Classification of Möbius isoparametric hypersurfaces in $\mathbb{S}^{4}$

Abstract

Let $M^{n}$ be an immersed umbilic-free hypersurface in the $(n+1)$-dimensional unit sphere $\mathbb{S}^{n+1}$, then $M^{n}$ is associated with a so-called Möbius metric $g$, a Möbius second fundamental form ${\bf B}$ and a Möbius form $\Phi$ which are invariants of $M^{n}$ under the Möbius transformation group of $\mathbb{S}^{n+1}$. A classical theorem of Möbius geometry states that $M^{n}$ $(n \geq 3)$ is in fact characterized by $g$ and ${\bf B}$ up to Möbius equivalence. A Möbius isoparametric hypersurface is defined by satisfying two conditions: (1) $\Phi \equiv 0$; (2) All the eigenvalues of ${\bf B}$ with respect to $g$ are constants. Note that Euclidean isoparametric hypersurfaces are automatically Möbius isoparametric, whereas the latter are Dupin hypersurfaces.

In this paper, we prove that a Möbius isoparametric hypersurface in $\mathbb{S}^{4}$ is either of parallel Möbius second fundamental form or Möbius equivalent to a tube of constant radius over a standard Veronese embedding of $\mathbb{R}P^{2}$ into $\mathbb{S}^{4}$. The classification of hypersurfaces in $\mathbb{S}^{n+1}$ $(n\ge2)$ with parallel Möbius second fundamental form has been accomplished in our previous paper \cite{6}. The present result is a counterpart of Pinkall's classification for Dupin hypersurfaces in $\mathbb{E}^{4}$ up to Lie equivalence.