Investigating the characteristics of Clifford hypersurfaces and the unit sphere via a minimal immersion in $ S^{n+1} $

Abstract

In this article, we find the different sufficient conditions for a compact minimal hypersurface $ M $ of the unit sphere $ S^{n+1}, n\in \mathbb{Z}^{+} $ to be the Clifford hypersurface $ S^{\ell }(\sqrt{\frac{\ell }{n}})\times S^{m}(\sqrt{\frac{m}{n}}), $ where $ \ell, m\in \mathbb{Z}^{+}, \; \ell +m = n $ or the sphere $ S^{n} $. This classification is achieved by applying constraints to the tangent and normal components of the immersion.

Keywords

• hypersurface
• minimal hypersurface
• clifford hypersurface
• conformal vector field
• curvature tensor field
• ricci operator