The $k$-almost Yamabe solitons and contact metric manifolds

We introduce the concept of a k-almost Yamabe soliton which extends naturally from Yamabe solitons. Our aim is to study the k-almost Yamabe soliton (g,V,k,λ) on a contact metric manifold M2n+1. Firstly, for a general contact metric manifold, it is proved that V is Killing if the potential vector field V is a contact vector field and that M is K-contact if V is collinear with Reeb vector field. Secondly, we prove that a compact K-contact manifold, admitting a k-almost Yamabe gradient soliton, is isometric to a standard unit sphere. Moreover, for a complete Sasakian manifold admitting a k-almost Yamabe soliton, we show that it is isometric to a standard unit sphere 𝕊2n+1(1) when n>1 and for n=1, M is also isometric to a standard unit sphere if it admits a closed k-almost Yamabe soliton. Finally, we consider a contact metric (κ,μ)-manifold with a nontrivial k-almost Yamabe gradient soliton and show that it is flat in dimension 3 and in higher dimension M is locally isometric to En+1×𝕊n(4). In the end, we construct two examples of contact metric manifolds with a k-almost Yamabe soliton.