The Einstein-Hilbert type action on foliations

The mixed gravitational field equations have been recently introduced for codimension one foliated spacetimes. These Euler-Lagrange equations for the total mixed scalar curvature (as analog of Einstein-Hilbert action) involve a new kind of Ricci curvature. In the work, based on variation formulas for the quantities of extrinsic geometry, we derive Euler-Lagrange equations of the action for arbitrary codimension foliations, in fact, for a closed Riemannian almost-product manifold and adapted variations of metric (i.e., preserving orthogonality of the distributions). Examples of critical metrics of the action are found among twisted products, isoparametric foliations and K-contact metrics.