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

We study invariant Einstein metrics on the Stiefel manifold \(V_k\mathbb{R}^n\cong \mathrm{SO}(n)/\mathrm{SO}(n-k)\) of all orthonormal \(k\)-frames in \(\mathbb{R}^n\). The isotropy representation of this homogeneous space contains equivalent summands, so a complete description of \(G\)-invariant m...

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Bibliographic Details
Published in:arXiv.org 2018-10
Main Authors: Arvanitoyeorgos, Andreas, Sakane, Yusuke, Statha, Marina
Format: Article
Language:English
Subjects:
Invariants
Isotropy
Manifolds (mathematics)
Online Access:Get full text
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Summary:We study invariant Einstein metrics on the Stiefel manifold \(V_k\mathbb{R}^n\cong \mathrm{SO}(n)/\mathrm{SO}(n-k)\) of all orthonormal \(k\)-frames in \(\mathbb{R}^n\). 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 \(V_{2p}\mathbb{R}^n\) as total space over a classical generalized flag manifolds with two isotropy summands and prove for \(2\le p\le \frac25 n-1\) it admits at least four invariant Einstein metrics determined by \(\mathrm{Ad}(\mathrm{U}(p) \times \mathrm{SO}(n-2p))\)-invariant scalar products. Two of the metrics are Jensen's metrics and the other two are new Einstein metrics.
ISSN:2331-8422
DOI:10.48550/arxiv.1810.01292