Classification of semi-parallel hypersurfaces of the product of two spheres

Abstract

It is known that Mendonça and Tojeiro (2013) [19] have established a complete classification of parallel submanifolds in the product manifold $Q_{k_1}^{n_1}\times Q_{k_2}^{n_2}$, where $Q_{k_1}^{n_1}$ (resp. $Q_{k_2}^{n_2}$) is an $n_1$-dimensional (resp. $n_2$-dimensional) real space form with constant curvature $k_1$ (resp. $k_2$). In this paper, motivated by this result with considering further generalization, we study those semi-parallel hypersurfaces in case $Q_{k_1}^{n_1}=S_{k_1}^{n_1}$ and $Q_{k_2}^{n_2}=S_{k_2}^{n_2}$ with $k_1,k_2>0$. As the main result, we classify semi-parallel hypersurfaces of $S_{k_1}^{n_1}\times S_{k_2}^{n_2}$ for $n_1,n_2\ge 2$.