The first Chern class of Riemannian 3-symmetric spaces: the classical case

Abstract

The existence of Einstein metrics compatible with J on a compact connected almost complex manifold $(M,J)$ is deeply concerned with its characteristic classes.Using the method of A. Borel and F. Hirzebruch,we prove that an irreducible simply connected (non-Kähler) compact Riemannian 3-symmetric space $(G/K,J,\langle,\;\rangle)$ is Einstein if and only if the first Chern class of $(G/K,J)$ vanishes.