On complete hypersurfaces with constant scalar curvature $n(n-1)$ in the unit sphere

Abstract

Let $M^n$ be an $n$-dimensional complete and locally conformally flat hypersurface in the unit sphere ${\mathbb{S}}^{n+1}$ with constant scalar curvature $n(n-1)$. We show that if the total curvature ${\left({\int }_M|H{|}^{n}dv\right)}^{\frac{1}{n}}$ of $M$ is sufficiently small, then $M^n$ is totally geodesic.

Typos and inaccuracies corrected, accepted by Kodai Mathematical Journal