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.