In this paper we classify real hypersurfaces all of whose geodesics orthogonal to the characteristic vector field are plane curves in complex projective or complex hyperbolic spaces.
"Real hypersurfaces some of whose geodesics are plane curves in nonflat complex space forms." Tohoku Math. J. (2) 57 (2) 223 - 230, 2005. https://doi.org/10.2748/tmj/1119888336