Weak $(p,k)$-Dirac manifolds

In this paper, we introduce the notion of a weak $(p,k)$-Dirac structure in $TM\oplus \Lambda^pT^*M$, where $0\leq k \leq p-1$. The weak $(p,k)$-Lagrangian condition has more informations than the $(p,k)$-Lagrangian condition and contains the $(p,k)$-Lagrangian condition. The weak $(p,0)$-Dirac structures are exactly the higher Dirac structures of order p introduced by N. Martinez Alba and H. Bursztyn in [23] and [6], respectively. The regular weak $(p,p-1)$-Dirac structure together with $(p,p-1)$-Lagrangian subspace at each point $m\in M$ have the multisymplectic foliation. Finally, we introduce the notion of weak $(p,k)$-Dirac morphism. We give the condition that a weak $(p,k)$-Dirac manifold is also a weak $(p,k)$-Dirac manifold after pulling back.