Open Access. © 2019 A. Lotta, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. Complex Manifolds 2019; 6:294–302 Research Article Open Access Antonio Lotta* Contact metric manifolds with large automorphism group and ( κ , µ )-spaces https://doi.org/10.1515/coma-2019-0015 Received February 19, 2019; accepted May 30, 2019 Abstract: We discuss the classification 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. Fi- nally, 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 (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. Keywords: contact metric manifold, (κ , µ)-space MSC: Primary 53C25, 53D10; Secondary 53C30 1 Introduction Among the contact metric spaces (M, φ, ξ , η, g), the so-called (κ , µ)-spaces form a special and significant class with remarkable geometric properties; they were originally defined by Blair, Koufogiorgos and Papan- toniou in 1995, by the following curvature condition: R(X , Y )ξ =(κId + µh)(η(Y )X − η(X)Y ), (1) where h = 1 2 L ξ φ and κ , µ are real numbers; see [2]. For the notation and basic facts of contact metric geometry we refer the reader to Blair’s book [1]. Of course the above condition is satisfied by the Sasakian manifolds (corresponding to h =0 and k = 1); in all that follows we shall deal exclusively with non Sasakian (κ , µ)- spaces. The main motivation for studying these manifolds was a previous result of Blair, stating that the unique simply connected, complete contact metric manifold of dimension 2n + 1, n ≥ 2, satisfying R(X , Y )ξ = 0 is the Riemannian product S n × R n+1 , where the metric on the sphere is chosen of curvature 4; this is the unique contact metric non Sasakian orientable hypersurface of C n+1 (see [17]). In the paper [2], it was established that every (κ , µ)-space is a CR manifold, and that its curvature tensor is completely determined by (1). Blair, Koufogiorgos and Papantoniou also exhibited models of type T 1 N(c), i.e. tangent sphere bundles over Riemannian manifolds N(c) with constant curvature c ∈ R, c ≠ 1. Moreover, they provided a complete classification in the 3-dimensional case. Later other characterizations of the (κ , µ)-spaces and related geometric results appeared in the literature. See for instance [5], [6], [12], [10], [11]. Concerning the classification problem in higher dimension, in [3] and [4] Boeckx proved that every (κ , µ)- space is locally homogeneous, and showed that, up to equivalence and D-homothetic deformations, the fam- *Corresponding Author: Antonio Lotta: Dipartimento di Matematica, Università di Bari Aldo Moro, Via Orabona 4, 70125 Bari, Italy, E-mail: antonio.lotta@uniba.it