From dual connections to gravitational field equations -- the curvature and Einstein tensors of the α - connection of a quasi-statistical manifold

We present a detailed review of the mathematical foundations of the theory of the statistical and quasi-statistical manifolds, which recently have found many applications in general relativity, quantum mechanics, and mathematical statistics. In particular, we fully develop, in a rigorous and coherent way, the formulas and concepts necessary for the understanding of the mathematical basis of the statistical and quasi-statistical manifolds, including the properties of the dual connections and of the equiaffine connections. For each geometrical structure the explicit expressions of the curvatures and the Einstein tensors are explicitly obtained. As possible applications to the field of gravitational theories we explicitly compute the curvatures of a family of $\alpha$-connections {$\nabla^{(\alpha)}:=\frac{1+\alpha}{2}\nabla +\frac{1-\alpha}{2}\nabla ^{*}$, where $\nabla :=\nabla ^{(1)}$ and $\nabla ^{*}:=\nabla ^{(-1)}$ } are the dual connections of a quasi-statistical manifold $M$. The Einstein vacuum field equations are also written down, and the physical relevance of the obtained results for gravity and cosmology is briefly discussed.