Homogeneous Einstein and Einstein–Randers metrics on Stiefel manifolds

Abstract

We study invariant Einstein metrics and Einstein–Randers metrics on the Stiefel manifold $V_k{R}^n=\mathrm{SO}\left(n\right)/\mathrm{SO}(n-k)$. We use a characterization for (nonflat) homogeneous Einstein–Randers metrics as pairs of (nonflat) homogeneous Einstein metrics and invariant Killing vector fields. It is well known that, for Stiefel manifolds the isotropy representation contains equivalent summands, so a complete description of invariant metrics is difficult. We prove, by assuming additional symmetries, that the Stiefel manifolds $V_{1+k}{R}^{1+2k}(k>2)$ and $V_6{R}^n(n\ge 8)$ admit at least four and six invariant Einstein metrics, respectively. Two of them are Jensen's metrics and the other two and four are new metrics. Also, we prove that $V_{{\ell }_1+{\ell }_2}{R}^n$ admit at least two invariant Einstein metrics, which are Jensen's metrics. Finally, we show that the previous mentioned Stiefel manifolds and $V_5{R}^n(n\ge 7)$ admit a certain number of non–Riemmanian Einstein–Randers metrics.