New homogeneous Einstein metrics on quaternionic Stiefel manifolds

Abstract

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)).

Equigeodesics on some classes of homogeneous spaces

“…The isotropy representation on this case according to (4) contains equivalent summands. We will describe a special class of invariant metrics in this space (for more details see for example [Sta1], [ArSaSt1] and [ArSaSt2]). The basic approach is to use an appropriate subgroup K of G, such that the special class of Ad K -invariant inner products, which are a subset of Ad H -invariant inner products, are diagonal.…”

Homogeneous Einstein manifolds based on symplectic triple systems

“…The family of homogeneous Einstein manifolds we described can be seen from the point of view in [2]. Namely, in our approach the para-quaternionic Kähler manifolds correspond with the symmetric space associated to the pair (g(T )0, g(T )1) in (5). Hence the manifolds we consider are the total spaces of principal bundles of standard basis on para--quaternionic Kähler symmetric spaces.…”

“…A variety of works about homogeneous Einstein metrics has been contributed by Arvanitoyeorgos and his collaborators. For instance, [4] finds which compact simple Lie groups admit non-naturally reductive Einstein metrics, [3] constructs explicit invariant non-Kähler Einstein metrics on generalized flag manifolds, and [5] obtains new invariant Einstein metrics on the quaternionic Stiefel manifold of all orthonormal p-frames in H n .…”

Homogeneous Einstein metrics on Stiefel manifolds associated to flag manifolds with two isotropy summands

“…For an overview about invariant Einstein metrics on quaternionic and complex Stiefel manifolds we refer to [ArSaSt3] and [ArSaSt2] respectively.…”