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 ${\mathcal{F}}^{\mathrm{\perp }}$, 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 ${\mathcal{F}}^{\mathrm{\perp }}$, if one of the foliations is totally umbilical. We also prove an integral formula for such foliations.