Submanifolds of a complex space form, whose geodesics lie in 1 - dimensional complex submanifolds

Abstract

It is proved, that if M is a connected, complete submanifold of a complex space form N and each geodesic of M lies in an 1-dimensional totally geodesic complex submanifold of N, then M is totally geodesic in N and is a real space form or a complex space form.