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, thenR=(n−1)(n−2)r2,0