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.



1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.