In the book "Introduction to symplectic topology" by McDuff and Salomon, it says that
Hypersurfaces of symplectic manifolds are coisotropic, so for $S \subset M$ it holds $\forall x \in S, (T_xS)^{\perp}\subset T_xS$ where $\perp$ is meant with respect to the symplectic form.
Now I am not sure if I really understand why.
As far as I see, in the book (also also in my lecture notes), we talked about the characteristic foliation:
First, we show that $L:=(TS)^{\perp}$ is a 1-dim. subbundle of $TS$.
Then, in the book it says that "It integrates to give a 1-dimensional foliation of $S$ called the characteristic foliation."
So to bring in some structure to my question:
1) I see it correctly that the fact that $L$ is a subbundle, so $(T_xS)^{\perp}$ is a 1-dim. subspace of $T_xS$ is already the proof for the claim that hypersurfaces are coisotropic (sorry if this is obvious)
2) What does the part "it integrates to" mean? How can $L$, a subbundle, be integrated?
Thanks in advance for any answer!
1) To show that $S$ is coisotropic, we have to check that $T_{x}S^{\perp}\subset T_{x}S$ for all $x\in S$. The observation $T_{x}S=(T_{x}S^{\perp})^{\perp}$ makes things easier; let's show that \begin{equation}\label{1}\tag{*} T_{x}S^{\perp}\subset(T_{x}S^{\perp})^{\perp}. \end{equation} If $v,w\in T_{x}S^{\perp}$, then there exists $\lambda\in\mathbb{R}$ such that $v=\lambda w$. This is true because $T_{x}S^{\perp}$ is one-dimensional. But then $$ \omega_{x}(v,w)=\lambda\omega_{x}(w,w)=0, $$ because $\omega_{x}$ is skew-symmetric. This proves the inclusion \eqref{1}.
2) A subbundle $D\subset TS$ is called integrable if each point $x\in S$ is contained in an immersed submanifold $L\subset S$ satisfying $T_{y}L=D_{y}$ for all $y\in S$. Such submanifolds are called integral with respect to $D$. So $D$ is integrable if each point of $S$ is contained in an integral submanifold of $D$.
Frobenius' theorem states that a distribution $D$ is integrable iff. it is involutive. It is a general fact that $TS^{\perp}$ is involutive whenever $S$ is coisotropic. In your specific example, things simplify even more: $TS^{\perp}$ has rank one, so it is automatically involutive.
The maximal integral submanifolds of an integrable distribution $D$ constitute a foliation. This is what is meant by "integrating a distribution to a foliation".