Weak $(p,k)$-Dirac manifolds

Abstract

In this paper, we introduce the notion of a weak (p, k)-Dirac structure in T M ⊕ Λ p T * M , where 0 ≤ k ≤ 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 ∈ 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 by pulling back.