α-connections in generalized geometry

We consider a family of α-connections defined by a pair of generalized dual quasistatistical connections (∇ˆ , ∇ˆ ∗) on the generalized tangent bundle (TM ⊕ T*M, hˇ) and determine their curvature, Ricci curvature and scalar curvature. Moreover, we provide the necessary and sufficient condition for ∇ˆ* to be an equiaffine connection and we prove that if h is symmetric and ∇h = 0, then (TM ⊕ T*M, h, ˇ ∇ˆ (α), ∇ˆ (−α)) is a conjugate Ricci-symmetric manifold. Also, we characterize the integrability of a generalized almost product, of a generalized almost complex and of a generalized metallic structure w.r.t. the bracket defined by the α-connection. Finally we study α-connections defined by the twin metric of a pseudo-Riemannian manifold, (M, g), with a non-degenerate g-symmetric (1, 1)-tensor field J such that d∇J = 0, where ∇ is the Levi-Civita connection of g.