`ISRN GeometryVolumeΒ 2012Β (2012), Article IDΒ 591296, 23 pageshttp://dx.doi.org/10.5402/2012/591296`
Research Article

On Some πΏ π -Finite-Type Euclidean Hypersurfaces

Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-134, Tehran, Iran

Received 24 May 2012; Accepted 26 July 2012

Academic Editors: I.Β Biswas, A.Β Ferrandez, and G.Β Martin

Copyright Β© 2012 Akram Mohammadpouri and S. M. B. Kashani. 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 study some -finite-type Euclidean hypersurfaces. We classify -1-type Euclidean hypersurfaces and -null-2-type Euclidean hypersurfaces with at most two distinct principal curvatures. We also prove that, under some mild restrictions, there exists no -null-3-type Euclidean hypersurface.

1. Introduction

The study of submanifolds of finite type began in the late seventies with Chen’s attempts to find the best possible estimate of the total mean curvature of compact submanifolds of Euclidean space and to find a notion of “degree” for submanifolds of Euclidean space (see [1] for details). Since then the subject has had a rapid development and so many mathematicians contribute to it; seeo the excellent survey of Chen [2]. By definition, an isometrically immersed submanifold is said to be of finite type if has a finite decomposition as , for some positive integer and , where is constant, and , , are nonconstant smooth maps, and is the Laplace operator of . If all 's are mutually different, then is said to be of -type. If, in particular, one of 's is zero, then is said to be of null -type.

As is well known, the Laplace operator of a hypersurface immersed into is an (intrinsic) second-order linear differential operator which arises naturally as the linearized operator of the first variation of the mean curvature for normal variations of the hypersurface. From this point of view, the Laplace operator can be seen as the first one of a sequence of operators , , where stands for the linearized operator of the first variation of the th mean curvature arising from normal variations of the hypersurface (see [3, 4]). These operators are given by for any , where denotes the th Newton transformation associated to the second fundamental from of the hypersurface and is the hessian of (see the next section for details).

In contrast to the operator , which is elliptic, in general the operators are not elliptic but they still share some nice properties with Laplacian of . Therefore, from this point of view, it seems natural and interesting to generalize the definition of finite-type hypersurface by replacing by and study the properties of such hypersurfaces. Having this idea, for the first time in [5], the second author inspired by the private communication with Alías, introduced such hypersurfaces and called them -finite-type hypersurfaces. In [5], he began to study generalized cylinders, ruled surfaces, and some revolution hypersurfaces from the point of view of -finiteness type.

In this paper which is a natural continuation of [5], we study some -finite-type hypersurfaces in the Euclidean space. The structure of the paper is as follows.

In Section 2, we give preliminaries.

In Section 3, we classify -1-type Euclidean hypersurfaces and -null-2-type Euclidean hypersurfaces with at most two distinct principal curvatures. We follow the work of Ferrández and Lucas [6], for Euclidean hypersurfaces of null-2-type. They proved that such hypersurfaces with at most two distinct principal curvatures are locally isometric to a generalized cylinder. This is a generalization of Chen's Theorem in [7], stating that null-2-type surfaces are circular cylinders. Here we generalize the result of [6] and prove that any Euclidean hypersurface of -null-2-type, , with at most two distinct principal curvatures, is locally isometric to a generalized cylinder. One of the important consequences of this theorem is the classification of conformally flat hypersurfaces of -null-2-type for . We also prove that there is no -null-2-type Euclidean hypersurface with at most two distinct principal curvatures. This is a generalization of Chen's result stating that there is no null-2-type plane curve [2, corollary of Theorem 7.3].

Section 4 is about -3-type Euclidean hypersurfaces. Here we prove that any -3-type surface has nonconstant Gaussian curvature. The result is an extension of Theorem 1 of [8], stating that any -type surface in has nonconstant mean curvature. Also motivated by the work of Ferrández and Lucas [6], stating that there is no Euclidean hypersurface of null-3-type with constant mean curvature and at most two principal curvatures, we prove that there is no Euclidean hypersurface of -null-3-type with constant and at most two distinct principal curvatures.

Our main results are the following theorems.

Theorem 1.1. minimal Euclidean hypersurfaces and open parts of hyperspheres are the only -1-type hypersurfaces in .

Theorem 1.2. There is no -null-2-type hypersurface in the Euclidean space , with at most two distinct principal curvatures.

Theorem 1.3. Let be an isometrically immersed Euclidean hypersurface with at most two distinct principal curvatures and multiplicities are greater than one. Then is of -null-2-type , if only if is isoparametric, so locally isometric to , .

Theorem 1.4. Let be an isometrically immersed Euclidean hypersurface with at most two distinct principal curvatures, one of them is simple. Then is of -null-2-type if only if is isoparametric, so locally isometric to for or for .

In Theorems 1.3 and 1.4, if we assume that is complete, then is globally isometric to a generalized cylinder.

Theorem 1.5. There is no -3-type surface in with constant Gaussian curvature.

Theorem 1.6. There is no hypersurface of -null-3-type in with constant and at most two distinct principal curvatures.

2. Preliminaries

In this section we recall some notions and prerequisites about (-finite type) hypersurfaces of the Euclidian space from [9, 10].

Consider an isometrically immersed hypersurface in the Euclidean space. We choose a local orthonormal frame in , with dual coframe , such that, at each point of , are tangent to and is the positively oriented unit normal vector. We will make use of the following convention on the ranges of indices: Then the structure equations of are given by When restricted to , we have and By Cartan's lemma, there exist functions such that This gives the second fundamental form of , as . The mean curvature is defined by . From (2.2)–(2.4) we obtain the structure equations of , (see [10]): and the Gauss equations where denotes the components of the Riemannian curvature tensor of .

Let denote the covariant derivative of . We have Thus, by exterior differentiation of (2.4), we obtain the Codazzi equation We choose such that Let be th mean curvature of , then we have And is called the Gauss-Kronecker curvature of . A hypersurface with zero th mean curvature in is called -minimal. To get more information about -minimal Euclidean hypersurfaces, the reader is referred to [11, 12].

The classical Newton transformations are defined inductively by the shape operator as for every , where denotes the identity transformation in . Equivalently, Note that by the Cayley-Hamilton theorem stating that any operator is annihilated by its characteristic polynomial, we have .

Since each is also a self-adjoint linear operator on each tangent plane which commutes with . Indeed, and can be simultaneously diagonalized: if are the eigenvectors of corresponding to the eigenvalues , respectively, then they are also the eigenvectors of with corresponding eigenvalues given by for every . We have the following formulae for the Newton transformations, [9]: where

Associated to each Newton transformation , we consider the second-order linear differential operator given by Here denotes the self-adjoint linear operator metrically equivalent to the Hessian of and is given by where , is the gradient of , and is the Levi-Civita connections on .

Now we recall the definition of an -finite-type hypersurface from [5], which is the basic notion of the paper. We also quote Proposition 2.2 from [5].

Definition 2.1. An isometrically immersed hypersurface is said to be of -finite type if has a finite decomposition , for some positive integer satisfying the condition that , , , where are smooth maps, , and is constant. If all 's are mutually different, is said to be of --type. An --type hypersurface is said to be null if some , is zero. The polynomial is called the minimal polynomial of for .

We should mention that similar to Proposition 1 of [13], if is of -finite type, then .

Proposition 2.2 (see [5]). If the isometrically immersed hypersurface is a generalized cylinder , then is of -null-1-type, if , and it is of -null-2-type, if .

3. πΏ π -2-Type Hypersurfaces

In this section we would like to follow [6] and consider -2-type hypersurfaces. In [6], it was proved that if is a nonminimal Euclidean hypersurface of 1- or 2-type and with constant mean curvature, then either (a) is an open piece of or (b) is of null-2-type. In Theorem 3.2 we obtain an extension of this result for -1 and 2-type Euclidean hypersurfaces. In Theorem 3.5, we prove that there is no -null-2-type hypersurface in with at most two distinct principal curvatures. It is a generalization of Chen's result stating that there is no null-2-type plane curve [2, corollary of Theorem 7.3]. In [7], Chen proved that null-2-type surfaces are circular cylinders. Later, Ferrández and Lucas got a generalization of Chen's theorem and showed that Euclidean hypersurfaces of null-2-type with at most two distinct principal curvatures are locally isometric to a generalized cylinder [6, Theorem 3.10]. In Theorems 3.11 and 3.12, we generalize this theorem and prove that any Euclidean hypersurface of -null-2-type , with at most two distinct principal curvatures is locally isometric to a generalized cylinder.

According to Takahashi's theorem [14], we know that minimal Euclidean hypersurfaces and open parts of hyperspheres are the only 1-type Euclidean hypersurfaces. In the next theorem, we show that -minimal hypersurfaces and open parts of hyperspheres are the only -1-type Euclidean hypersurfaces.

Theorem 3.1. -minimal Euclidean hypersurfaces and open parts of hyperspheres are the only -1-type hypersurfaces in .

Proof. In Theorem 1 of [9] Alías and Gürbüz have classified hypersurfaces in satisfying the general condition , where is a matrix and . Section 3 of [9] and Proposition 2.2 show that -minimal hypersurfaces and open parts of hyperspheres are the only hypersurfaces satisfying the condition , for some real constant .

Now we can state the next theorem.

Theorem 3.2. Let be an isometrically immersed hypersurface in the Euclidean space. If it is of -1-type or -2-type for some and the th mean curvature of is a nonzero constant, then one of the following two cases occurs:(a)is an open piece of ,(b) is of -null-2-type.

Proof. The case was proved by Chen and Lue [15, Theorem 1] and Ferrández and Lucas [6, Corollary 3.2], so that we may consider the case . Here, we will follow the techniques introduced by Chen and Lue [15, Theorem 1] for our context. According to Theorem 3.1, -minimal hypersurfaces and open parts of hyperspheres are the only -1-type hypersurfaces. If is of -2-type, the position vector of in has the following spectral decomposition: so We also have
The formula (3.4) holds since is a nonzero constant, see [9]. Therefore, by using (3.2), (3.3), and (3.4), we obtain
From (3.5) we have either or is normal to at every point of . If , then is of -null-2-type. If is normal to , then is a positive constant. In this case, is an open piece of centered at ; therefore, is of -1-type, which is not.

As a corollary we get the following result.

Corollary 3.3. If is an -2-type Euclidean hypersurface with constant th mean curvature, then is of null -2-type, in particular, if then the mean curvature is a nonzero constant.

Proof. By Theorem 3.2, is null, and we have the following spectral decomposition for the position vector of in : Since and is null, from formula (3.5) we have From (3.7) we obtain . Since and are nonzero and constant, so is also nonzero and constant.

To express an extension of Chen's result in [2] stating that there is no null-2-type plane curve, we need the next lemma.

Lemma 3.4. Let be an isometrically immersed Euclidean hypersurface which is of -null-2-type, then the Gauss-Kronecker curvature of is nonzero and constant.

Proof. Let us consider the open set , our objective is to show that is empty. From [9], we have From this relation and (3.2) we obtain that Therefore on we get But by the Cayley-Hamilton theorem we have , so which jointly with (3.10) yields on , which is a contradiction.

Now, we state one of the main results of the section as follows.

Theorem 3.5. There is no -null-2-type hypersurface in , with at most two distinct principal curvatures.

Proof. If is an -null-2-type hypersurface in with at most two principal curvatures, by applying Corollary 3.3 and Lemma 3.4 we obtain that the mean and Gauss-Kronecker curvatures are constant. Since is of -null-2-type, we conclude that has exactly two constant principal curvatures. From [16], is an open piece of for some , so the Gauss-Kronecker curvature of is zero; therefore, is of -null-1-type, which is a contradiction.

In [7], Chen proved that null 2-type surfaces are circular cylinders. In sharp contrast to this result, in the next corollary, we claim that there is not any -null-2-type surface at all! It is an obvious consequence of Theorem 3.5.

Corollary 3.6. There is no -null-2-type surface in .

By Theorem 3.5 and the following theorems from [17, 18], we can prove that there is no -null-2-type compact hypersurface in with everywhere nonzero sectional curvature.

Theorem 3.7 (see [17]). Let be an isometrically immersed compact hypersurface in the Euclidean space, then the following conditions are equivalent:(i) for some and ,(ii) and ,(iii) has parallel second fundamental form (i.e., is isoparametric with at most two principle curvatures),
where is the sectional curvature of .

The next theorem shows that there is no compact isometrically immersed hypersurface in with everywhere nonpositive sectional curvature.

Theorem 3.8 (see [18]). Let be a compact -dimensional Riemannian manifold, and let be a complete simply connected Riemannian manifold of dimension less than 2. If the sectional curvatures and of and satisfy , then cannot be immersed in .

Now we can express our result.

Proposition 3.9. There is no -null-2-type compact hypersurface in the Euclidean space with everywhere nonzero sectional curvature.

Proof. By Theorem 3.8, we assume that . By lemma 4.3, is nonzero and constant. So we get from Theorem 3.7 that is isoparametric with at most two distinct principle curvatures, and this is a contradiction with Theorem 3.5.

By considering two different cases we prove that -null-2-type Euclidean hypersurfaces with at most two distinct principal curvatures are circular cylinders. Case  I: The multiplicities are greater than one. Case  II: one of the principal curvatures is simple.

We use the following Lemma from [19], for the proof of claim.

Lemma 3.10. (see [19], Theorem 2 and its corollary). Let be an -dimensional hypersurface in the Euclidean space such that multiplicities of principal curvatures are constant. Then the distribution of the space of principal vectors corresponding to each principal curvature is completely integrable. In particular, if the multiplicity of a principal curvature is greater than one, then this principal curvature is constant on each integral submanifold of the corresponding distribution of the space of principal vectors.

Theorem 3.11. Let be an isometrically immersed hypersurface with at most two distinct principal curvatures and multiplicities are greater than one. Then is of -null-2-type if only if is isoparametric so locally isometric to , .

Proof. Let is of -null-2-type. If is totally umbilical, then is a piece of or . By using Theorem 3.1, and are of -1-type, so is not totally umbilical. Therefore has two distinct principal curvatures of multiplicities and , .
Let us consider the open set Our objective is to show that is empty.
Consider to be a local orthonormal frame of principal directions of on such that for every . We assume that Therefore from (2.13) we have with
So we get We obtain from (2.10) that Since is of -null-2-type, the position vector field satisfies the following equation for some constant , So by using the formulae of and from [9], we can write From (3.19) we get Since , it follows from the inductive definition of that (3.20) is equivalent to Therefore, writing we see that (3.21) is equivalent to for every . So there is no loss of generality, assuming that
Let us denote the integral submanifolds through corresponding to and by and , respectively. From Lemma 3.10, we know that is constant on . Equations (3.16), (3.17), and (3.24) imply that is constant on . Again by Lemma 3.10, we get that is constant on . It now follows from [20, page 182, I] that is locally isometric to the Riemannian product of the maximal integral manifolds and . Therefore, is constant on . By the same assertion, we know that is constant on , so is constant on , which is a contradiction. Hence is constant and nonzero on . From (3.19), we obtain that ; therefore, is constant. By the fact that has two principal curvatures and , are constant, we get that the principal curvatures are constant. So is isoparametric.
A classical result of Segre [16] states that isoparametric hypersurfaces in are locally isometric to , and circular cylinder. On the other hand, since is of -null-2-type, by using Proposition 2.2, we conclude that is locally isometric to , .

Theorem 3.12. Let be an isometrically immersed Euclidean hypersurface with at most two distinct principal curvatures, one of them is simple. Then is of -null-2-type, , if only if is isoparametric, so locally isometric to or for .

Proof. Let be of -null-2-type . If is totally umbilic, then is a piece of or . By using Theorem 3.1, and are of -1-type, so cannot be totally umbilic. Therefore suppose has two distinct principal curvatures of multiplicities and .
Let us consider the open set.
We want to prove that is empty. If , then we express as a polynomial in (the nonsimple principal curvature of ) with constant coefficients, after that we express as a constant multiple of the simple principal curvature of . By using Otsuki's Lemma (Lemma 3.10), the structure equations of , and the fact that is of -null-2-type, we get that satisfies a polynomial with constant coefficients. So is constant; hence, is constant, a contradiction with . So is empty.
Here is the detailed treatment of the proof.
With the assumption that , consider to be a local orthonormal frame of principal directions of on such that for every . We assume Therefore from (2.10) we have with So we get We obtain from (2.13) that Since is of -null-2-type, the position vector field satisfies the following equation for some constant , So by using the formulae of and from [9], we can write From (3.32) we get Since , it follows from the inductive definition of that (3.33) is equivalent to Therefore, by the formula we see that (3.34) is equivalent to for every . Hence, for every such that on we get So for the expression in (3.35) we consider two cases.
Case  1. , by (4.31), we obtain that Case  2. , so on we have for some . By (4.31) and using the fact that are the eigenvalues of and the formula of , we obtain that Both cases require the same calculation, so we consider just Case I.
By Lemma 3.10, let us denote the maximal integral submanifold through , corresponding to by . We write Then Lemma 3.10 implies that . We can assume that on , then (3.30) and (3.38) yield By means of (2.7) and (2.9), we obtain that We adopt the notational convention that .
From (3.41) and (3.42), we have Combining this with (2.8) and the formula we obtain from (3.41) Therefore we have
Notice that we may consider to be locally a function of the parameter , where is the arc length of an orthogonal trajectory of the family of the integral submanifolds corresponding to . We may put .
Thus, for , we have so from(3.45), we get According to the structure equations of and (3.48), we may compute Then we obtain from the two equalities above that Combining (3.51) with (3.41), we have Let us define a function , by , then (3.52) reduces to Integrating (3.53), we obtain where is the constant of integration.
Equation (3.54) is equivalent to Now we use the definition of to compute . So we need to compute , , , and .
From (3.51) we have By using (3.48) and (3.56) we obtain Now we compute and : From (3.57) and (3.58), we get Since is of -null-2-type, hence from (3.32), we get Combining (3.58) and (3.60), we have Equation (3.52) is equivalent to Thus, putting together (3.61) and (3.62) one has We deduce, using (3.55), (3.63), and (3.38), that is locally constant on , which is a contradiction with the definition of . Hence is constant and nonzero on . From the discussion as in the last part of the proof of the Theorem 3.11, we get the result.

An important consequence of the theorem is the classification of conformally flat hypersurfaces of -null-2-type for .

Definition 3.13 (see [21]). A Riemannian manifold is called conformally flat if it is locally conformally equivalent to a Euclidean space , that is, if every point of has a neighborhood which is conformal to an open set in the Euclidean space . A submanifold of the Euclidean space is said to be conformally flat if with the induced metric for is conformally flat.

The dimension of the hypersurface plays an important role in the study of conformally flat Euclidean hypersurfaces. For , the existence of isothermal coordinates means that any Riemannian surface is conformally flat. For , the result of Cartan-Schouten stats that a conformally flat hypersurface is characterized with two principal curvatures that one multiplicity at least (see [21] for more details). This significant fact is crucial in our classification of -null-2-type conformally flat Euclidean hypersurfaces for .

As a simple consequence of Theorem 3.12, we obtain the following nice corollary which is an extension of the result of [22].

Corollary 3.14. Let be a conformally flat hypersurface of , . Then is of -null-2-type, if only if is locally isometric to the cylinder .

4. πΏ π β 1 -3-Type Hypersurfaces in β π + 1

In Theorem 1 of [8], Hasanis and Vlachos proved that there is no 3-type surface in with constant mean curvature. Later Ferrández and Lucas proved that there is no Euclidean hypersurface of null 3-type with constant mean curvature with at most two distinct principal curvatures [6, Theorem 3.5]. Here we follow Hasanis and Vlachos's work to consider -3-type surfaces. Also we get a generalization of Theorem 3.5 of [6] and prove that there is no Euclidean hypersurface of -null-3-type with constant and at most two distinct principal curvatures.

Theorem 4.1. There is no -3-type surface in with constant Gaussian curvature.

Proof. Let be the position vector of an -3-type surface , and set for the corresponding distance function. Let be the unit normal vector field of . We decompose the position vector of as follows where is the tangential component, then the gradient of on is given by . By taking covariant derivative of (4.1) and using the Gauss and Weingarten formula, we have for every tangent vector field . Therefore by using (2.14) and (2.15) we have If has constant Gaussian curvature , it is nonzero. In fact, if , then according to Theorem 3.1, must be of -1-type. So we assume that has nonzero constant Gaussian curvature and is of -3-type. By computing , , one finds that Since is of -3-type, we have where , , . Taking into account (4.4), and comparing the tangential and normal components of in (4.5), (4.6) we obtain the following useful equations: For brevity we set . We distinguish the following two cases.
Case  I . That is is of -null-3-type, then, by (4.7), . By the Cayley-Hamilton theorem, we have , and from the inductive definition of we get that So Since we conclude that is constant, so is isoparametric, then by Theorem 3.1, has exactly two constant principal curvatures, therefore from Lemma 2.B of [17] we know that one of the principal curvatures has to be zero, so , this is a contradiction.
Case II . Then from (4.7) and (4.9) we obtain that so we have ; hence, is constant, therefore from which, by using (4.3), one finds that from which, by using (4.8) we obtain where we set Since , by (4.11), (4.14) we have . If is identically zero on an open subset , then should be a sphere, thus of -1-type, a contradiction. Hence is a principal direction with corresponding principal curvature . Therefore, the other principal curvature is , thus the Gaussian curvature has to be or Substituting from (4.15) in (4.16), we get the following polynomial equation for : This shows that is constant, a contradiction. So is nonconstant.

We use the following theorem from [10] to prove of Theorem 4.3.

Theorem 4.2 (see [10]). Let be an -dimensional hypersurface in a Euclidean space , with constant th mean curvature and two distinct principal curvatures. If the multiplicities are greater than one, then M is locally isometric to , .

The next theorem will generalize the Ferrández and Lucas result in [6].

Theorem 4.3. There is no hypersurface of -null-3-type in with constant and at most two distinct principal curvatures.

Proof. Let be the position vector of an -null-3-type hypersurface , and let be the unit normal vector field of . If has constant , it is nonzero. In fact, if , then according to Theorem 3.1, must be of -1-type. So, we assume that has nonzero constant and is of -null-3-type.
By computing , one finds that Since is of -null-3-type, we have where , , .
We prove the theorem in three steps.
Step I . From (4.18) and comparing the tangential and normal components of in (4.19), (4.20) and by the Cayley-Hamilton theorem we obtain the following useful equation: Since , we conclude that is constant; Therefore, has at most two constant principal curvatures. From [16] and Theorem 3.1, is an open piece of for some , so Gauss-Kronecker curvature of is zero, therefore is of -null-1-type, which is a contradiction.
Step II . has at most two distinct principal curvatures with multiplicities greater than one.
The conclusion is directly obtained from Theorem 4.2 and Proposition 2.2.
Step III . has two distinct principal curvatures, one of them is simple.
By (4.18) and comparing the tangential and normal components of in (4.19), (4.20) and using the definition of , we obtain that It gives Consider to be a local orthonormal frame of principal directions of on such that for every . We assume that Therefore from (2.13) we have with
So we get We obtain from (2.10) that We can write we see that (4.29) is equivalent to for every .
If is constant, by the fact that has two principal curvature and , are constant, we conclude that is isoparametric, by using [16] and Proposition 2.2, this is a contradiction. Therefore, for every such that we get So by using (4.27) and (4.28), we conclude that is isoparametric and this is a contradiction.

From Theorem 4.3, we can easily obtain the following corollary.

Corollary 4.4. There is no -null-3-type conformally flat Euclidean hypersurface with constant .

References

1. B. Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Mathematics, World Scientific Publishing, Singapore, 1984.
2. B. Y. Chen, βA report on submanifolds of finite type,β Soochow Journal of Mathematics, vol. 22, no. 2, pp. 117β337, 1996.
3. R. C. Reilly, βVariational properties of functions of the mean curvatures for hypersurfaces in space forms,β Journal of Differential Geometry, vol. 8, no. 3, pp. 465β477, 1973.
4. H. Rosenberg, βHypersurfaces of constant curvature in space forms,β Bulletin des Sciences Mathématiques, vol. 117, no. 2, pp. 211β239, 1993.
5. S. M. B. Kashani, βOn some ${L}_{1}$-fnite type (hyper)surfaces in ${ℝ}^{n+1}$,β Bulletin of the Korean Mathematical Society, vol. 46, no. 1, pp. 35β43, 2009.
6. A. Ferrández and P. Lucas, βNull finite type hypersurfaces in space forms,β Kodai Mathematical Journal, vol. 14, no. 3, pp. 406β419, 1991.
7. B. Y. Chen, βNull 2-type surfaces in E3 are circular cylinders,,β Kodai Mathematical Journal, vol. 11, no. 2, pp. 295β299, 1988.
8. Th. Hasanis and Th. Vlachos, βSurfaces of finite type with constant mean curvature,β Kodai Mathematical Journal, vol. 16, no. 2, pp. 244β252, 1993.
9. L. J. Alías and N. Gürbüz, βAn extension of Takahashi theorem for the linearized operators of the higher order mean curvatures,β Geometriae Dedicata, vol. 121, pp. 113β127, 2006.
10. G. Wei, βComplete hypersurfaces in a Euclidean space ${ℝ}^{n+1}$ with constant th mean curvature,β Differential Geometry and Its Applications, vol. 26, no. 3, pp. 298β306, 2008.