Statistical structures, alpha-connections and Generalized Geometry

The main purpose of this paper is to describe how statistical structures fit perfectly into Generalized Geometry. Firstly, we will briefly present the properties of generalized pseudo-calibrated almost complex structures induced by statistical structures. Then we will characterize the integrability of generalized almost complex structures with respect to the bracket defined by the $\alpha$-connection, finding conditions under which the concept of integrability is $\alpha$-invariant. Finally, we consider a pair of generalized dual quasi-statistical connections $(\hat{\nabla},\hat{\nabla}^*)$ on the generalized tangent bundle $TM\oplus T^*M$ and we provide conditions for $TM\oplus T^*M$ with the $\alpha$-connections $(\hat{\nabla}^{(\alpha)},\hat{\nabla}^{(-\alpha)})$ induced by $(\hat{\nabla},\hat{\nabla}^*)$ to be conjugate Ricci-symmetric.