# Non-naturally reductive Einstein metrics on $\mathrm{SO}(n)$

## TL;DR

This paper proves that for all sufficiently large n, the special orthogonal group SO(n) admits multiple distinct non-naturally reductive Einstein metrics, expanding understanding of geometric structures on Lie groups.

## Contribution

It establishes the existence of numerous non-naturally reductive Einstein metrics on SO(n) for n ≥ 10, providing new examples and advancing the classification of Einstein metrics on Lie groups.

## Key findings

- At least 2([ (n-1)/3 ] - 2) such metrics exist for each n ≥ 10
- The metrics are non-naturally reductive and left-invariant on SO(n)
- The result applies to all sufficiently large n, specifically n ≥ 10

## Abstract

In this article, we prove that every compact simple Lie group $SO(n)$ for $n\geq 10$ admits at least $2\left([\frac{n-1}{3}]-2\right)$ non-naturally reductive left-invariant Einstein metrics.