Homogeneous Einstein-like metrics on symmetric spaces

Abstract

In this paper, we study homogeneous Einstein-like metrics on the compact irreducible symmetric space M, which is not isometric to a compact Lie group and has rank greater than 1. Whenever there exists a closed proper subgroup G′ of G = Isom0(M) acting transitively on M, we find all G′-invariant $${\mathcal A}$$ -metrics and $${\mathcal B}$$ -metrics on M. More precisely, we prove that G′-invariant metrics on M must be $${\mathcal A}$$ -metrics, and G′-invariant $${\mathcal B}$$ -metrics on M are always Einstein.