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

Abstract

We prove that the number of complex invariant Einstein metrics on the flag manifold $M_{n_1,n_2,n_3}=S{O}_{2(n_1+n_2+n_3)+1}/U_{n_1}\times U_{n_2}\times S{O}_{2n_3+1}$ is equal to 132, except when 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.

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