TANGENT DIRAC STRUCTURES AND SUBMANIFOLDS

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 once again prove a result of Xu: the submanifold N has a normal bundle which is a coisotropic submanifold of TM with the tangent Poisson structure if and only if 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, make new remarks about it and establish its characteristic, local formulas for various interesting classes of submanifolds of a Dirac manifold.

Contact co-isotropic CR submanifolds of a pseudo-Sasakian manifold

It is proved that any co-isotropic submanifoldMof a pseudo-Sasakian manifoldM˜(U,ξ,η˜,g˜)is a CR submanifold (such submanfolds are called CICR submanifolds) with involutive vertical distributionν1. The leavesM1ofD1are isotropic andMisν1-totally geodesic. IfMis foliate, thenMis almost minimal. IfMis RicciD1-exterior recurrent, thenMreceives two contact Lagrangian foliations. The necessary and sufficient conditions forMto be totally minimal is thatMbe contactD1-exterior recurrent.