Invariant Einstein metrics on SU(N) and complex Stiefel manifolds
Abstract
We study existence of invariant Einstein metrics on complex Stiefel manifolds G/K = (+m+n)/(n) and the special unitary groups G = (+m+n). We decompose the Lie algebra g of G and the tangent space p of G/K, by using the generalized flag manifolds G/H = (+m+n)/(()×(m)×(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 ((()×(m)×(n))-invariant scalar products on g and p respectively. Then we compute their Ricci tensor for such metrics. We prove existence of (((1)×(2)×(2))-invariant Einstein metrics on V3C5=(5)/(2), (((2)×(2)×(2))-invariant Einstein metrics on V4C6=(6)/(2), and (((m)×(m)×(n))-invariant Einstein metrics on V2mC2m+n=(2m+n)/(n). We also prove existence of (((1)×(2)×(2))-invariant Einstein metrics on the compact Lie group (5), which are not naturally reductive. The Lie group (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 (4+n) admits two non naturally reductive (((2)×(2)×(n)))-invariant Einstein metrics for 2 ≤ n ≤ 25, and four non naturally reductive Einstein metrics for n 26. This extends previous results of K.~ Mori about non naturally reductive Einstein metrics on (4+n) (n ≥ 2).