Classification of Möbius homogeneous hypersurfaces in a 5-dimensional sphere

Let x : Mⁿ → Sⁿ⁺¹ be an immersed hypersurface in the (n+1)-dimensional sphere Sn+1. If, for any two points p; q 2 Mn, there exists a Möbius transformation φ : Sⁿ⁺¹ → Sⁿ⁺¹ such that φ o x(Mⁿ) = x(Mⁿ) and φ o x(p) = x(q), then the hypersurface is called a Möbius homogeneous hypersurface. In this paper, We classify completely the Möbius homogeneous hypersurfaces in the 5-dimensional sphere S5 up to a Möbius transformation.