On solutions to equations with partial Ricci curvature

We consider a problem of prescribing the partial Ricci curvature on a locally conformally flat manifold (Mn,g)(Mn,g) endowed with the complementary orthogonal distributions D1D1 and D2D2. We provide conditions for symmetric (0,2)(0,2)-tensors TT of a simple form (defined on MM) to admit metrics g̃, conformal to gg, that solve the partial Ricci equations. The solutions are given explicitly. Using above solutions, we also give examples to the problem of prescribing the mixed scalar curvature related to DiDi. In aim to find “optimally placed” distributions, we calculate the variations of the total mixed scalar curvature (where again the partial Ricci curvature plays a key role), and give examples concerning minimization of a total energy and bending of a distribution.