The paper introduces the notion of Riemannian submersion for the modeling and control of certain types of redundant robotic chains. In the robotics literature, the redundant case is normally treated only in numerical terms, as the need to resort to pseudoinversion techniques is usually considered a barrier to the use of analytic or geometric methods. Using Riemannian submersions, however, we can single out a particular type of inverse, the horizontal lift, with distinguished properties with respect to the infinitely many possible others. Quite remarkably, for a wide class of robotic chains, characterized by the vanishing of the curvature tensor, such horizontal lift coincides with the curve obtained from the Moore-Penrose pseudoinverse of the Jacobian.