A Class of Special Hypersurfaces in Real Space Forms

Yan Zhao,Ximin Liu

doi:10.1155/2016/8796938

Abstract

We define the generalized golden- and product-shaped hypersurfaces in real space forms. A hypersurfaceMin real space formsRn+1,Sn+1, andHn+1is isoparametric if it has constant principal curvatures. Based on the classification of isoparametric hypersurfaces, we obtain the whole families of the generalized golden- and product-shaped hypersurfaces in real space forms.

Highlights

The golden ratio, which sometimes is called golden number, golden section, golden proportion, or golden mean, has many applications in many parts of mathematics, natural sciences, music, art, philosophies, and computational science [1]
It is interesting to notice that they are hyperspheres, a hyperbolic hyperplane in the hyperbolic case, or a generalized Clifford torus in the spherical space form, and this can be a new motivation to see the sphere as a golden-shaped surface
Based on the classification of isoparametric hypersurfaces, we obtain the whole families of the generalized golden- and productshaped hypersurfaces in real space forms

Summary

Introduction

The golden ratio, which sometimes is called golden number, golden section, golden proportion, or golden mean, has many applications in many parts of mathematics, natural sciences, music, art, philosophies, and computational science [1]. The notion of golden structure on a manifold M was introduced in [4, 5] as a (1, 1)-tensor field on M which satisfies the equation. All golden (as well as product) hypersurfaces in real space forms are parallel hypersurfaces. The metallic structure on a manifold M is a (1, 1)-tensor field on M satisfying the equation. The notion of a metallic shaped hypersurface was defined and the full classification of metallic shaped hypersurfaces in real space forms was obtained in [8]. We define the generalized golden- and product-shaped hypersurfaces in real space forms. Based on the classification of isoparametric hypersurfaces, we obtain the whole families of the generalized golden- and productshaped hypersurfaces in real space forms

Generalized Golden-Shaped Hypersurfaces

Generalized Product-Shaped Hypersurfaces