Bulletin of the
Korean Mathematical Society BKMS

In this article, we first introduce the notion of commuting Ricci tensor and pseudo-anti commuting Ricci tensor for Hopf hypersurfaces in the homogeneous nearly K\"{a}hler $\mathbb{S}^3\times\mathbb{S}^3$ and prove that the mean curvature of hypersurface is constant under certain assumptions. Next, we prove the nonexistence of Ricci soliton on Hopf hypersurface with potential Reeb vector field, which improves a result of Hu et al.~on the nonexistence of Einstein Hopf hypersurfaces in the homogeneous nearly K\"{a}hler $\mathbb{S}^3\times\mathbb{S}^3$.

Keywords: Nearly K\"ahler $\mathbb{S}^3\times\mathbb{S}^3$, Hopf hypersurface, commuting Ricci tensor, pseudo-anti commuting Ricci tensor, Ricci soliton

MSC numbers: 53B25, 53B35, 53C30, 53C42

Supported by: This research was supported by Science Foundation of China University of Petroleum-Beijing (Nos. 2462020XKJS02, 2462020YXZZ 004).