Möbius Homogeneous Hypersurfaces with One Simple Principal Curvature in $\mathbb{S}^{{n + 1}}$

Abstract

Let Möb($\mathbb{S}^{{n + 1}}$) denote the Möbius transformation group of $\mathbb{S}^{{n + 1}}$. A hypersurface f: ${M^n} \to \mathbb{S}^{{n + 1}}$ is called a Möbius homogeneous hypersurface, if there exists a subgroup $G \triangleleft {\text{M}}\ddot o{\text{b}}{(^{n + 1}})$ such that the orbit G(p) = {ϕ(p) ∣ ϕ ∈ G} = f (Mⁿ),p ∈ f (Mⁿ). In this paper, we classify the Möbius homogeneous hypersurfaces in $\mathbb{S}^{{n + 1}}$ with at most one simple principal curvature up to a Möbius transformation.