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          \scriptsize {\it \small Symplectic geometry, lecture 12 \hfil
  \tiny M. Verbitsky }}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Symplectic geometry\\[15mm]
\small lecture 12: Foliations and holonomy}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf HSE, room 306, 16:20,
\\[2mm] 
October 13, 2021
}
\end{center}

\newpage

{\bf \blue Frobenius bracket}

\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 Frobenius theorem and foliations}


{\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: $B= T_\pi M$.}

\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 Holonomy of a foliation with compact leaves}

\definition
Let ${\cal F}$ be a smooth foliation on $M$.
Suppose that all leaves of ${\cal F}$ are compact
(in this case ${\cal F}$ is called {\bf\blue a foliation
with compact leaves}). 


Let $F\subset M$ be a compact leaf of ${\cal F}$, and
$U$ its tubular neighbourhood. Denote by $\pi$ a
smooth retraction of $U$ to $F$. For $U$ sufficiently
small, we may assume that $\pi$ is locally a diffeomorphism
on each leaf of ${\cal F}$. Then $\pi$ restricted
to a compact leaf $F_1 \subset F$ is a covering.
In particular, every path $\gamma \subset F$
can be lifted to a covering $\gamma_1 \in F_1$.
Let $S:= \pi^{-1}(x)$, where $x$ is the 
starting point of a loop $\gamma:\; [0,1] \arrow F$.
Then $\gamma_1$ is uniquely determined by
$\gamma_1(0)$ and gives a $H_\gamma:\; S \arrow S$
mapping $\gamma_1(0)$ to $\gamma_1(1)$.
{\bf \red This construction defines a group homomorphism
$\pi_1(F) \arrow \Diff_x(S_x)$,} where $S_x$
is a germ of $S$ in $x$, and $\Diff_x(S_x)$
denotes the group of diffeomorphisms of this 
germ.

\definition {\bf \blue (Ehresmann)}\\
The homomorphism $\pi_1(F) \arrow \Diff_x(S_x)$
is called {\bf \blue the holonomy of the foliation
${\cal F}$ in $F$.}

\remark Holonomy is well defined for any 
leaf of a foliation, {\bf \red compactness of its leaves
is not necessary. }
Moreover, a germ of a foliation
in a neighbourhood of a closed leaf {\bf \red is uniquely
(up to a diffeo) determined by its holonomy.}


\newpage

{\bf \blue Holonomy of a foliation with compact leaves}

\remark
Holonomy of a foliation with compact leaves 
is finite in dimension 3, by a theorem of
D. B. A. Epstein. However, in dimension 5
D. Sullivan {\bf \red produced an $S^1$-foliation
on $S^5$ with infinite holonomy.}

\exercise
Let $G$ be a compact Lie group acting on
a manifold $M$. Prove that
{\bf \purple all orbits have dimension $\dim G$
if and only if for some basis
$g_1, ... g_n \in \Lie G$
the corresponding vector fields 
on $M$ are linearly independent everywhere.}

\definition
In this case, the action of $G$ on $M$
is called {\bf\blue locally free.}

\theorem
Let $G$ be a compact Lie group which 
locally freely acts on a manifold $M$,
and ${\cal F}$ the corresponding foliation,
with its leaves being the orbits of 
$G$. {\bf \red Then the holonomy of ${\cal F}$
is finite.} Moreover, the leaf space
$M/G$ locally in a neighbourhood of 
$[F]$ {\bf \red is homeomorphic to 
$\R^n/\Gamma$, where $\Gamma$ is the 
holonomy of ${\cal F}$ in $F$. }

\newpage

{\bf \blue Holonomy of a foliation with transitive group
action on its leaves }


\theorem
Let $G$ be a compact Lie group with 
locally freely acts on a manifold $M$,
and ${\cal F}$ the corresponding foliation,
with its leaves being the orbits of 
$G$. {\bf \red Then the holonomy of ${\cal F}$
is finite.} Moreover, the leaf space
$M/G$ locally in a neighbourhood of 
$[F]$ {\bf \red is homeomorphic to 
$\R^n/\Gamma$, where $\Gamma$ is the 
holonomy of ${\cal F}$ in $F$. }

\pstep
Choose a $G$-invariant Riemannian metric on $M$
by taking any Riemannian metric and averaging it with $G$.
Then for any $r\in \R^{>0}$,  an $r$-neighbourhood
$U$ of a leaf $F$ is $G$-invariant. Therefore,
{\bf \purple $U$ contains the whole leaf $F_1$ of ${\cal F}$
if it contains a point of $F_1$.}

Let $\pi:\; U \arrow F$ a smooth retraction,
$S:= \pi^{-1}(x)$, and $\Gamma$ the holonomy of 
${\cal F}$ in $F$. Consider a leaf $F_1$ of ${\cal F}$
passing through $x_1 \in S$. {\bf \purple Then 
$F_1 \cap S= \Gamma\cdot x_1$, hence 
$U/G= S/\Gamma$.} We proved the second
claim of the theorem.


{\bf \green Step 2:} 
To see that $\Gamma$ is positive, we choose
$\sigma$ inverse to the Riemannian geodesic (exponential)
map in the direction orthogonal to $F$. 
Consider the map $\mu:\; S\times G\arrow U$ mapping $(s, g)$ 
to $g(s)$. This map is by construction surjective
and each point $x_1\in S\cap F_1$ has precisely
$\Gamma_{F_1}$ preimages. Let $\Gamma_F$ be the
subgroup of $G$ fixing $x\in F$. Then
$\mu(\Gamma_F)$ maps $S\subset U$ to itself, and
this action coincides with $\Gamma$.
Since $G$ is compact, $\Gamma_F$ is 
finite.
\endproof




\end{document}



{\bf \green Step 2:}
A leaf $F\in M/{\cal F}$ 
is called {\bf\blue submersive} if 
$\pi:\; M \arrow M/{\cal F}$ is a locally trivial fibration
in a neighbourhood of $F$.
Clearly, {\bf \purple a leaf $F$ is submersive
if and only if its holonomy is trivial.}
{\bf \red We prove that the set of submersive leaves is open and 
dense in $M/G$. } 

{\bf \green Step 3:} For any closed leaf
$F'$ in $U$, the set $F' \cap S$ can be identified
with the holonomy orbit of a point in $F'\cap S$.
{\bf \purple The number of intersections with $S$ is semicontinuous
as a function on $M/G$,} in the topology on
the orbit space considered as a set of closed subsets in $M$.

Indeed,  if a  leaf $F_\infty$ is a Hausdorff limit of a sequence
of leaves $F_1, F_2, ...$,
then $F_\infty\cap S$ is the Hausdorff limit
of $F_i\cap S$, hence the number of points
$\myhash(F_\infty\cap S)$ in
$F_\infty\cap S$ is bounded by 
$\overline\lim_i \myhash(F_i\cap S)$.


{\bf \green Step 4:}
{\bf \purple Submersive leaves
are the leaves where this intersection number is 
a local maximum.} Indeed, 
in a neighbourhood $U_1$
of a submersive leaf $F'$, the set $S$ intersects it only once,
because $\pi$ is locally trivial and $S\cap U_1$ is
a section of $\pi$. Let $s_1, ..., s_k$ be the intersection
points of $F'$ and $S$, and $U_1, ..., U_k$
sufficiently small neighbourhoods of these points.
Any submersive leaf close enough to $F'$ intersects 
each of $U_i$ exactly once, hence {\bf \purple the number
$\myhash(F'\cap S)$ is locally constant
(and, by semicontinuity, maximal) on
the set of submersive leaves.}
By semicontinuity, {\bf \red the set of submersive
leaves is open in $U/G$.}
