Research Article | Open Access

# Weingarten and Linear Weingarten Type Tubular Surfaces in

**Academic Editor:**Victoria Vampa

#### Abstract

We study tubular surfaces in Euclidean 3-space satisfying some equations in terms of the Gaussian curvature, the mean curvature, the second Gaussian curvature, and the second mean curvature. This paper is a completion of Weingarten and linear Weingarten tubular surfaces in Euclidean 3-space.

#### 1. Introduction

Let and be smooth functions on a surface in Euclidean 3-space . The Jacobi function formed with is defined by where and . In particular, a surface satisfying the Jacobi equation with respect to the Gaussian curvature and the mean curvature on a surface is called a Weingarten surface or a -surface. Also, if a surface satisfies a linear equation with respect to and , that is, , ,, then it is said to be a linear Weingarten surface or a -surface [1].

When the constant , a linear Weingarten surface reduces to a surface with constant Gaussian curvature. When the constant , a linear Weingarten surface reduces to a surface with constant mean curvature. In such a sense, the linear Weingarten surfaces can be regarded as a natural generalization of surfaces with constant Gaussian curvature or with constant mean curvature [1].

If the second fundamental form of a surface in is nondegenerate, then it is regarded as a new pseudo-Riemannian metric. Therefore, the Gaussian curvature is the second Gaussian curvature on [1].

For a pair , of the curvatures , , and of in , if satisfies by , then it said to be a -Weingarten surface or -linear Weingarten surface, respectively [1].

Several geometers have studied -surfaces and -surfaces and obtained many interesting results [1–9]. For the study of these surfaces, Kühnel and Stamou investigated ruled -Weingarten surfaces in Euclidean 3-space [7, 9]. Also, Baikoussis and Koufogiorgos studied helicoidal -Weingarten surfaces [10]. Dillen, and sodsiri, and Kühnel, gave a classification of ruled -Weingarten surfaces in Minkowski 3-space , where [2–4]. Koufogiorgos, Hasanis, and Koutroufiotis investigated closed ovaloid -linear Weingarten surfaces in [11, 12]. Yoon, Blair and Koufogiorgos classified ruled -linear Weingarten surfaces in [8, 13, 14]. Ro and Yoon studied tubes in Euclidean 3-space which are , , -Weingarten, and linear Weingarten tubes, satisfying some equations in terms of the Gaussian curvature, the mean curvature, and the second Gaussian curvature [1].

Following the Jacobi equation and the linear equation with respect to the Gaussian curvature , the mean curvature , the second Gaussian curvature , and the second mean curvature , an interesting geometric question is raised: classify all surfaces in Euclidean 3-space satisfying the conditions where , and .

In this paper, we would like to contribute the solution of the above question by studying this question for tubes or tubular surfaces in Euclidean 3-space .

#### 2. Preliminaries

We denote a surface in by Let be the standard unit normal vector field on a surface defined by where . Then, the first fundamental form and the second fundamental form of a surface are defined by, respectively, where [14]. On the other hand, the Gaussian curvature and the mean curvature are respectively. From Brioschi's formula in a Euclidean 3-space, we are able to compute and of a surface by replacing the components of the first fundamental form , , and by the components of the second fundamental form , , and , respectively [14]. Consequently, the second Gaussian curvature of a surface is defined by and the second mean curvature of a surface is defined by where and stand for “” and “”, respectively, and , where are the coefficients of the second fundamental form [3, 4].

*Remark 2.1. *It is well known that a minimal surface has a vanishing second Gaussian curvature, but that a surface with the vanishing second Gaussian curvature need not to be minimal [14].

#### 3. Weingarten Tubular Surfaces

*Definition 3.1 .. *Let be a unit-speed curve. A tubular surface of radius about is the surface with parametrization, where , are the principal normal and the binormal vectors of , respectively [1].

The curvature and the torsion of the curve are denoted by , . Then, Frenet formula of is defined by [1]. Furthermore, we have the natural frame given by

The components of the first fundamental form are where .

On the other hand, the unit normal vector field is obtained by

As , is the sign of such that if , then and if , then . From this, the components of the second fundamental form of are given by

If the second fundamental form is nondegenerate, , that is, , and are nowhere vanishing. In this case, we can define formally the second Gaussian curvature and the second mean curvature on On the other hand, the Gauss curvature , the mean curvature , the second Gaussian curvature and the second mean curvature are obtained by using (2.5), (2.6) and (2.7) as follows: and where the coefficients are

Differentiating , , , and with respect to and , after straightforward calculations, we get, and where are and where the coefficients are

Now, we consider a tubular surface in satisfying the Jacobi equation . By using (3.9), (3.13), and (3.15), we obtain in the following form: with respect to the Gaussian curvature and the second mean curvature , where Then, by , (3.17) becomes

Hence, we have the following theorem.

Theorem 3.2. *Let be a tubular surface defined by (3.1) with nondegenerate second fundamental form. is a -Weingarten surface if and only if is a tubular surface around a circle or a helix.*

*Proof. *Let us assume that is a -Weingarten surface, then the Jacobi equation (3.19) is satisfied. Since polynomial in (3.19) is equal to zero for every , all its coefficients must be zero. Therefore, the solutions of are , and , that is, is a tubular surface around a circle or a helix, respectively.

Conversely, suppose that is a tubular surface around a circle or a helix, then it is easily to see that is satisfied for the cases both , and , . Thus M is a -Weingarten surface.

We suppose that a tubular surface with nondegenerate second fundamental form in is -Weingarten surface. From (3.10), (3.13), and (3.15), is
with respect to the variable , where

Then, by , (3.22) becomes in following form:

Thus, we state the following theorem.

Theorem 3.3. *Let be a tubular surface defined by (3.1) with nondegenerate second fundamental form. is a -Weingarten surface if and only if M is a tubular surface around a circle or a helix.*

*Proof. *Considering and by using (3.13), one can obtaine the solutions , , and , of the equations for all . Thus, it is easly proved that is a -Weingarten surface if and only if *M* is a tubular surface around a circle or a helix.

We consider a tubular surface is -Weingarten surface with nondegenerate second fundamental form in . By using (3.11), (3.12), (3.13), and (3.15), is where

Since , then (3.23) becomes in following form:

Hence, we have the following theorem.

Theorem 3.4. *Let be a tubular surface defined by (3.1) with nondegenerate second fundamental form. is a -Weingarten surface if and only if M is a tubular surface around a circle or a helix.*

*Proof. *It can be easly proved similar to Theorems 3.2 and 3.3.

Consequently, we can give the following main theorem for the end of this part.

Theorem 3.5. *Let , and let be a tubular surface defined by (3.1) with nondegenerate second fundamental form. is a -Weingarten surface if and only if M is a tubular surface around a circle or a helix.*

Thus, the study of Weingarten tubular surfaces in 3-dimensional Euclidean space is completed with [1].

#### 4. Linear Weingarten Tubular Surfaces

In last part of this paper, we study on , , , , , , , and linear Weingarten tubular surfaces in . , , and linear Weingarten tubes are studied in [1].

Let , ,, , and be constants. In general, a linear combination of , , and can be constructed as

By the straightforward calculations, we obtained the reduced form of (4.1) where the coefficients are

Then, , , , , and are zero for any . If or , from , one has . Hence, we can give the following theorems.

Theorem 4.1. *Let . Then, there are no -linear Weingarten tubular surfaces in Euclidean 3-space defined by (3.1) with nondegenerate second fundamental form.*

Theorem 4.2. *Let Then, there are no -linear Weingarten tubular surfaces in Euclidean 3-space defined by (3.1) with nondegenerate second fundamental form.*

Theorem 4.3. *Let be a tubular surface defined by (3.1) with nondegenerate second fundamental form. Then, there are no -linear Weingarten surface in Euclidean 3-space.*

Consequently, the study of linear Weingarten tubular surfaces in 3-dimensional Euclidean space is completed with [1].

#### Acknowledgments

The authors would like to thank the referees for the helpful and valuable suggestions.

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#### Copyright

Copyright © 2011 Yılmaz Tunçer et al. 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.