New homogeneous Einstein metrics on quaternionic Stiefel manifolds

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 $\mathrm{Sp}(n)/(\mathrm{U}(p)\times \mathrm{Sp}(n-p))$.

19 pages