On the Ricci curvature of homogeneous metrics on noncompact homogeneous spaces (Q1577146)

The author considers noncompact homogeneous spaces \(G/H\) with noncompact semisimple Lie group \(G\) and compact Lie group \(H\). Let \(K\) be a maximal compact subgroup of \(G\). We have the Cartan decomposition \(g=k \oplus p'\) and the orthogonal decomposition with respect to the Killing form \(k = h \oplus p''\). The author proves that there are no Einstein homogeneous metrics on \(G/H\) which provide orthogonality of the modules \(p'\) and \(p''\). In particular, \(SO(a+b, c+d)/ SO(a) \times SO(b) \times SO(c) \times SO(d)\) and \(Sp(a+b, c+d)/ Sp(a) \times Sp(b) \times Sp(c) \times Sp(d)\) have no homogeneous Einstein metrics.