  We obtain integral formulas for a metric-affine space equipped with two
complementary orthogonal distributions. The integrand depends on the Ricci and
mixed scalar curvatures and invariants of the second fundamental forms and
integrability tensors of the distributions. The formulas under some conditions
yield splitting of manifolds (including submersions and twisted products) and
provide geometrical obstructions for existence of distributions and foliations
(or compact leaves of them).
