A substantial proper submanifold of a Riemannian symmetric space is called a curved Lie triple if its tangent space at every point is invariant under the curvature tensor of , i.e. a sub-Lie triple. E.g. any complex submanifold of complex projective space has this property. However, if the tangent Lie triple is irreducible and of higher rank, we show a certain rigidity using the holonomy theorem of Berger and Simons: must be intrinsically locally symmetric. In fact we conjecture that is an extrinsically symmetric isotropy orbit. We are able to prove this conjecture provided that a tangent space of is also a tangent space of such an orbit.
"Higher rank curved Lie triples." J. Math. Soc. Japan 54 (3) 551 - 564, July, 2002. https://doi.org/10.2969/jmsj/1191593908