A rigidity theorem for hypersurfaces with positive Möbius Ricci curvature in $S^{n+1}$

Abstract

Let $M^{n}(n\geq 3)$ be an immersed hypersurface without umbilic points in the $(n+1)$-dimensional unit sphere $S^{n+1}$. Then $M^{n}$ is associated with a so-called Möbius form $\Phi$ and a Möbius metric $g$ which are invariants of $M^{n}$ under the Möbius transformation group of $S^{n+1}$. In this paper, we show that if $\Phi$ is identically zero and the Ricci curvature $Ric_{g}$ is pinched: $(n-1)(n-2)/n^{2}\leq Ric_{g}\leq$ $(n^{2}-2n+5)(n-2)/[n^{2}(n-1)]$, then it must be the case that $n=2p$ and $M^{n}$ is Möbius equivalent to $S^{p}(1/\sqrt{2})\times S^{p}(1/\sqrt{2})$.