Homogeneous Einstein metrics on Stiefel manifolds associated to flag manifolds with two isotropy summands

Abstract

We study invariant Einstein metrics on the Stiefel manifold $V_k{\mathbb{R}}^n\cong \mathrm{SO}(n)/\mathrm{SO}(n-k)$ of all orthonormal $k$-frames in ${\mathbb{R}}^n$. The isotropy representation of this homogeneous space contains equivalent summands, so a complete description of $G$-invariant metrics is not easy. In this paper we view the manifold $V_{2p}{\mathbb{R}}^n$ as total space over a classical generalized flag manifolds with two isotropy summands and prove for $2\le p\le \frac{2}{5}n-1$ it admits at least four invariant Einstein metrics determined by $\mathrm{Ad}(\mathrm{U}(p)\times \mathrm{SO}(n-2p))$-invariant scalar products. Two of the metrics are Jensen's metrics and the other two are new Einstein metrics.

19 pages. arXiv admin note: substantial text overlap with arXiv:1810.00655, arXiv:1311.1579