Narrow Results

First, we want to give a complete classification of Yamabe solitons and gradient Yamabe solitons for real hypersurfaces in the complex hyperbolic two-plane Grassmannians $G_2^{*}({ℂ}^{m+2})$. Next, as an application we also give a complete classification of quasi-Yamabe and gradient quasi-Yamabe solitons on real hypersurfaces in the complex hyperbolic two-plane Grassmannians $G_2^{*}({ℂ}^{m+2})$.

We characterize almost co-Kähler manifolds with gradient Yamabe, gradient Einstein and quasi-Yamabe solitons. It is proved that if the metric of a $(\kappa ,\mu )$-almost co-Kähler manifold $M^{2n+1}$ is a gradient Yamabe soliton, then $M^{2n+1}$ is either $K$-almost co-Kähler or $N(\kappa )$-almost co-Kähler or the metric of $M^{2n+1}$ is a trivial gradient Yamabe soliton. A $(\kappa ,\mu )$-almost co-Kähler manifold with gradient Einstein soliton is $K$-almost co-Kähler. Finally, it is shown that an almost co-Kähler manifold admitting a quasi-Yamabe soliton, whose soliton vector is pointwise collinear with the Reeb vector field of the manifold, is $K$-almost co-Kähler. Consequently, some results of almost co-Kähler manifolds are deduced.

In this paper, we give a complete classification of Yamabe solitons and gradient Yamabe solitons on real hypersurfaces in the complex quadric $Q^m=S{O}_{m+2}/S{O}_{2}S{O}_m$. In the following, as an application, we show a complete classification of quasi-Yamabe and gradient quasi-Yamabe solitons on Hopf real hypersurfaces in the complex quadric $Q^m$.