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Step 1:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 $U/G$.}