Complex invariant Einstein metrics on $SO_{2(n_1+n_2+n_3)+1}/U_{n_1} \times U_{n_2} \times SO_{2n_3+1}$ and Ricci-flat manifolds

Abstract

We prove that the number of complex invariant Einstein metrics on the flagwhen the parameters n 1 , n 2 , n 3 satisfy one of some algebraic equations. Also the family of (real) non-flat Ricci-flat metrics on the Euclidean spaces will be constructed using the method of Inonu-Wigner contractions of Lie algebras.