In the present paper we study contact metric manifolds whose characteristic vector field ξ belonging to the k-nullity distribution. First we consider concircularly pseudosymmetric N(k)-contact metric manifolds of dimension (2n + 1). Beside these, we consider Ricci solitons and gradient Ricci solitons on three dimensional N(k)-contact metric manifolds. As a consequence we obtain several results. Finally, an example is given. © 2018 Tamkang University. All Rights Reserved.