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

Abstract We discuss the classifiation of simply connected, complete (κ, µ)-spaces from the point of view of homogeneous spaces. In particular, we exhibit new models of (κ, µ)-spaces having Boeckx invariant -1. Finally, we prove that the number (n+1)(n+2)2 ${{(n + 1)(n + 2)} \over 2}$ 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 (κ, µ)-space. The same conclusion holds also in dimension 7 provided the almost CR structure of the contact metric manifold under consideration is integrable.