  We study a variational problem on a smooth manifold with a decomposition of
the tangent bundle into $k>2$ subbundles (distributions), namely, we consider
the integrated sum of their mixed scalar curvatures as a functional of adapted
pseudo-Riemannian metric (keeping the pairwise orthogonality of the
distributions) and contorsion tensor, defining a linear connection. This
functional allows us to generalize the class of Einstein metrics in the
following sense: if all of the distributions are one-dimensional, then it
coincides with the geometrical part of the Einstein-Hilbert action restricted
to adapted metrics. We prove that metrics in pairs metric-contorsion critical
for our functional make all of the distributions totally umbilical. We obtain
examples and obstructions to existence of those critical pairs in some special
cases: twisted products with statistical connections; semi-symmetric
connections and 3-Sasaki manifolds with metric-compatible connections.
