Non-naturally reductive Einstein metrics on SO(n)

Abstract

In this article, we obtain that the compact simple Lie group $SO(4k+3)(k\ge 8)$ admits at least two new non-naturally reductive $Ad\left(SO\right(k+1)\times SO(k+1)\times SO(k+1)\times SO(k\left)\right)$-invariant Einstein metrics, and the compact simple Lie group $SO(4k+6)(k\ge 11)$ admits at least two new non-naturally reductive $Ad\left(SO\right(k+2)\times SO(k+2)\times SO(k+2)\times SO(k\left)\right)$-invariant Einstein metrics. Furthermore, we explore the isometry problem for these Einstein metrics. Finally, we prove that there are at least two families of non-naturally reductive invariant Einstein-Randers metrics on $SO\left(n\right)(n\ge 35)$ and $SO\left(n\right)(n\ge 50)$ respectively.