We introduce the concept of monotone point-to-set field in Riemannian manifold and give a characterization, that make clear in this definition the occult geometric meaning. We will show that the sub-differential operator of a Riemannian convex function is a monotone point-to-set field. The concept of directional derivative, which appears already in other publications, plays an important role in the proof of the result above. We study some of its properties, in particular, we obtain the chain rule, which is fundamental in our work. Some topological consequences of the existence of strictly monotone point-to-set fields are presented.