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

TL;DR

This paper proves that connected, complete submanifolds of complex space forms with geodesics confined to 1-dimensional complex submanifolds are necessarily totally geodesic and are either real or complex space forms.

Contribution

It establishes a classification result linking geodesic behavior to the global geometry of submanifolds in complex space forms.

Findings

M is totally geodesic in N

M is a real or complex space form

Geodesic confinement implies strong geometric restrictions

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.