The main goal of this present paper is to bring the results proved by Barbosa, Kenmotsu and Oshikiri (1991) and its ideas to a perspective where the Ricci curvature is bounded from below. For instance, for a foliation by CMC hypersurfaces on a compact (without boundary) Riemannian manifold $M^{ n+1}$ with Ricci curvature bounded from below by $-nK_0\leq 0$ and such that the mean curvature $H$ of the leaves of the foliation satisfies $|H|\geq \sqrt{ K_0}$, we prove that $|H|\equiv \sqrt{ K_0}$ and all the leaves are totally umbilical. This gives, in particular, a generalization for the result proved by Barbosa, Kenmotsu and Oshikiri (1991), where the above result was proved in the case $K_0=0$. We also obtain a proof of the following: for a foliation by CMC hypersurfaces on a compact (without boundary) Riemannian manifold $M$ with Ricci curvature bounded from below by $-nK_0\leq 0$, the mean curvature $H$ of the leaves of the foliation satisfies $|H|\leq \sqrt{ K_0}$. Furthermore, if the foliation contains a leaf $L$ whose absolute mean curvature is $|H_L|=\sqrt{ K_0}$, then either $K_0=0$ and all the leaves of $\mathfrak{ F}$ are totally geodesic, or $K_0>0$ and there is a totally umbilical leaf.
Categories: math.DG