On hypersurfaces of $\mathbb{S}^2\times \mathbb{S}^2$ and $\mathbb{H}^2\times \mathbb{H}^2$ with recurrent Ricci tensor

Abstract In this paper, we investigate hypersurfaces of $\mathbb{S}^2\times \mathbb{S}^2$ and $\mathbb{H}^2\times \mathbb{H}^2$ with recurrent Ricci tensor. As the main result, we prove that a hypersurface in $\mathbb{S}^2\times \mathbb{S}^2$ (resp. $\mathbb{H}^2\times \mathbb{H}^2$ ) with recurrent Ricci tensor is either an open part of $\Gamma \times \mathbb{S}^2$ (resp. $\Gamma \times \mathbb{H}^2$ ) for a curve $\Gamma$ in $\mathbb{S}^2$ (resp. $\mathbb{H}^2$ ), or a hypersurface with constant sectional curvature. The latter has been classified by H. Li, L. Vrancken, X. Wang, and Z. Yao very recently.