In this paper we classify all non-trivial Riemannian submersions with connected fibers from any of the simply connected, rank-one projective spaces. The result follows from results of Gromoll, Grove, Wilking, Becker, Casson, Gottlieb, Schultz, Ucci, and Wolf, together with results of the author.
"Metric Fibrations from Simply Connected Rank---One Projective Spaces." Tokyo J. Math. 35 (2) 505 - 512, December 2012. https://doi.org/10.3836/tjm/1358951335