Contact metric manifolds with large automorphism group and (κ, μ)-spaces

We discuss the classification of simply connected, complete (kappa, mu)-spaces from the point of view of homogeneous spaces. In particular, we exhibit new models of (kappa, mu)-spaces having Boeckx invariant-1. Finally, we prove that the number (n+1)(n+2)/2 is the maximum dimension of the automorphism group of a contact metric manifold of dimension 2n + 1, n >= 2, whose symmetric operator h has rank at least 3 at some point; if this dimension is attained, and the dimension of the manifold is not 7, it must be a (kappa, mu)-space. The same conclusion holds also in dimension 7 provided the almost CR structure of the contact metric manifold under consideration is integrable.