Journal of Function Spaces
Volume 2016 (2016), Article ID 8796938, 6 pages
http://dx.doi.org/10.1155/2016/8796938

Research Article

A Class of Special Hypersurfaces in Real Space Forms

Yan Zhao1,2 and Ximin Liu¹

¹School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
²Henan Light Industry School, Zhengzhou 450000, China

Received 15 January 2016; Revised 18 May 2016; Accepted 26 May 2016

Academic Editor: Hugo Leiva

Copyright © 2016 Yan Zhao and Ximin Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We define the generalized golden- and product-shaped hypersurfaces in real space forms. A hypersurface in real space forms , , and is 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.

1. 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]. In the past few years, the golden ratio has played a more and more significant role in modern physical research and atomic physics [2]. The golden ratio also has interesting properties in topology of four-manifolds, in conformal field theory, in mathematical probability theory, in Cantorian spacetime [3], and in differential geometry.

The notion of golden structure on a manifold was introduced in [4, 5] as a -tensor field on which satisfies the equation

where is the usual Kronecker tensor field of . It attracts many authors’ attentions to focus on a class of well-known objects, namely, hypersurfaces in real space forms. 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. A well-known result of parallelism for the shape operators satisfying a quadratic equation is applied also for the spherical framework. More precisely, all golden (as well as product) hypersurfaces in real space forms are parallel hypersurfaces.

Crasmareanu and Hreţcanu derived a natural correspondence between golden structures and almost product structures in [4]. Recently, the golden- and product-shaped hypersurfaces in real space forms were defined and the whole families of golden- and product-shaped hypersurfaces were obtained in [6]. Similar to the golden structure on a manifold , in [7], Crasmareanu and Hreţcanu defined the metallic structure on a manifold . The metallic structure on a manifold is a -tensor field on satisfying the equation

where is the usual Kronecker tensor field of and are positive integers [7]. In the present study, 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].

In this paper, 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 product-shaped hypersurfaces in real space forms.

2. Generalized Golden-Shaped Hypersurfaces

Let be an oriented embedded hypersurface of the real space form and, for a certain normal field , let be the associated shaped operator (see [9]); then will be the principal curvatures of (see [10] for the semi-Euclidean case). We will take and consequently , , respectively, .

Definition 1. is called a golden-shaped hypersurface if is a golden structure; that is, , where is the identity on the tangent bundle of .

Definition 2. is called a metallic shaped hypersurface if the shaped operator is a metallic structure.

In this paper, we give the definition of generalized golden-shaped hypersurfaces.

Definition 3. is called a generalized golden-shaped hypersurface if is a generalized golden structure; that is, , where is the identity on the tangent bundle of and are real constants satisfying .

Cartan’s Identity. Let be an isoparametric hypersurface with distinct principal curvatures , having respective multiplicities . If , for each , ,

The important equation, known as Cartan’s identity, is crucial in Cartan’s work on isoparametric hypersurfaces.

In the case , there are at most two distinct principal curvatures and if there are two, then one of them must be zero.

In the case , there are at most two distinct principal curvatures and if there are two, then they are reciprocals of each other.

In the case and if there are two distinct principal curvatures, then they are negative reciprocals of each other.

Suppose is a generalized golden-shaped hypersurface. It follows that the principal curvatures of , as eigenvalues of , are and if . The principal curvatures of , as eigenvalues of , especially, are if . According to [11], the manifold is an isoparametric hypersurface and this yields the following theorems.

Theorem 4. The only generalized golden-shaped hypersurfaces of are as follows.
Case 1 (If ).
(1) If ,

(2) If ,

which is totally geodesic and is isometric to . In this case, the second fundamental form is zero.
Case 2 (If ).
(1) If ,

(2) If , there are the following three cases:

(i); in this case, the second fundamental form is zero; is totally geodesic and is isometric to ;(ii) ;(iii) for ,

 with .

Proof.  
Case 1. Suppose and ; we distinguish the following two cases.
(1) If and , we get and ; then is totally umbilical and .
(2) If and , we get and ; then is totally umbilical and which is totally geodesic and is isometric to .
Case 2. Suppose ; we distinguish the following cases.
(1) If and , we get . Then, is totally umbilical and which presents the first above sphere.
If and , we get . Then, is totally umbilical too and which presents the second above sphere.
(2) If , then and , , or , .

(i)If , we get . Then is totally umbilical and . In this case, the second fundamental form is zero; is totally geodesic and is isometric to .(ii)If , we get . Then is totally umbilical and .(iii)For , if and , then and . We get with .

Recall that the hyperbolic space in the upper half-space model is defined as

and the isoparametric hypersurfaces of it are [12, P. 252].

(i) , for with ; is totally geodesic and is in fact just .

(ii) , for with , where ; then ; in this case is isometric to .

(iii) with ; is isometric to .

(iv) with , where ; in this case is isometric to .

(v) , for with , where .

Theorem 5. The only generalized golden-shaped hypersurfaces of are as follows.
Case 1 (If ).
(i) If , for ,

In this case ; is totally geodesic and is in fact just .
(ii) If , for ,

which is isometric to .
(iii) If ,

which is isometric to .
(iv) If ,

which is isometric to .
Case 2 (If ).
(i) Suppose ; there are the following four cases.

(1)If , for ,

 In this case ; is totally geodesic and is in fact just .(2)If     , for ,

 which is isometric to .(3)If   ,

 which is isometric to .(4)If   . we get and :

 which is isometric to .

(ii) Suppose ; there are the following four cases.

(1)If , for ,

 In this case ; is totally geodesic and is in fact just .(2)If     , for ,

 which is isometric to .(3)If   ,

 which is isometric to .(4)If   ,

 which is isometric to .

(iii) Suppose , and , for .
If ,

with .

Proof.  
Case 1. Suppose , ; we distinguish the following four cases.
(i) If , we get and . Then for

In this case ; is totally geodesic and is in fact just .
(ii) If , we get and . Then, for , which is isometric to .
(iii) If , we get and . Then with which is isometric to .
(iv) If , we get and . Then which is isometric to .
Case 2 (If ).
(i) Suppose ; there are the following four cases.

(1)If , we get . That is to say, and . Then, for , . In this case ; is totally geodesic and is in fact just .(2)If     , we get and . Then, for , , which is isometric to .(3)If   , we get and . Then , which is isometric to .(4)If     , we get and . Then , which is isometric to .

(ii) Suppose ; there are the following four cases.

(1)If , we get . That is to say, and . Then, for , In this case ; is totally geodesic and is in fact just .(2)If     , we get and . Then, for , , which is isometric to .(3)If   , we get and . Then , which is isometric to .(4)If     , we get and . Then , which is isometric to .

(iii) Suppose , and , for .
If , we get and , with .

In the sequel, let be a hypersurface of which is isoparametric of type ; that is, the constant principal curvatures of are . It is known (see [12, 13]) that and for we obtain the following.

(i) For , is umbilical: for with . is isometric to .

(ii) For , is the generalized Clifford torus with and . From [14, p. 86] we have and with .

Theorem 6. The only generalized golden-shaped hypersurfaces of are as follows.
Case 1 (If ). Consider the following:

which is isometric to .
Case 2 (If ).
(i) For ,

which is isometric to ;

which is isometric to .
(ii) If and for ,

with and for .

Proof.  
Case 1. Suppose , ; we get and . Then which is isometric to .
Case 2. Suppose ; we distinguish the following two cases:
(i) If , we get and which is isometric to .
If , we get and which is isometric to .
(ii) If and, for , we get and , for , , , and . Consider

Then with .

Remark 7. Theorems 4, 5, and 6 are the generalization of the results in [6]. That is, when and , these three theorems are the same as theorems in [6].

3. Generalized Product-Shaped Hypersurfaces

In [4], it is proved that there exists a natural relationship between golden structures and almost product structures and this naturally leads to the following definition.

Definition 8. is called product-shaped hypersurface of if .

In this paper, we give the definition of generalized product-shaped hypersurface of .

Definition 9. is called generalized product-shaped hypersurface of if , where is the identity on the tangent bundle of and is a constant satisfying .

It follows that the principal curvatures of , as eigenvalues of , are the golden mean and if . The principal curvatures of , as eigenvalues of , especially, are the golden mean if . With similar computations as in Section 2, we obtain the following theorem.

Theorem 10. The product-shaped hypersurfaces are as follows:
(1) For

(i)if ,

 which is totally geodesic and is isometric to . In this case, the second fundamental form is zero;(ii)if ,

(2) For

(i)if , for ,

 in this case ; is totally geodesic and is in fact just ;(ii)if ,

 for which is isometric to ;(iii)if ,

 which is isometric to ;(iv) if ,

 which is isometric to .

(3) For

(i)if ,

 which is isometric to ;(ii)if ,

 which is isometric to . If , especially, and, for ,

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by NSFC (nos. 11371076 and 11431009).

References

S. K. Sen and R. P. Agarwal, “Golden ratio in science, as random sequence source, its computation and beyond,” Computers and Mathematics with Applications, vol. 56, no. 2, pp. 469–498, 2008.

R. Heyrovska, “The Golden ratio, ionic and atomic radii and bond lengths,” Molecular Physics, vol. 103, no. 6–8, pp. 877–882, 2005.

L. Marek-Crnjac, “The golden mean in the topology of four-manifolds, in conformal field theory, in the mathematical probability theory and in Cantorian space-time,” Chaos, Solitons and Fractals, vol. 28, no. 5, pp. 1113–1118, 2006.

M. Crasmareanu and C.-E. Hreţcanu, “Golden differential geometry,” Chaos, Solitons and Fractals, vol. 38, no. 5, pp. 1229–1238, 2008.

C.-E. Hretcanu and M.-C. Crasmareanu, “Applications of the golden ratio on Riemannian manifolds,” Turkish Journal of Mathematics, vol. 33, no. 2, pp. 179–191, 2009.

M. Crasmareanu, C.-E. Hreţcanu, and M.-I. Munteanu, “Golden- and product-shaped hypersurfaces in real space forms,” International Journal of Geometric Methods in Modern Physics, vol. 10, no. 4, Article ID 1320006, 2013.

C.-E. Hreţcanu and M. Crasmareanu, “Metallic structures on riemannian manifolds,” Revista de la Union Matematica Argentina, vol. 54, no. 2, pp. 15–27, 2013.

C. Özgür and N. Yilmaz Özgür, “Classification of metallic shaped hypersurfaces in real space forms,” Turkish Journal of Mathematics, vol. 39, no. 5, pp. 784–794, 2015.

G. Rotondaro, G. Sparano, and G. Vilasi, “Geometrical aspects of mixed torsionless tensor fields,” International Journal of Geometric Methods in Modern Physics, vol. 3, no. 7, pp. 1381–1394, 2006.

A. C. Coken, “On Euler's theorem in semi-Euclidean spaces ,” International Journal of Geometric Methods in Modern Physics, vol. 8, no. 5, pp. 1117–1129, 2011.

T. E. Cecil, “Isoparametric and Dupin hypersurfaces,” Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), vol. 4, article 062, 2008.

P. J. Ryan, “Hypersurfaces with parallel Ricci tensor,” Osaka Journal of Mathematics, vol. 8, pp. 251–259, 1971.

P. J. Ryan, “Homogeneity and some curvature conditions for hypersurfaces,” Tohoku Mathematical Journal, vol. 21, no. 3, pp. 363–388, 1969.

M. Djoric and M. Okumura, CR Submanifolds of Complex Projective Space, vol. 19 of Developments in Mathematics, Springer, New York, NY, USA, 2010.