A Möbius scalar curvature rigidity on compact conformally flat hypersurfaces in ${\mathbb{S}}^{n+1}$

Abstract

In this paper, we study conformally flat hypersurfaces of dimension $n(\ge 4)$ in ${\mathbb{S}}^{n+1}$ using the framework of Möbius geometry. First, we classify and explicitly express the conformally flat hypersurfaces of dimension $n(\ge 4)$ with constant Möbius scalar curvature under the Möbius transformation group of ${\mathbb{S}}^{n+1}$. Second, we prove that if the conformally flat hypersurface with constant Möbius scalar curvature $R$ is compact, then $$R=(n-1)(n-2)r^2,0<r<1,$$ and the compact conformally flat hypersurface is Möbius equivalent to the torus $${\mathbb{S}}^1(\sqrt{1-r^2})\times {\mathbb{S}}^{n-1}(r)\hookrightarrow {\mathbb{S}}^{n+1}.$$

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