  In the paper we prove integral formulae for a Riemannian manifold endowed
with $k>2$ orthogonal complementary distributions, which generalize well-known
formula for $k=2$ and give applications to splitting and isometric immersions
of Riemannian manifolds, in particular, multiply warped products, and to
hypersurfaces with $k>2$ distinct principal curvatures of constant
multiplicities.
