The Einstein–Hilbert type action on pseudo-Riemannian almost-product manifolds

We develop variation formulas for the quantities of extrinsic geometry of almost-product pseudo-Riemannian manifolds, and we consider the variations of metric preserving orthogonality of the distributions. These formulas are applied to study the Einstein–Hilbert type actions for the mixed scalar curvature and the extrinsic scalar curvature of a distribution. The Euler–Lagrange equations for these variations are derived in full generality and in several particular cases (foliations that are integrable plane fields, conformal submersions, etc.). The obtained Euler–Lagrange equations generalize the results for codimension-one foliations to the case of arbitrary codimension, and admit a number of solutions, e.g., twisted products and isoparametric foliations.