Tohoku Mathematical Journal
2020
June
SECOND SERIES VOL. 72, NO. 2

Tohoku Math. J.
72 (2020), 161-210

Title INVARIANT EINSTEIN METRICS ON $\mathrm{SU}(n)$ AND COMPLEX STIEFEL MANIFOLDS

Author Andreas Arvanitoyeorgos, Yusuke Sakane and Marina Statha

(Received July 19, 2018, revised September 12, 2018)
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 $V_3\bb{C}^{5}=\operatorname{SU}(5)/\operatorname{SU}(2)$, $\operatorname{Ad}(\operatorname{S}(\operatorname{U}(2)\times\operatorname{U}(2)\times\operatorname{U}(2))$-invariant Einstein metrics on $V_4\bb{C}^{6}=\operatorname{SU}(6)/\operatorname{SU}(2)$, and $\operatorname{Ad}(\operatorname{S}(\operatorname{U}(m)\times\operatorname{U}(m)\times\operatorname{U}(n))$-invariant Einstein metrics on $V_{2m}\bb{C}^{2m+n}=\operatorname{SU}(2m+n)/\operatorname{SU}(n)$. 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 $\operatorname{SU}(4+n)$ ($n \geq 2$).

Mathematics Subject Classification. Primary 53C25; Secondary 53C30, 13P10, 65H10, 68W30.
Key words and phrases. Homogeneous space, Einstein metric, Stiefel manifold, special unitary group, invariant metric, isotropy representation, Gröobner basis.

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