Deformation of integral coisotropic submanifolds in symplectic manifolds

In this paper we prove the unobstructedness of the deformation of integral coisotropic submanifolds in symplectic manifolds, which can be viewed as a natural generalization of results of Weinstein [4] for Lagrangian submanifolds. Let (X,ω) be a symplectic manifold. A submanifold Y ⊂ (X,ω) is coisotropic if rank(ω|Y ) = 2 dim(Y ) − dimX . The closedness of ω implies that ker(ω|Y ) defines an integrable distribution on the coisotropic submanifold Y . The corresponding foliation F is called null foliation. Y is called integral if the leaves of the null foliation F are all closed and form a fibration π : Y → S. Denote Fs = π (s) for s ∈ S. Real hypersurfaces and Lagrangian submanifolds in (X,ω) are examples of coisotropic submanifolds. For a Lagrangian submanifold L ⊂ (X,ω), according to the work of Weinstein ([4]), there exists an open neighborhood U ⊂ (X,ω) of L, and a symplectic open embedding (U, ω) → (T L, ωT∗L) such that L ⊂ U is mapped to the zero section of T L, where ωT∗L is the canonical symplectic form on T L. The moduli space of Lagrangian sumanifolds in (X,ω) near L modulo Hamiltonian deformation can be canonically identified with H(L,R). (Y, ωY ) is called a pre-symplectic manifold if ωY is a closed 2-form on Y with constant rank. Gotay’s coisotropic neighborhood theorem [1] implies that there exists a symplectic neighborhood (U, ω) containing Y such that ω|Y = ωY . Moreover, for another such symplectic neighborhood (U , ω) containing Y , by shrinking U and U ′ suitably, there exists a symplectomorphism (U, ω) → (U , ω) that fixes Y . A coisotropic submanifold Y ⊂ (X,ω) naturally gives Partially supported by NSF Grant DMS-0104150.