Homogeneous Einstein metrics on Stiefel manifolds associated to flag manifolds with two isotropy summands. (November 2020) - Arvanitoyeorgos, Andreas; Sakane, Yusuke; Statha, Marina
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 V 2 p R n as total space over a classical generalized flag manifold 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 − 2 p ) ) -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.