Certain Results on K-Contact and (k, μ)-Contact Manifolds

Inspired by a result of Boyer and Galicki, we prove that a complete K-contact gradient soliton is compact Einstein and Sasakian. For the non-gradient case we show that the soliton vector field is a Jacobi vector field along the geodesics of the Reeb vector field. Next we show that among all complete and simply connected K-contact manifolds only the unit sphere admits a non-Killing holomorphically planar conformal vector field (HPCV). Finally we show that, if a (k, mu)-contact manifold admits a non-zero HPCV, then it is either Sasakian or locally isometric to E-3 or En+1 x S-n (4).