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

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