It is known that the connected totally geodesic complex submanifolds of a projective space ${\rm P}V$ equipped with a Fubini-Study metric are precisely the projective subspaces ${\rm P}Z$, where $Z \subseteq V$ is a complex subspace. Direct implication is obvious, and if for the converse we assume that $N\subseteq {\rm P}V$ satisfies the assumptions, we fix $L \in N$ and a basis $H_1,\ldots, H_k$ for $T_LN \subseteq T_L({\rm P}V)\cong {\rm Hom}(L,L^\perp)$, and let $Z = L \oplus \bigoplus_{i=1}^k H_i[L]$.
For Grassmannians, it is again easy to see that every ${\rm Gr}_k(Z)\subseteq {\rm Gr}_k(V)$ is connected totally geodesic and complex, but the same argument for the converse seems to fail. One reason being that if $N\subseteq {\rm Gr}_k(V)$ is a Grassmannian, then $k$ must divide the dimension of $N$.
Looking around I have found papers here and there discussing particular cases where $\dim V = 4$ and $k=2$, but I'd like to know if there's any reference discussing this more general situation.
Thanks.
