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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
