Abstract:In this talk, I will address the following two problems:
Let $(M,\omega)$ be a symplectic manifold, $N\sub M$ a coisotropic submanifold, and $\phi:M\to M$ a map. A leaf-wise fixed point of $\phi$ is a point $x\in N$ such that $\phi(x)$ lies in the isotropic leaf through $x$.
Problem A: Find conditions on $(M,\om,N,\phi)$ under which there are leaf-wise fixed points and give a lower bound on their number.
To formulate the second problem, recall that a presymplectic structure on a manifold is a closed two-form of constant corank. We say that a presymplectic manifold $(M',\omega')$ embeds into a presymplectic manifold $(M,\omega)$ iff there exists an embedding $\phi:M'\to M$ such that $\phi^*\omega=\omega'$.
Problem B: Find conditions on $(M,M',\omega,\omega')$ under which $(M',\omega')$ does not embed into $(M,\omega)$.
I will define a Maslov map for coisotropic submanifolds, which naturally generalizes the Lagrangian Maslov index and twice the first Chern number. This gives rise to a notion of monotonicity of a coisotropic submanifold. I will give some solution to Problems A and B in the monotone case.