We show that, in the complex hyperquadric, the intersection of two real forms, which are certain totally geodesic Lagrangian submanifolds, is an antipodal set whose cardinality attains the smaller 2-number of the two real forms. As a corollary of the result, we know that any real form in the complex hyperquadric is a globally tight Lagrangian submanifold.
"The intersection of two real forms in the complex hyperquadric." Tohoku Math. J. (2) 62 (3) 375 - 382, 2010. https://doi.org/10.2748/tmj/1287148617