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Homogeneous Einstein metrics on Stiefel manifolds

Andreas Arvanitoyeorgos — 1996

Commentationes Mathematicae Universitatis Carolinae

A Stiefel manifold V k 𝐑 n is the set of orthonormal k -frames in 𝐑 n , and it is diffeomorphic to the homogeneous space S O ( n ) / S O ( n - k ) . We study S O ( n ) -invariant Einstein metrics on this space. We determine when the standard metric on S O ( n ) / S O ( n - k ) is Einstein, and we give an explicit solution to the Einstein equation for the space V 2 𝐑 n .

Metrics with homogeneous geodesics on flag manifolds

Dimitri V. Alekseevsky;  Andreas Arvanitoyeorgos — 2002

Commentationes Mathematicae Universitatis Carolinae

A geodesic of a homogeneous Riemannian manifold ( M = G / K , g ) is called homogeneous if it is an orbit of an one-parameter subgroup of G . In the case when M = G / H is a naturally reductive space, that is the G -invariant metric g is defined by some non degenerate biinvariant symmetric bilinear form B , all geodesics of M are homogeneous. We consider the case when M = G / K is a flag manifold, i.eȧn adjoint orbit of a compact semisimple Lie group G , and we give a simple necessary condition that M admits a non-naturally reductive...

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