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.

Let M be an almost cosymplectic manifold such that the Reeb vector field is Killing. In this paper, it is proved that if the metric of M satisfying the $\eta$-Einstein condition is an $\eta$-Ricci soliton, then either M is Ricci flat or the potential vector field is an infinitesimal contact transformation. Also, a concrete example of cosymplectic $3$-manifold admitting non-trivial Ricci and $\eta$-Ricci solitons is constructed.

We investigate geometric properties of 3-dimensional real hypersurfaces with Aξ = 0 in a complex 2-dimensional nonflat complex space form from the view-points of their shape operators, Ricci tensors and *-Ricci tensors.