Classification of hypersurfaces with constant Möbius curvature in S ^m+1

Let x: M ^m → S ^m+1 be an m-dimensional umbilic-free hypersurface in an (m + 1)-dimensional unit sphere S ^m+1, with standard metric I = dx · dx. Let II be the second fundamental form of isometric immersion x. Define the positive function. Then positive definite (0,2) tensor g = ρ ²}I is invariant under conformal transformations of S ^m+1 and is called Möbius metric. The curvature induced by the metric g is called Möbius curvature. The purpose of this paper is to classify the hypersurfaces with constant Möbius curvature.