  We write down the local equations that characterize the submanifolds N of a
Dirac manifold M which have a normal bundle that is either a coisotropic or an
isotropic submanifold of TM 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 TM 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.
