The intersection of two real forms in Hermitian symmetric spaces of compact type

Abstract

We show that the intersections of two real forms, certain totally geodesic Lagrangian submanifolds, in Hermitian symmetric spaces of compact type are antipodal sets. The intersection number of two real forms is invariant under the replacement of the two real forms by congruent ones. If two real forms are congruent, then their intersection is a great antipodal set of them. It implies that any real form in Hermitian symmetric spaces of compact type is a globally tight Lagrangian submanifold. Moreover we describe the intersection of two real forms in the irreducible Hermitian symmetric spaces of compact type.