Certain Results on $N(k)-$Contact Metric Manifolds and Torse-Forming Vector Fields

Abstract

In this paper, we study $N(k)-$contact metric manifolds endowed with a torse-forming vector field and give some characterizations for such manifolds. Then, we deal with $N(k)-$contact metric manifolds admitting a Ricci soliton and find that the potential vector field $V$ of the Ricci soliton is a constant multiple of $\xi$. Also, we obtain a necessary condition for a torse-forming vector field to be recurrent and Killing on $M$.