Certain Contact Metrics as Ricci Almost Solitons

Abstract

We show that if a compact K-contact metric is a gradient Ricci almost soliton, then it is isometric to a unit sphere S2n+1. Next, we prove that if the metric of a non-Sasakian (κ, μ)-contact metric is a gradient Ricci almost soliton, then in dimension 3 it is flat and in higher dimensions it is locally isometric to En+1 × Sn (4). Finally, a couple of results on contact metric manifolds whose metric is a Ricci almost soliton and the potential vector field is point wise collinear with the Reeb vector field of the contact metric structure were obtained.