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

摘要

We study invariant Einstein metrics on the Stiefel manifold VkRn congruent to SO(n)/SO(n - k) of all orthonormal k-frames in 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 V2pRn as total space over a classical generalized flag manifold with two isotropy summands and prove for 2 = p = 2/5n - 1 it admits at least four invariant Einstein metrics determined by Ad(U(p) x 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 Grobner bases and geometrical arguments. (C) 2019 Elsevier Ltd. All rights reserved.