Invariant Einstein metrics on generalized flag manifolds of $Sp(n)$ and $SO(2n)$

TL;DR

This paper explicitly constructs new invariant Einstein metrics on generalized flag manifolds of $Sp(n)$ and $SO(2n)$, solving the algebraic Einstein equations and analyzing their isometry properties.

Contribution

It provides explicit solutions for invariant Einstein metrics on these manifolds and analyzes their isometric classifications, advancing understanding of Einstein geometry on flag manifolds.

Findings

New invariant Einstein metrics on $Sp(n)$ and $SO(2n)$ flag manifolds

Explicit solutions to Einstein equations for type $Sp(n)$

Analysis of isometry classes of the constructed metrics

Abstract

It is well known that the Einstein equation on a Riemannian flag manifold $(G/K,g)$ reduces to an algebraic system if $g$ is a $G$-invariant metric. In this paper we obtain explicitly new invariant Einstein metrics on generalized flag manifolds of $Sp(n)$ and $SO(2n)$; and we compute the Einstein system for generalized flag manifolds of type $Sp(n)$. We also consider the isometric problem for these Einstein metrics.