Möbius isoparametric hypersurfaces with three distinct principal curvatures, II

Hu, Zejun; Zhai, Shujie

doi:10.2140/pjm.2011.249.343

Cited by 13 publications

(3 citation statements)

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“…and therefore Möbius curvatures are finer than Lie curvatures. On a seemingly different thread, Hu, Li, and Wang [15,16,17] classified the hypersurfaces with vanishing Möbius form and constant Möbius principal curvatures, which are called Möbius isoparametric hypersurfaces, provided that the dimension of the hypersurface or the number of distinct principal curvatures is small. Until Rodrigues and Tenenblat [39] observed that an oriented hypersurface is a…”

Section: Introductionmentioning

confidence: 99%

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Möbius curvature, Laguerre curvature and Dupin hypersurface

Qing

Wang

2017

Advances in Mathematics

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In this paper we show that a Dupin hypersurface with constant Möbius curvatures isMöbius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat Laguerre isoparametric hypersurface. These results solve the major issues related to the conjectures of Cecil et al on the classification of Dupin hypersurfaces.

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Section: Introductionmentioning

confidence: 99%

“…Hence one easily derives Recall from [15,16,20], an immersed hypersurface is said to be a Möbius isoparametric hypersurface if its Möbius form vanishes and its Möbius principal curvatures are all constant.…”

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confidence: 99%

Möbius curvature, Laguerre curvature and Dupin hypersurface

Qing

Wang

2017

Advances in Mathematics

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“…Since the publication of [23], the study of Möbius geometry of submanifolds in S n+p has made a lot of progress and many interesting results were obtained. Among them, we have the classification of submanifolds with particular Möbius invariants, such as the Möbius characterization of hypersurfaces with constant mean curvature and constant scalar curvature [17], the classification of Möbius isotropic submanifolds [10,19], the classification of Möbius isoparametric (respectively, Blaschke isoparametric) hypersurfaces with three distinct principal curvatures [11,14,15].…”

Section: Introductionmentioning

confidence: 99%

On submanifolds with parallel Möbius second fundamental form in the unit sphere

Zhai

Wang

2014

Int. J. Math.

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In this paper, we study umbilic-free submanifolds of the unit sphere with parallel Möbius second fundamental form. As one of our main results, we establish a complete classification for such submanifolds under the additional condition of codimension two. Int. J. Math. 2014.25. Downloaded from www.worldscientific.com by MICHIGAN STATE UNIVERSITY on 02/07/15. For personal use only. 1450062-2 Int. J. Math. 2014.25. Downloaded from www.worldscientific.com by MICHIGAN STATE UNIVERSITY on 02/07/15. For personal use only. Submanifolds with parallel Möbius second fundamental form (d) the image of σ of a cone overwhere H n−k ( √ 1 + a 2 ) denotes the (n − k)-dimensional hyperbolic space of constant sectional curvature −1/(1 + a 2 ), and S k (a) denotes the k-dimensional Euclidean sphere of radius a.Remark 1.1. The two maps σ and τ mentioned above are the canonical conformal diffeomorphisms whose definitions are presented in (2.2) and (2.3), respectively. Whereas the notion about "cone" is referred to Definition 3.1.When compared with the hypersurfaces situation, the problem on the classification of higher codimension umbilic-free submanifolds of the unit sphere with parallel Möbius second fundamental form is now still open, which obviously would be much more complicated.In this paper, as our continuous attempts working on the above problem, we made the discovery that the assumption "parallel Möbius second fundamental form" implies that the submanifold is of parallel Blaschke tensor. This latter property plays an important role for us to know further about submanifolds with parallel Möbius second fundamental form. Indeed, as our main result, on the basis of Ferus, Backes-Reckziegel and Takeuchi's classification of (Euclidean) parallel submanifolds in space forms, and by using the particular properties of submanifolds of codimension two, a complete classification of Möbius parallel submanifolds can be established for p = 2 case, which can be stated as follows:Theorem 1.2. Let x : M n → S n+2 be an n-dimensional umbilic-free submanifold with parallel Möbius second fundamental form. Then x is Möbius equivalent to an open part of one of the following submanifolds:(i) an umbilic-free submanifold with parallel second fundamental form in S n+2 , (ii) the image of σ of an umbilic-free submanifold with parallel second fundamental form in the Euclidean space R n+2 , (iii) the image of τ of an umbilic-free submanifold with parallel second fundamental form in the hyperbolic space H n+2 , (iv) the image of σ of a cone over some k-dimensional umbilic-free submanifold with parallel second fundamental form in S k+2 (k ≤ n − 1).Remark 1.2. In above theorem, we made the assumption of codimension two that is used, first of all, in Lemma 5.1 to reduce the distinct number of eigenvalues of A to no more than four. We noticed, which Prof. Xiang Ma also observed and kindly informed us, that in higher codimensional case similar results can still be obtained, but the problem at the moment is that subsequently geometric discussions become much more ...

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Dupin Hypersurfaces

Cecil¹,

Ryan²

2015

Springer Monographs in Mathematics

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Möbius isoparametric hypersurfaces with three distinct principal curvatures, II

Cited by 13 publications

References 24 publications

Möbius curvature, Laguerre curvature and Dupin hypersurface

Möbius curvature, Laguerre curvature and Dupin hypersurface

On submanifolds with parallel Möbius second fundamental form in the unit sphere

Dupin Hypersurfaces

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