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We study a variational problem on a smooth manifold with a decomposition ofthe tangent bundle into $k>2$ subbundles (distributions), namely, we considerthe integrated sum of their mixed scalar curvatures as a functional of adaptedpseudo-Riemannian metric (keeping the pairwise orthogonality of thedistributions) and contorsion tensor, defining a linear connection. Thisfunctional allows us to generalize the class of Einstein metrics in thefollowing sense: if all of the distributions are one-dimensional, then itcoincides with the geometrical part of the Einstein-Hilbert action restrictedto adapted metrics. We prove that metrics in pairs metric-contorsion criticalfor our functional make all of the distributions totally umbilical. We obtainexamples and obstructions to existence of those critical pairs in some specialcases: twisted products with statistical connections; semi-symmetricconnections and 3-Sasaki manifolds with metric-compatible connections.

Variational problem on a metric-affine almost product manifold