Tangent Dirac Structures and Poisson Dirac Submanifolds

The local equations that characterize the submanifolds N of a Dirac manifold M is an isotropic (coisotropic) submanifold of TM endowed with the tangent Dirac structure. In the Poisson case which is 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 this paper we have proved a theorem in the general Poisson case that the fixed point set MG has a natural induced Poisson structure that implies a Poisson-Dirac submanifolds, where G×M?M be a proper Poisson action. DOI: http://dx.doi.org/10.3329/dujs.v62i1.21955 Dhaka Univ. J. Sci. 62(1): 21-24, 2014 (January)

TOTALLY NON-COISOTROPIC DISPLACEMENT AND ITS APPLICATIONS TO HAMILTONIAN DYNAMICS

In this paper, we prove the Conley conjecture and the almost existence theorem in a neighborhood of a closed nowhere coisotropic submanifold under certain natural assumptions on the ambient symplectic manifold. Essential to the proofs is a displacement principle for such submanifolds. Namely, we show that a topologically displaceable nowhere coisotropic submanifold is also displaceable by a Hamiltonian diffeomorphism, partially extending the well-known non-Lagrangian displacement property.

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.