Variational problem on a metric-affine almost product manifold

Abstract

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.