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

TL;DR

This paper investigates invariant Einstein metrics on Stiefel manifolds, revealing at least four such metrics for certain dimensions, including two new Einstein metrics beyond known Jensen's metrics.

Contribution

It provides a complete description of invariant Einstein metrics on specific Stiefel manifolds viewed as total spaces over flag manifolds with two isotropy summands, identifying new Einstein metrics.

Findings

Existence of at least four invariant Einstein metrics on $V_{2p}\mathbb{R}^n$ for certain $p$ and $n$

Two of the metrics are Jensen's known metrics

Two are newly discovered Einstein metrics

Abstract

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.