Problem
Problem 6.141 Let $M$ be an $n$-dimensional $(n \geqslant 3)$, totally umbilical submanifold of a $2 m$-dimensional complex space form $(\widetilde{M}, g, J)$ of holomorphic sectional curvature $c \neq 0$. Prove that $M$ is one of the following submanifolds:
(i) A complex space form holomorphically immersed in $\tilde{M}$ as a totally geodesic submanifold.
(ii) A real space form (i.e. a not necessarily simply connected space of constant curvature) immersed in $\widetilde{M}$ as a totally real and totally geodesic submanifold.
(iii) A real space form immersed in $\widetilde{M}$ as a totally real submanifold with nonvanishing parallel mean curvature vector.
The relevant theory is developed, for instance, in Chen and Ogiue [7].