ANTIPODAL SETS OF COMPACT SYMMETRIC SPACES AND THE INTERSECTION OF TOTALLY GEODESIC SUBMANIFOLDS

A real form in a Hermitian symmetric space M of compact type is the fixed point set of an involutive anti-holomorphic isometry of M, which is connected and a totally geodesic Lagrangian submanifold. We prove that the intersection of two real forms is an antipodal set, in which the geodesic symmetry at each point is the identity. Using this we investigate the intersection of two real forms in irreducible M as well as non-irreducible M and determine the intersection numbers of them. This is a survey article on the joint research with Hiroyuki Tasaki.