Non naturally reductive Einstein metrics on the orthogonal group via real flag manifolds

TL;DR

This paper constructs new Einstein metrics on the special orthogonal groups that are not naturally reductive, using real flag manifolds and symmetry assumptions to expand the known metric landscape.

Contribution

It introduces a novel method for obtaining non naturally reductive Einstein metrics on SO(n) via real flag manifolds and symmetry constraints.

Findings

New invariant Einstein metrics on SO(n)

Metrics are not naturally reductive

Method uses real flag manifolds and symmetry assumptions

Abstract

We obtain new invariant Einstein metrics on the compact Lie groups $\SO(n)$ which are not naturally reductive. This is achieved by using the real flag manifolds $\SO(k_1+\cdots +k_p)/\SO(k_1)\times\cdots\times\SO(k_p)$ and by imposing certain symmetry assumptions in the set of all left-invariant metrics on $\SO(n)$.