A contact manifold (M, ω) is said to be homogeneous if there is a connected Lie group G acting transitively as a group of diffeomorphisms on M which leave the contact form ω invariant. As is well known, this class extends the class of contact manifolds given by odd-dimensional spheres. If g is a metric associated to ω and G is a group acting transitively as a group of isometries which leave ω invariant, then (ω, g) is called a homogeneous contact Riemannian structure on M. When (M, ω) is a compact homogeneous contact manifold, by the Boothby-Wang fibration one can consider a homogeneous Sasakian structures (ω, g) on M. In this context Goldberg showed that the sphere is the only simply connected homogeneous contact manifold which can be equipped with an invariant contact metric of positive sectional curvature (we note that a homogeneous Riemannian manifold is complete and hence compact when its sectional curvatures are positive). More recently, it has been proved in [13], [14] that the spheres S~3, S~5 and the Stiefel manifold T~1(S~3) are the only compact simply connected n-dimensional manifolds, n = 3, 5, which admit a homogeneous contact structure.
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