Reverse Ricci-Curvature Bounds for Riemannian Submersions and Riemannian Maps

Abstract

In this paper, we establish, for the first time, upper bounds of the Ricci--curvature for Riemannian submersions along the vertical distribution as well as along both the vertical and horizontal distributions. We derive their general forms and provide precise geometric characterisations of the equality cases. Furthermore, we obtain lower bounds of the Ricci--curvature for Riemannian maps, together with their general formulations and complete geometric characterisations of the equality cases. As applications, we apply these results to Riemannian submersions from real and complex space forms onto Riemannian manifolds, and to Riemannian maps from Riemannian manifolds into real and complex space forms.