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We develop variation formulas for the quantities of extrinsic geometry foradapted variations of metrics on almost-product (e.g. foliated) Riemannianmanifolds, and apply them to study the total mixed scalar curvature of adistribution -- analogue of the classical Einstein-Hilbert action. The mixedscalar curvature ${\rm S}_{\,\rm mix}$ is the averaged sectional curvature overall planes that contain vectors from both distributions of an almost-productstructure and the variations we consider preserve orthogonality of thedistributions. We derive the directional derivative $D J_{\,\rm mix}$ (of thetotal ${\rm S}_{\,\rm mix}$) for adapted variations of metrics on closedalmost-product manifolds and foliations of arbitrary dimension. The obtainedEuler-Lagrange equations are presented in two equiva\-lent forms: in terms ofextrinsic geometry and intrinsically using the partial Ricci tensor. Certainly,these mixed field equations admit amount of solutions (e.g., twisted products).

The mixed Einstein-Hilbert action and extrinsic geometry of foliated manifolds