We consider invariant Einstein metrics on the Stiefel manifold V q R n of all orthonormal q-frames in R n . This manifold is diffeomorphic to the homogeneous space SO(n)/ SO(n − q) and its isotropy representation contains equivalent summands. We prove, by assuming additional symmetries, that V 4 R n (n ≥ 6) admits at least four SO(n)-invariant Einstein metrics, two of which are Jensen's metrics and the other two are new metrics. Moreover, we prove that V 5 R 7 admits at least six invariant Einstein metrics, two of which are Jensen's metrics and the other four are new metrics.
We obtain new invariant Einstein metrics on the compact Lie groups SO(n) (n ≥ 7) which are not naturally reductive. This is achieved by imposing certain symmetry assumptions in the set of all left-invariant metrics on SO(n) and by computing the Ricci tensor for such metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gröbner bases.
We consider invariant Einstein metrics on the quaternionic Stiefel manifolds VpH n of all orthonormal p-frames in H n . This manifold is diffeomorphic to the homogeneous space Sp(n)/ Sp(n − p) and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on VpH n ∼ = Sp(n)/ Sp(n − p), where n = k1 + k2 + k3 and p = n − k3. We view VpH n as a total space over the genaralized Wallach space Sp(n)/(Sp(k1) × Sp(k2) × Sp(k3)) and over the generalized flag manifold Sp(n)/(U(p) × Sp(n − p)).
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