Classification of Möbius isoparametric hypersurfaces in the unit six-sphere

Abstract

An immersed umbilic-free hypersurface in the unit sphere is equipped with three Möbius invariants, namely, the Möbius metric, the Möbius second fundamental form and the Möbius form. The fundamental theorem of Möbius submanifolds geometry states that a hypersurface of dimension not less than three is uniquely determined by the Möbius metric and the Möbius second fundamental form. A Möbius isoparametric hypersurface is defined by two conditions that it has vanishing Möbius form and has constant Möbius principal curvatures. It is well-known that all Euclidean isoparametric hypersurfaces are Möbius isoparametrics, whereas the latter are Dupin hypersurfaces. In this paper, combining with previous results, a complete classification for all Möbius isoparametric hypersurfaces in the unit six-sphere is established.

Möbius isoparametric hypersurfaces with three distinct principal curvatures, II

“…As a counterpart to the Cecil-Ryan conjecture for Dupin hypersurfaces, which states that a compact embedded Dupin hypersurface in a space form is Lie equivalent to an Euclidean isoparametric hypersurface, C. P. Wang conjectured that any compact embedded Möbius isoparametric hypersurface in ‫ޓ‬ n+1 is Möbius equivalent to an Euclidean isoparametric hypersurface. Pinkall and Thorbergsson [1989] and Miyaoka and Ozawa [1989], have constructed counterexamples to the Cecil-Ryan conjecture, but we point out that the classifications of Möbius isoparametric hypersurfaces in 2005;Hu and Zhai 2008;Li et al 2002] and this paper strengthen Wang's conjecture.…”

“…By relaxing the restriction that γ = 2, local classifications for all Möbius isoparametric hypersurfaces in ‫ޓ‬ 4 , ‫ޓ‬ 5 and ‫ޓ‬ 6 were established in [Hu and Li 2005], and [Hu and Zhai 2008], respectively. It was shown that a Möbius isoparametric hypersurface in ‫ޓ‬ 4 is either of parallel Möbius second fundamental form or Möbius equivalent to the Euclidean isoparametric hypersurface in ‫ޓ‬ 4 with three distinct principal curvatures, that is, a tube of constant radius over a standard Veronese embedding of ‫ޒ‬P 2 into ‫ޓ‬ 4 .…”

On the Blaschke isoparametric hypersurfaces in the unit sphere with three distinct Blaschke eigenvalues

“…The study of these invariants is important in Möbius geometry of submanifolds, and many interesting results were obtained in recent years. Among them are various classification theorems of submanifolds with special Möbius invariants, e.g., we have the classification of all immersed hypersurfaces in S n+1 with parallel Möbius second fundamental forms [9], and that of the so-called Möbius isoparametric hypersurfaces [7,8,[10][11][12][13].…”

Classification of Laguerre isoparametric hypersurfaces in R6

“…Then x is Laguerre equivalent to an open part of one of the following hypersurfaces : the oriented hypersurface given by Example 3.4, or the image of τ of the oriented hypersurface given by Example 3.5 , or the image of σ of a hypersurface in with mean curvature radius and , or the image of τ of a hypersurface in with mean curvature radius and . Remark From Theorem , we see that the examples of case (3) and case (4) in the Main Theorem have the same Laguerre second fundamental form with the Laguerre isoparametric hypersurfaces , respectively, thus they are also Laguerre isoparametric hypersurfaces. Remark From the proof of our Main Theorem, we see that the Laguerre isoparametric hypersurfaces in have parallel Laguerre second fundamental form if they are not the Laguerre isotropic hypersurfaces: case (3) and case (4) in the Main Theorem. But, the Laguerre isoparametric surfaces in and the Laguerre isoparametric hypersurfaces in have parallel Laguerre second fundamental form (see ). Remark In Möbius geometry of hypersurfaces, we should notice that Hu, Zhai and Li, Peng classified the Möbius isoparametric hypersurfaces and the Blaschke isoparametric hypersurfaces in the unit spheres completely.…”