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In analogy to the concept of a non-metric dual connection, which is essentialin defining statistical manifolds, we develop that of a torsion dualconnection. Consequently, we illustrate the geometrical meaning of such atorsion dual connection and show how the use of both connections preserves thecracking of parallelograms in spaces equipped with a connection and its torsiondual. The coefficients of such a torsion dual connection are essentiallycomputed by demanding a vanishing mutual torsion among the two connections. Forthis manifold we then prove two basic Theorems. In particular, if bothconnections are metric-compatible we show that there exists a specific $3$-formmeasuring how the connection and its torsion dual deviate away from theLevi-Civita one. Furthermore, we prove that for these torsion dual manifoldsflatness of one connection does not necessary impose flatness on the other butrather that the curvature tensor of the latter is given by a specificdivergence. Finally, we give a self-consistent definition of the mutualcurvature tensor of two connections and subsequently define the notion of acurvature dual connection.

On a Torsion/Curvature Analogue of Dual Connections and Statistical Manifolds