Invariant Einstein metrics on $\mathrm{SU}(n)$ and complex Stiefel manifolds

Abstract

We study existence of invariant Einstein metrics on complex Stiefel manifolds $G/K = \operatorname{SU}(\ell+m+n)/\operatorname{SU}(n) $ and the special unitary groups $G = \operatorname{SU}(\ell+m+n)$. We decompose the Lie algebra $\frak g$ of $G$ and the tangent space $\frak p$ of $G/K$, by using the generalized flag manifolds $G/H = \operatorname{SU}(\ell+m+n)/\operatorname{S}(\operatorname{U}(\ell)\times\operatorname{U}(m)\times\operatorname{U}(n))$. We parametrize scalar products on the 2-dimensional center of the Lie algebra of $H$, and we consider $G$-invariant and left invariant metrics determined by $\operatorname{Ad}(\operatorname{S}(\operatorname{U}(\ell)\times\operatorname{U}(m)\times\operatorname{U}(n))$-invariant scalar products on $\frak g$ and $\frak p$ respectively. Then we compute their Ricci tensor for such metrics. We prove existence of $\operatorname{Ad}(\operatorname{S}(\operatorname{U}(1)\times\operatorname{U}(2)\times\operatorname{U}(2))$-invariant Einstein metrics on $V_3\mathbb{C}^{5}=\operatorname{SU}(5)/\operatorname{SU}(2)$, $\operatorname{Ad}(\operatorname{S}(\operatorname{U}(2)\times\operatorname{U}(2)\times\operatorname{U}(2))$-invariant Einstein metrics on $V_4\mathbb{C}^{6}=\operatorname{SU}(6)/\operatorname{SU}(2)$, and $\operatorname{Ad}(\operatorname{S}(\operatorname{U}(m)\times\operatorname{U}(m)\times\operatorname{U}(n))$-invariant Einstein metrics on $V_{2m}\mathbb{C}^{2m+n}=\operatorname{SU}(2m+n)/\operatorname{SU}(n)$. We also prove existence of $\operatorname{Ad}(\operatorname{S}(\operatorname{U}(1)\times\operatorname{U}(2)\times\operatorname{U}(2))$-invariant Einstein metrics on the compact Lie group $\operatorname{SU}(5)$, which are not naturally reductive. The Lie group $\operatorname{SU}(5)$ is the special unitary group of smallest rank known for the moment, admitting non naturally reductive Einstein metrics. Finally, we show that the compact Lie group $\operatorname{SU}(4+n)$ admits two non naturally reductive $\operatorname{Ad}(\operatorname{S}(\operatorname{U}(2)\times\operatorname{U}(2)\times\operatorname{U}(n)))$-invariant Einstein metrics for $ 2 \leq n \leq 25$, and four non naturally reductive Einstein metrics for $n\ge 26$. This extends previous results of K. Mori about non naturally reductive Einstein metrics on $\operatorname{SU}(4+n)$ ($n \geq 2$).