Communications of the
Korean Mathematical Society CKMS

Consider a gradient Einstein-type metric in the setting of $K$-contact manifolds and $(\kappa,\mu)$-contact manifolds. First, it is proved that, if a complete $K$-contact manifold admits a gradient Einstein-type metric, then $M$ is compact, Einstein, Sasakian and isometric to the unit sphere $\mathbb{S}^{2n+1}$. Next, it is proved that, if a non-Sasakian $(\kappa,\mu)$-contact manifolds admits a gradient Einstein-type metric, then it is flat in dimension $3$, and for higher dimension, $M$ is locally isometric to the product of a Euclidean space $\mathbb{E}^{n+1}$ and a sphere $\mathbb{S}^n(4)$ of constant curvature $+4$.

Keywords: Einstein-type manifolds, $K$-contact manifolds, Sasakian manifold, $(\kappa,\mu)$-contact manifold, Einstein manifold

MSC numbers: 53C25, 53C20, 53D15