A Möbius scalar curvature rigidity on compact conformally flat hypersurfaces in Sⁿ⁺¹

Let x:Mⁿ→Sⁿ⁺¹ be an immersed hypersurface without umbilical point, one can define the Möbius metric g on x which is invariant under the Möbius transformation group of Sⁿ⁺¹. The scalar curvature R with respect to g is called the Möbius scalar curvature. In this paper, we study conformally flat hypersurfaces with constant Möbius scalar curvature in Sⁿ⁺¹. First, we classify locally the conformally flat hypersurfaces of dimension n(≥4) with constant Möbius scalar curvature under the Möbius transformation group of Sⁿ⁺¹. Second, we prove that if an umbilic-free conformally flat hypersurface of dimension n(≥4) with constant Möbius scalar curvature R is compact, then R=(n−1)(n−2)r²,0<r<1, and the compact conformally flat hypersurface is Möbius equivalent to the torus S¹(1−r²)×Sⁿ⁻¹(r)↪Sⁿ⁺¹.