Characterization of Hermitian Symmetric Spaces by Fundamental Forms

Theorem A (Landsberg) Let H be an irreducible compact Hermitian symmetric space of rank 2, different from the hyperquadric Qn ⊂ Pn+1. Let H ⊂ PN be a minimal non-degenerate equivariant embedding, equivalently, an embedding of H in PN defined by the complete linear system associated to the ample generator of Pic(H) ∼= Z. Let M ⊂ PN be a (not necessarily closed) complex sub-manifold with dim(M) = dim(H) and x ∈ M be a point in a neighborhood of which all the integer-valued differential invariants of M remain constant. If the second fundamental form of M at x is isomorphic to the second fundamental form of H at a point, then M is projective-linearly equivalent to an open subset of H.