Homogeneous Einstein metrics on Stiefel manifolds associated to flag manifolds with two isotropy summands. (November 2020)

• Record Type: Journal Article
• Authors: Arvanitoyeorgos, Andreas; Sakane, Yusuke; Statha, Marina
• Is Part Of: Journal of symbolic computation. Volume 101(2020)
• Page Start: 189
• Page End: 201
• Publication Date: 2020-11
• Subjects: primary 53C25 -- secondary 53C30 13P10 65H10 68W30; Homogeneous space -- Einstein metric -- Stiefel manifold -- Generalized flag manifold -- Isotropy representation -- Gröbner basis; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2019.08.001
• Languages: English
• ISSNs: 0747-7171

Abstract

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.