Narrow Results

We show that the scalar curvature of a $K$-contact Ricci soliton is constant and satisfies sharp bounds. Next we show that the scalar curvature of a ($2n+1$)-dimensional $K$-contact Ricci almost soliton is equal to $2n(2n+1)$ plus the divergence of a global vector field. Finally, we show that, if a complete connected Sasakian or $\eta$-Einstein $K$-contact manifold of dimension $>3$ is a proper Ricci almost soliton, then it is isometric to a unit sphere.

We prove that a Ricci almost soliton on a Kenmotsu manifold of dimension $>3$ reduces to an expanding Ricci soliton satifying certain condition on the potential vector field or on the soliton function. Next, we show that any Ricci almost soliton on a Kenmotsu manifold is trivial (Einstein) if the soliton vector leaves the contact form $\eta$ invariant. Finally, we classify (locally) a Kenmotsu manifold admitting an almost Yamabe soliton. Some examples have been constructed of almost Yamabe solitons on different class of warped product spaces.