New homogeneous Einstein metrics on quaternionic Stiefel manifolds

TL;DR

This paper discovers new invariant Einstein metrics on quaternionic Stiefel manifolds by analyzing their geometric structure as homogeneous spaces, expanding the known solutions in differential geometry.

Contribution

It introduces novel Einstein metrics on quaternionic Stiefel manifolds using their descriptions as homogeneous spaces with equivalent isotropy summands.

Findings

New Einstein metrics on quaternionic Stiefel manifolds

Metrics constructed via homogeneous space decompositions

Enhanced understanding of Einstein metrics on complex geometric structures

Abstract

We consider invariant Einstein metrics on the quaternionic Stiefel manifolds $V_p\mathbb{H} ^n$ of all orthonormal $p$-frames in $\mathbb{H}^n$. This manifold is diffeomorphic to the homogeneous space $\mathrm{Sp}(n) / \mathrm{Sp}(n-p)$ and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on $V_p\mathbb{H}^n \cong \mathrm{Sp}(n)/\mathrm{Sp}(n-p)$, where $n = k_1 + k_2 + k_3$ and $p = n-k_3$. We view $V_p\mathbb{H}^n$ as a total space over the generalized Wallach space $\mathrm{Sp}(n) / (\mathrm{Sp}(k_1) \times \mathrm{Sp}(k_2) \times \mathrm{Sp}(k_3))$ and over the generalized flag manifold $\mathrm{Sp}(n) / (\mathrm{U}(p) \times \mathrm{Sp}(n-p))$.