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

2018/10/11 by Andreas Arvanitoyeorgos, Yusuke Sakane, Marina Statha
Mathematics · #math.DG #msc:13P10 #msc:53C25 #msc:53C30 #msc:65H10 #msc:68W30

paper · pdf · doi:10.1016/j.jsc.2019.08.001

published as J. Symbolic Comput. 101 (2020) 189-201 · 19 pages. arXiv admin note: substantial text overlap with arXiv:1810.00655, arXiv:1311.1579

arxiv created 2018/10/11 · arxiv updated 2020/06/12

Abstract

We study invariant Einstein metrics on the Stiefel manifold V_kℝⁿ≅ 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 V_2pℝⁿ as total space over a classical generalized flag manifolds with two isotropy summands and prove for 2≤ p≤ \frac25 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.

Citations

• New homogeneous Einstein metrics on quaternionic Stiefel manifolds
• New homogeneous Einstein metrics on Stiefel manifolds