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A Riemannian manifold endowed with $k>2$ orthogonal complementarydistributions (called here a Riemannian almost $k$-product structure) appearsin such topics as multiply warped products, the webs composed of severalfoliations, and proper Dupin hypersurfaces of real space-forms. In the paper,we consider the mixed scalar curvature of such structure for $k>2$, deriveEuler-Lagrange equations for the Einstein-Hilbert type action with respect toadapted variations of metric, and present them in a nice form of Einsteinequation.

The Einstein-Hilbert type action on almost $k$-product manifolds