On solutions to equations with partial Ricci curvature

We consider a problem of prescribing the partial Ricci curvature on a locally conformally flat manifold ( M n , g ) endowed with the complementary orthogonal distributions D 1 and D 2 . We provide conditions for symmetric ( 0 , 2 ) -tensors T of a simple form (defined on M ) to admit metrics g , conformal to g , 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 D i . 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.