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.



Totally geodesic submanifolds of complex Grassmannians
There is an extensive literature on totally geodesic submanifolds of symmetric spaces. A good place to start to read about this (and references to the preceeding literature) would be, for example,


Bang-yen Chen and Tadashi Nagano, *Totally geodesic submanifolds of symmetric spaces, I*, Duke Math Journal **44** (1977), 745–755.


They discuss various complex Grassmannians and their totally geodesic submanifolds in particular examples.


