COMPLEX STRUCTURES, TOTALLY REAL AND TOTALLY GEODESIC SUBMANIFOLDS OF COMPACT 3-SYMMETRIC SPACES, AND AFFINE SYMMETRIC SPACES

Abstract

We construct invariant complex structures of a compact 3-symmetric space by means of the canonical almost complex structure of the underlying manifold and some involutions of a Lie group. Moreover, by making use of graded Lie algebras and some invariant structures of affine symmetric spaces, we classify half dimensional, totally real and totally geodesic submanifolds of a compact 3-symmetric space with respect to each invariant complex structure.