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Keywords:
Riemannian geometry; homogeneous spaces; Einstein metrics; Stiefel manifolds
Riemannian geometry; homogeneous spaces; Einstein metrics; Stiefel manifolds
Summary:
A Stiefel manifold $V_k\bold R^n$ is the set of orthonormal $k$-frames in $\bold R^n$, and it is diffeomorphic to the homogeneous space $SO(n)/SO(n-k)$. We study $SO(n)$-invariant Einstein metrics on this space. We determine when the standard metric on $SO(n)/SO(n-k)$ is Einstein, and we give an explicit solution to the Einstein equation for the space $V_2\bold R^n$.
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