# Totally geodesic submanifolds of complex Grassmannians

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.