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

TL;DR

This paper determines the exact number of complex invariant Einstein metrics on a specific class of flag manifolds and constructs Ricci-flat metrics on Euclidean spaces via Lie algebra contractions.

Contribution

It establishes the precise count of invariant Einstein metrics on certain flag manifolds and introduces a method to construct Ricci-flat metrics using Lie algebra contractions.

Findings

Number of complex invariant Einstein metrics is 132 for most parameter values.

Constructs Ricci-flat metrics on Euclidean spaces through Inonu-Wigner contractions.

Identifies algebraic conditions where the metric count differs.

Abstract

We prove that the number of complex invariant Einstein metrics on the flag manifold $M_{n_1,n_2,n_3}=SO_{2(n_1+n_2+n_3)+1}/U_{n_1} \times U_{n_2} \times SO_{2n_3+1}$ 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.