\bf \blue involutive}, if its Frobenius form vanishes. \newpage {\bf \blue Smooth submersions} \definition Let $\pi:\; M \arrow M'$ be a smooth map of manifolds. This map is called {\bf \blue submersion} if at each point of $M$ the differential $D\pi$ is surjective, and {\bf \blue immersion} if it is injective. \claim Let $\pi:\; M \arrow M'$ be a submersion. Then each $m\in M$ has a neighbourhood $U\cong V\times W$, where $V, W$ are smooth and {\bf \red $\pi \restrict U$ is a projection of $V\times W=U\subset M$ to $W\subset M'$ along $V$.} \exercise {\bf \purple Deduce this result from the inverse function theorem.} \exercise {\bf \blue (``Ehresmann's fibration theorem'')}\\ Let $\pi:\; M \arrow M'$ be a smooth submersion of compact manifolds. Prove that $\pi$ is a locally trivial fibration. \definition {\bf \blue Vertical tangent space} $T_\pi M\subset TM$ of a submersion $\pi:\; M \arrow M'$ is the kernel of $D\pi$. \claim Let $\pi:\; M \arrow M'$ be a submersion and $T_\pi M\subset TM$ the vertical tangent space. {\bf \red Then $T_\pi M$ is an involutive subbundle.} \proof $D_\pi([X, Y])= [D_\pi(X), D_\pi(Y)]=0$ for any $X, Y \in \ker D_\pi$. \endproof \newpage {\bf \blue Frobenius theorem (statement)} {\bf \green Frobenius Theorem:} Let $B\subset TM$ be a sub-bundle. Then $B$ is involutive if and only if each point $x\in M$ has a neighbourhood $U\ni x$ and {\bf \red a smooth submersion $U\stackrel \pi \arrow V$ such that $B$ is its vertical tangent space: $B= T_\pi M$.} \remark The implication {\bf \blue ``$B=T_\pi M$'' $\Rightarrow$ ``Frobenius form vanishes''} was proven above. \definition The fibers of $\pi$ are called {\bf \blue leaves}, or {\bf \blue integral submanifolds} of the distribution $B$. Globally on $M$, {\bf \blue a leaf of $B$} is a maximal connected manifold $Z\hookrightarrow M$ which is immersed to $M$ and tangent to $B$ at each point. A distribution for which Frobenius theorem holds is called {\bf \blue integrable}. If $B$ is integrable, the set of its leaves is called {\bf \blue a foliation}. The leaves are manifolds which are immersed to $M$, but not necessarily closed. \newpage {\bf \blue Frobenius theorem: existence of integral submanifolds} \remark To prove the Frobenius theorem for $B\subset TM$, {\bf \purple it suffices to show that each point is contained in an interal submanifold.} In this case, the smooth submersion $U\stackrel \pi \arrow V$ is a projection to the leaf space of the distribution. \remark {\bf \red When $B$ is 1-dimensional} (in this case one says that {\bf \blue $B$ has rank 1}, denoted $\rk B=1$), {\bf \red Frobenius theorem follows from existence of the diffeomorphism flow associated with a vector field.} Indeed, locally we may assume that $B$ admits a non-degenerate section $v$. Let $V_t:\; M \times \R \arrow M$ be the corresponding flow of diffeomorphisms. Then $Z_m:=V_t(\{m\}\times \R$ is tangent to $v$ everywhere, hence it is a 1-dimensional manifold immersed in $M$. Clearly, $Z_m$ is a leaf this distribution. {\bf \purple Since $B$ is a tangent to a foliation, it is integrable.} Further on we shall need the following exercise. \exercise Let $V_t= e^{tv}$ be a diffeomorphism flow on $M$, and $F\subset TM$ a vector bundle. Assume that $[v, F]\subset F$. {\bf \red Prove that $V_t$ preserves $F\subset TM$.} \newpage {\bf \blue Flow of diffeomorphisms} \definition Let $f:\; M \times [a,b]\arrow M$ be a smooth map such that for all $t\in [a,b]$ the restriction $f_t:= f\restrict{M\times\{t\}}:\; M \arrow M$ is a diffeomorphism. Then $f$ is called {\bf\blue a flow of diffeomorphisms}. \claim Let $V_t$ be a flow of diffeomorphisms, $f\in C^\infty M$, and $V^*_t(f)(x):= f(V_t(x))$. Consider the map $\frac d{dt}V_t\restrict{t=c}:\; C^\infty M \arrow C^\infty M$, with $\frac d{dt}V_t\restrict{t=c}(f)= (V_c^{-1})^*\frac{dV_t}{dt}\restrict{t=c}f$. {\bf \red Then $f\arrow (V_t^{-1})^*\frac d{dt}V_t^*f$ is a derivation} (that is, a vector field). \proof $\frac d{dt}V_t^*(fg)= V_t^*(f)\frac d{dt}V_t^* g+ \frac d{dt}V_t^* f V_t^*(g)$ by the Leignitz rule, giving \[ (V_t^{-1})^*\frac d{dt}V_t^*(fg)= f (V_t^{-1})^*\frac d{dt}V_t^*g + g (V_t^{-1})^*\frac d{dt}V_t^*f. \] \endproof\\ \definition The vector field $\frac d{dt}V_t\restrict{t=c}$ is called {\bf\blue the vector field tangent to a flow of diffeomorphisms $V_t$ at $t=c$.} \claim Let $V_t$ be a flow of diffeomorphisms and $X_t$ the corresponding vector field. {\bf \red Then for any $\eta\in \Lambda^* M$, one has $\frac d{dt}V_t^*(\eta)= \Lie_{X_t}(\eta)$.} \proof The operators $\frac d{dt}V_t^*$ and $\Lie_{X_t}$ are equal on functions, satisfy the Leibitz identity and commute with $d$. \endproof \newpage {\bf \blue Flow of diffeomorphisms obtained from vector fields} \exercise Let $M$ be a compact manifold, and $\Psi:\; C^\infty M \arrow C^\infty M$ is a ring automorphism. Prove that {\bf \purple $\Psi$ is induced by an action of a diffeomorphism of $M$.} \theorem Let $M$ be a compact manifold, and $X_t\in TM$ a family of vector fields smoothly depending on $t\in [0,a]$. {\bf \red Then there exists a unique diffeomorphism flow $V_t$, $t\in [0,a]$, such that $V_0=\Id$ and $\frac d{dt}V_t^*=X_t$.} \pstep Given $f\in C^\infty M$, we can solve an equation $\frac d{dt}W_t(f)=\Lie_{X_t}(f)$ (here $\Lie_{X_t}(f)$ denotes the derivative along the vector field). The solution $W_t(f)$ exists for all $t\in [0,x]$ and is unique by Peano theorem on existence and uniqueness of solutions of ODE. {\bf \green Step 2:} Since \[ \frac d{dt}W_t(fg)= \Lie_{X_t}(f) g + \Lie_{X_t}(g)f = \frac d{dt}(W_t(f)W_t(g)), \] $W_t$ is multiplicative. Also, it is invertible. Applying the previous exercise, we obtain that $W_t$ is a diffeomorphism. \endproof \remark If the vector field $X_t=X$ is independent from $t$, the corresponding diffeomorphism flow {\bf \blue is often denoted as $e^{tX}$.} \newpage {\bf \blue Distributions preserved by a vector field} \exercise Let $v, v'$ be commuting vector fields. {\bf \red Then the corresponding diffeomorphism flows $e^{tv}$ and $e^{tv'}$ commute.} \claim Let $V_t= e^{tv}$ be a diffeomorphism flow on $M$, and $F\subset TM$ a vector bundle. Assume that $[v_t, F]\subset F$. {\bf \red Then $V_t$ preserves $F\subset TM$.} \pstep Since a non-degenerate vector fields can be linearized, {\bf \red we can always assume that the vector field $v_t$ is a coordinate vector field, $v_t = x_1$.} Since the statement is local, we can always assume that $M$ is an open subset in $\R^n$ with coordinates $x_1, ..., x_n$. {\bf \green Step 2:} Let $R_1, ..., R_r$ be a basis in $F$. We write $R_i= \sum f_{ij} \frac{d}{dx_j}$; then $[v_t, R_i]= \sum \frac{df_{ij}}{dx_i} \frac{d}{dx_j}$. In these terms $[v_t, F]\subset F$ is written as $\frac{d R_i}{dx_1}= \sum a_{ij} R_j$, where $a_{ij}$ are appropriate functions on $M$. \newpage {\bf \blue Distributions preserved by a vector field (2)} \claim Let $V_t= e^{tv}$ be a diffeomorphism flow on $M$, and $F\subset TM$ a vector bundle. Assume that $[v_t, F]\subset F$. {\bf \red Then $V_t$ preserves $F\subset TM$.}\\ {\bf \green Step 2:} Let $R_1, ..., R_r$ be a basis in $F$. We write $R_i= \sum f_{ij} \frac{d}{dx_j}$; then $[v_t, R_i]= \sum \frac{df_{ij}}{dx_1} \frac{d}{dx_j}$. In these terms $[v_t, F]\subset F$ is written as $\frac{d R_i}{dx_1}= \sum a_{ij} R_j$, where $a_{ij}$ are appropriate functions on $M$.\\ {\bf \green Step 3:} Let $F_i(t):= V_t(R_i)\in TM$. By definition, $V_t=e^{t d/dx_1}$, hence $\frac {d}{dt} V_t(R_i)=[d/dx_1, V_t(R_i)]$. Then, $F_i(t)$ is a solution of a vector-valued differential equation \[ \frac {d}{dt} F_i(t)=[d/dx_1, F_i(t)], \ \ \ (*) \] with initial values $F_i(t)=R_i$. The solution can be found as $F_i(t) = \sum_{j=1}^r b_{ij}(t) R_j$ because \[ \left [d/dx_1, \sum_{j=1}^r b_{ij}(t) R_j\right] = \sum_{j=1}^r b_{ij}(t)\sum_{k=1}^r a_{jk} R_k + \frac{b_{ij}}{dx_1} R_j. \] Then $F_i(t) = \sum_{j=1}^r b_{ij}(t) R_j$ is a solution of (*) if for all $i, j=1, ..., r$, we have \[ \frac{db_{ij}(t)}{dt} = \frac{db_{ij}}{dx_1}+\sum_{i=1}^r \sum_{k=1}^rb_{ik} a_{kj}. \] This differential equation has a unique solution with a given initial value. \endproof \newpage {\bf \blue Basic sub-bundles (1)} \definition Let $B\subset TM$ be an involutive sub-bundle. A sub-bundle $F\subset TM$ is called {\bf \blue basic} for $B$ if $F\supset B$ and for all $b\in B, b'\in F$, one has $[b, b']\in F$. %\remark One should think of basic sub-bundles as of {\bf \purple sub-bundles %preserved by all diffeomorphisms obtaned from exponentiation %of a vector field $v\in B$.} \lemma Let $B\subset TM$ be an integrable distribution, $\pi:\; M \arrow M_1$ projection to the leaf space of $B$, and $F\supset B$ a sub-bundle of $TM$ containing $B$. Then the following conditions are equivalent: {\bf \red (a) $F$ is basic for $B$.\\ (b) There exists a sub-bundle $F_1\subset TM_1$ such that $\pi^{-1}(F_1) = F$.} \proof Next slide. \newpage {\bf \blue Basic sub-bundles (2)} \lemma Let $B\subset TM$ be an integrable distribution, $\pi:\; M \arrow M_1$ projection to the leaf space of $B$, and $F\supset B$ a sub-bundle of $TM$ containing $B$. Then the following conditions are equivalent: {\bf \red (a) $F$ is basic for $B$.\\ (b) There exists a sub-bundle $F_1\subset TM_1$ such that $\pi^{-1}F_1 = F$.} \pstep Consider coordinates $x_1, ..., x_n$ on $M$ such that $x_{k+1}=\pi^*(x_{k+1}', ..., x_n=\pi^*(x_n)$, where $x_i', i= k+1, k+2,..., n$ are coordinates on $M_1$, and $\frac {d}{dx_1}, ..., \frac {d}{dx_k}$ generate $B$. Locally such coordinates always exist, because $B$ is integrable. Denote by $G$ a subgroup of $\Diff(M)$ obtained by exponents of $\frac {d}{dx_1}, ..., \frac {d}{dx_k}$. Since $[B, F]\subset F$, the corresponding diffeomorphisms preserve $F$. {\bf \purple Therefore, $F$ is a $G$-invariant sub-bundle of $TM$.} {\bf \green Step 2:} {\bf \purple Any $G$-invariant sub-bundle $F\supset B$ is obtained as $\pi^{-1}(F_1)$ for some sub-bundle $F_1 \subset TM_1= M/G$.} Indeed, since the action of $G_1$ is free, the bundle $F$ is generated over $C^\infty M$ by $G$-invariant sections. However, any $G$-invariant bundle $F$ containing $B$ is generated by $G$-invariant sections, which can be lifted from $M/G$ {\bf \purple (check this)}. {\bf \green Step 3:} Conversely, if $F$ is lifted from $M_1=M/G$, it is $G$-invariant, hence $e^{tb}(b')\subset F$, and this gives $[b, b']\subset F$ {\bf \purple (check this)}. \endproof \newpage {\bf \blue Frobenius theorem (proof)} {\bf \green Frobenius Theorem:} Let $B\subset TM$ be a sub-bundle. Then $B$ is involutive if and only if each point $x\in M$ has a neighbourhood $U\ni x$ and {\bf \red a smooth submersion $U\stackrel \pi \arrow V$ such that $B$ is its vertical tangent space: $B= T_\pi M$.} \pstep Consider a rank 1 sub-bundle $B_1 \subset B$. Using the diffeomorphism flow as above, we prove that $B_1$ is integrable. Since $[B_1, B]\subset B$, the bundle $B$ is basic with respect to $B_1$. {\bf \purple Therefore, $B= \pi^{-1}(B')$ for some $B'\subset TM_1$, where $M_1$ is the leaf space of $B_1$.} {\bf \green Step 2:} Let $\pi:\; M \arrow M_1$ be the projection to the leaf space. Then $B= \pi^{-1}(B')$, where $\rk B'=\rk B-1$. Using induction in $\rk B$, we can assume that $B'$ is integrable. Let $\pi_0:\; M_1 \arrow M_0$ be the projection to the leaf space of $B'$, defined locally in $M$. {\bf \purple Then $\pi\circ\pi_0:\; M \arrow M_0$ is the projection to the leaf space of $B$.} \endproof \end{document}