Orthogonal foliations on riemannian manifolds

In this work, we find an equation that relates the Ricci curvature of a riemannian manifold \(M\) and the second fundamental forms of two orthogonal foliations of complementary dimensions, \(\mathcal{F}\) and \(\mathcal{F}^{\bot}\), defined on \(M\). Using this equation, we show a sufficient condition for the manifold M to be locally a riemannian product of the leaves of \(\mathcal{F}\) and \(\mathcal{F}^{\bot}\), if one of the foliations is totally umbilical. We also prove an integral formula for such foliations.