Möbius geometry of hypersurfaces with constant mean curvature and scalar curvature (Q1411982)

The authors prove the following classification theorem for a hypersurface \(x: M^m \rightarrow \mathbb{S}^{m+1}\) without umbilics in the \((m+1)\)-dimensional unit sphere \(\mathbb{S}^{m+1}\): if for \(x\) a certain (invariant) 1-form \(\Phi = 0\) and \(A + \lambda g + \mu B = 0\), where \(A\) is the symmetric \((0, 2)\) Blaschke tensor of \(x\), \(B\) is the (Möbius) second fundamental tensor of \(x\), \(g\) is the (Möbius) Riemannian metric, and \(\lambda\) and \(\mu\) are some functions, then \(\lambda\) and \(\mu\) are constants and \(x\) is conformally equivalent (Möbius-equivalent) to either a hypersurface with constant mean and scalar curvatures or the preimage of a stereographic projection of a hypersurface with constant mean and scalar curvatures in the Euclidean space \(\mathbb{R}^{m+1}\) or the image of the standard conformal map \(\tau: \mathbb{H}^{m+1} \rightarrow \mathbb{S}^{m+1}_{+}\) of a hypersurface with constant mean and scalar curvatures in the hyperbolic space \(\mathbb{H}^{m+1}\).