Pseudo-Poisson Nijenhuis Manifolds

We introduce the notion of pseudo-Poisson Nijenhuis manifolds. These manifolds are generalizations of Poisson Nijenhuis manifolds defined by Magri and Morosi [13]. We show that there is a one-to-one correspondence between the pseudo-Poisson Nijenhuis manifolds and certain quasi-Lie bialgebroid structures on the tangent bundle as in the case of Poisson Nijenhuis manifolds by Kosmann-Schwarzbach [7]. For that reason, we expand the general theory of the compatibility of a 2-vector field and a (1, 1)-tensor. We also introduce pseudo-symplectic Nijenhuis structures, and investigate properties of them. In particular, we show that those structures induce twisted Poisson structures [18].