Möbius homogeneous hypersurfaces with two distinct principal curvatures in \(S^{n+1}\) (Q363498)

The authors provide a classification of Möbius homogeneous hypersurfaces of the conformal \((n+1)\)-sphere, that is, hypersurfaces on which the group of Möbius transformations of \(S^{n+1}\) that fix the hypersurface acts transitively, and with two distinct principal curvatures. In particular, Theorem 1.3 states that such a hypersurface is Möbius equivalent to either a product of a sphere and a space form with appropriately weighted curvatures or to a cylinder over a logarithmic spiral in \(\mathbb R^2\). The combination of this result with a previous classification result by the third author [Nagoya Math. J. 139, 1--20 (1995; Zbl 0863.53034)] yields a complete classification of Möbius homogeneous hypersurfaces of the conformal \(4\)-sphere (Corollary 1.4). NEWLINENEWLINENEWLINETo prove the theorem, the authors employ the third author's methodology from [Manuscr. Math. 96, No. 4, 517--534 (1998; Zbl 0912.53012)], in particular the system of Möbius invariants for hypersurfaces deduced there. A key argument is a characterization of ``logarithmic spiral cylinders'' as Möbius homogeneous hypersurfaces with two distinct principal curvatures and non-vanishing ``Möbius form'' (Theorem 1.2).