Tangent Dirac structures and submanifolds by Izu Vaisman

We write down the local equations that characterize the sub-manifolds N of a Dirac manifold M which have a normal bundle that is either a coisotropic or an isotropic submanifold of T M endowed with the tangent Dirac structure. In the Poisson case, these formulas prove again a result of Xu: the submanifold N has a normal bundle which is a coisotropic submanifold of T M with the tangent Poisson structure iff N is a Dirac submanifold. In the presymplectic case it is the isotropy of the normal bundle which characterizes the corresponding notion of a Dirac submanifold. On the way, we give a simple definition of the tangent Dirac structure, we make new remarks about it, and we establish characteristic, local formulas for various interesting classes of submanifolds of a Dirac manifold.