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

Abstract

We study invariant Einstein metrics on the Stiefel manifold VkRn≅SO(n)/SO(n−k) of all orthonormal k-frames in Rn. 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 V2pRn as total space over a classical generalized flag manifold with two isotropy summands and prove for 2≤p≤25n−1 it admits at least four invariant Einstein metrics determined by Ad(U(p)×SO(n−2p))-invariant scalar products. Two of the metrics are Jensen's metrics and the other two are new Einstein metrics. The Einstein equation reduces to a parametric system of polynomial equations, which we study by combining Gröbner bases and geometrical arguments.