The Mixed Scalar Curvature of Almost-Product Metric-Affine Manifolds

We study two Einstein–Hilbert type actions on an almost-product metric-affine manifold, considered as functionals of the contorsion tensor. The first one is the total mixed scalar curvature of the linear connection, and the second one is based on a new type of curvature, recently introduced by B. Opozda for statistical structures. We deduce Euler–Lagrange equations of the actions and examine critical contorsion tensors associated with general and distinguished classes of connections, e.g. metric, statistical and adapted. The existence of such critical tensors depends on simple geometric properties of the almost-product structure, expressed only in terms of the Levi-Civita connection.