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 R n ∼ = 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 V2pR n as total space over a classical generalized flag manifolds with two isotropy summands and prove for 2 ≤ p ≤ 2 5 n−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.

Homogeneous Einstein manifolds

“…In the decade following Wang's extensive survey [93], the bulk of new results for compact homogeneous Einstein manifolds has been the construction of new types of Einstein metrics. In some cases, these have been non-naturally reductive Einstein metrics [101,23,102], the classification of Einstein metrics for some special homogeneous spaces [86,24], and the generation of new examples of Einstein metrics on particular homogeneous spaces, sometimes with the aid of computer algebra systems, [7,8,26]. For a detailed treatment of Einstein metrics on flag manifolds, see [6].…”