Cosymplectic-Nijenhuis structures on Lie groupoids

Aïssa Wade
2016 Banach Center Publications  
This paper introduces cosymplectic-Nijenhuis structures on smooth manifolds and proposes alternative odd-dimensional counterparts of symplectic-Nijenhuis groupoids, called cosymplectic-Nijenhuis groupoids. We discuss the correspondence between cosymplectic groupoids and integrable coPoisson manifolds. Moreover, we investigate the integrability problem for coPoisson manifolds equipped with a compatible Nijenhuis operator. As a result, we obtain a one-to-one correspondence between
more » ... een cosymplectic-Nijenhuis groupoids and integrable coPoisson-Nijenhuis manifolds. Introduction. It is well known that there is a correspondence between contact structures and symplectic structures via the symplectization scheme. Furthermore, contact manifolds are often considered as odd-dimensional analogues of symplectic manifolds. An alternative odd-dimensional counterpart of a symplectic structure is the notion of a cosymplectic structure. Recall that a cosymplectic structure on a (2n + 1)-dimensional manifold M is defined by a pair (ω, η) consisting of a closed 2-form ω and a closed 1-form η such that ω n ∧ η is a volume element on M . Cosymplectic manifolds were initiated by Libermann in the late 1950's [24] . They were further studied by Lichnerowicz in his papers [26, 27] in which he called them canonical manifolds. In [31], Marle discussed Lie group actions on a cosymplectic manifold. The reader is referred to Blair's paper [5] for another concept of a cosymplectic structure which is different but related to Libermann's notion. Precisely, a cosymplectic manifold in the sense of Blair is a cosymplectic manifold in the sense of Libermann together with
doi:10.4064/bc110-0-19 fatcat:rahaflhpovdprmdb6fgkh45xta