Invariant Einstein metrics on real flag manifolds with two or three isotropy summands

TL;DR

This paper investigates the existence of invariant Einstein metrics, including non-diagonal ones, on real flag manifolds with two or three isotropy summands, expanding understanding of geometric structures on these spaces.

Contribution

It proves the existence of non-diagonal Einstein metrics on real flag manifolds with two or three isotropy summands, highlighting new geometric possibilities.

Findings

Existence of non-diagonal Einstein metrics on certain real flag manifolds.

Classification of invariant Einstein metrics based on isotropy summands.

Extension of known results to non-diagonal homogeneous Riemannian metrics.

Abstract

We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible sub-representations. In this situation, one can have equivalent sub-modules, leading to the existence of non-diagonal homogeneous Riemannian metrics. In particular, we prove the existence of non-diagonal Einstein metrics on real flag manifolds.