Five-dimensional homogeneous contact manifolds and related problems (Q1177410)

A contact manifold \(M\) is said to be homogeneous if there is a connected Lie group \(G\) acting transitively and effectively as a group of diffeomorphisms on \(M\) which leave the contact form invariant. The authors prove that a five-dimensional, compact, simply connected and homogeneous contact manifold is diffeomorphic to \(\mathbb{S}^ 5\) or \(\mathbb{S}^ 2\times\mathbb{S}^ 3\). In both cases, it is a globally \(\varphi\)- symmetric space with respect to the underlying invariant Sasakian structure. The non-simply connected case is also considered.