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Mathematical Problems in Engineering
Volume 2011, Article ID 191849, 11 pages
http://dx.doi.org/10.1155/2011/191849
Research Article

Weingarten and Linear Weingarten Type Tubular Surfaces in E3

1Department of Mathematics, Arts and Science Faculty, Usak University, 64200 Usak, Turkey
2Department of Mathematic Education and RINS, Gyeongsang National University, Jinju 660701, Republic of Korea

Received 5 January 2011; Revised 30 March 2011; Accepted 26 April 2011

Academic Editor: Victoria Vampa

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.

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 𝑓 and 𝑔 be smooth functions on a surface 𝑀 in Euclidean 3-space E3. The Jacobi function Φ(𝑓,𝑔)formed with 𝑓,𝑔is defined by𝑓Φ(𝑓,𝑔)=det𝑠𝑓𝑡𝑔𝑠𝑔𝑡,(1.1) where 𝑓𝑠=𝜕𝑓/𝜕𝑠 and 𝑓𝑡=𝜕𝑓/𝜕𝑡. In particular, a surface satisfying the Jacobi equation Φ(𝐾,𝐻)=0 with respect to the Gaussian curvature 𝐾 and the mean curvature 𝐻on a surface 𝑀 is called a Weingarten surface or a 𝑊-surface. Also, if a surface satisfies a linear equation with respect to 𝐾 and 𝐻, that is, 𝑎𝐾+𝑏𝐻=𝑐, (𝑎,𝑏,𝑐)(0,0,0),𝑎,𝑏,𝑐𝐼𝑅, then it is said to be a linear Weingarten surface or a LW-surface [1].

When the constant 𝑏=0, a linear Weingarten surface 𝑀 reduces to a surface with constant Gaussian curvature. When the constant 𝑎=0, a linear Weingarten surface 𝑀 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 II of a surface 𝑀 in E3 is nondegenerate, then it is regarded as a new pseudo-Riemannian metric. Therefore, the Gaussian curvature 𝐾II is the second Gaussian curvature on 𝑀[1].

For a pair (𝑋,𝑌), 𝑋𝑌,of the curvatures 𝐾, 𝐻, 𝐾II and 𝐻II of 𝑀 in E3, if 𝑀satisfies Φ(𝑋,𝑌)=0 by 𝑎𝑋+𝑏𝑌=𝑐, then it said to be a (𝑋,𝑌)-Weingarten surface or (𝑋,𝑌)-linear Weingarten surface, respectively [1].

Several geometers have studied 𝑊-surfaces and 𝐿𝑊-surfaces and obtained many interesting results [19]. For the study of these surfaces, Kühnel and Stamou investigated ruled (𝑋,𝑌)-Weingarten surfaces in Euclidean 3-space E3 [7, 9]. Also, Baikoussis and Koufogiorgos studied helicoidal (𝐻,𝐾II)-Weingarten surfaces [10]. Dillen, and sodsiri, and Kühnel, gave a classification of ruled (𝑋,𝑌)-Weingarten surfaces in Minkowski 3-space E31, where (𝑋,𝑌){𝐾,𝐻,𝐾II} [24]. Koufogiorgos, Hasanis, and Koutroufiotis investigated closed ovaloid (𝑋,𝑌)-linear Weingarten surfaces in E3 [11, 12]. Yoon, Blair and Koufogiorgos classified ruled (𝑋,𝑌)-linear Weingarten surfaces in E3 [8, 13, 14]. Ro and Yoon studied tubes in Euclidean 3-space which are (𝐾,𝐻), (𝐾,𝐾II), (𝐻,𝐾II)-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 𝐾, the mean curvature 𝐻, the second Gaussian curvature 𝐾II, and the second mean curvature 𝐻II, an interesting geometric question is raised: classify all surfaces in Euclidean 3-space satisfying the conditionsΦ(𝑋,𝑌)=0,𝑎𝑋+𝑏𝑌=𝑐,(1.2) where 𝑋,𝑌{𝐾,𝐻,𝐾II,𝐻II}, 𝑋𝑌 and (𝑎,𝑏,𝑐)(0,0,0).

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 E3.

2. Preliminaries

We denote a surface 𝑀 in E3 by 𝑀𝑚(𝑠,𝑡)=1(𝑠,𝑡),𝑚2(𝑠,𝑡),𝑚3.(𝑠,𝑡)(2.1) Let 𝑈be the standard unit normal vector field on a surface 𝑀 defined by𝑀𝑈=𝑠𝑀𝑡𝑀𝑠𝑀𝑡,(2.2) where 𝑀𝑠=𝜕𝑀(𝑠,𝑡)/𝜕𝑠. Then, the first fundamental form Iand the second fundamental form II of a surface 𝑀are defined by, respectively,I=𝐸𝑑𝑠2+2𝐹𝑑𝑠𝑑𝑡+𝐺𝑑𝑡2,II=𝑒𝑑𝑠2+2𝑓𝑑𝑠𝑑𝑡+𝑔𝑑𝑡2,(2.3) where𝐸=𝑀𝑠,𝑀𝑠,𝐹=𝑀𝑠,𝑀𝑡,𝐺=𝑀𝑡,𝑀𝑡,𝑒=𝑀𝑠,𝑈𝑠=𝑀𝑠𝑠,𝑈,𝑓=𝑀𝑠,𝑈𝑡=𝑀𝑠𝑡,𝑈,𝑔=𝑀𝑡,𝑈𝑡=𝑀𝑡𝑡,𝑈,(2.4) [14]. On the other hand, the Gaussian curvature 𝐾 and the mean curvature 𝐻are 𝐾=𝑒𝑔𝑓2𝐸𝐺𝐹2,𝐻=𝐸𝑔2𝐹𝑓+𝐺𝑒2𝐸𝐺𝐹2,(2.5) respectively. From Brioschi's formula in a Euclidean 3-space, we are able to compute 𝐾II and 𝐻II of a surface by replacing the components of the first fundamental form 𝐸, 𝐹, and 𝐺 by the components of the second fundamental form 𝑒, 𝑓, and 𝑔, respectively [14]. Consequently, the second Gaussian curvature 𝐾II of a surface is defined by𝐾II=1||||𝑒𝑔𝑓22||||||||||12𝑒𝑡𝑡+𝑓𝑠𝑡12𝑔𝑠𝑠12𝑒𝑠𝑓𝑠12𝑒𝑡𝑓𝑡12𝑔𝑠1𝑒𝑓2𝑔𝑡||||||||||||||||||||01𝑓𝑔2𝑒𝑡12𝑔𝑠12𝑒𝑡1𝑒𝑓2𝑔𝑠||||||||||,𝑓𝑔(2.6) and the second mean curvature 𝐻II of a surface is defined by𝐻II1=𝐻2||detII||𝑖,𝑗𝜕𝜕𝑢𝑖||detII||𝐿𝑖𝑗𝜕𝜕𝑢𝑗ln||𝐾||,(2.7) where 𝑢𝑖 and 𝑢𝑗 stand for “𝑠” and “𝜃=𝑡”, respectively, and 𝐿𝑖𝑗=(𝐿𝑖𝑗)1, where 𝐿𝑖𝑗 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 𝛼[𝑎,𝑏]E3 be a unit-speed curve. A tubular surface of radius 𝜆>0 about 𝛼 is the surface with parametrization[],𝑀(𝑠,𝜃)=𝛼(𝑠)+𝜆𝑁(𝑠)cos𝜃+𝐵(𝑠)sin𝜃(3.1)𝑎𝑠𝑏, where 𝑁(𝑠), 𝐵(𝑠) are the principal normal and the binormal vectors of 𝛼, respectively [1].

The curvature and the torsion of the curve 𝛼 are denoted by 𝜅, 𝜏. Then, Frenet formula of 𝛼(𝑠) is defined by𝑇𝑁𝐵=𝑇𝑁𝐵,0𝜅0𝜅0𝜏0𝜏0(3.2) [1]. Furthermore, we have the natural frame {𝑀𝑆,𝑀𝜃} given by𝑀𝑠𝑀=(1𝜆𝜅cos𝜃)𝑇𝜆𝜏sin𝜃𝑁+𝜆𝜏cos𝜃𝐵,𝜃=𝜆sin𝜃𝑁+𝜆cos𝜃𝐵.(3.3)

The components of the first fundamental form are𝐸=𝜆2𝜏2+𝜎2,𝐹=𝜆2𝜏,𝐺=𝜆2,(3.4) where 𝜎=1𝜆𝜅cos𝜃.

On the other hand, the unit normal vector field 𝑈 is obtained by 𝑀𝑈=𝑠𝑀𝜃𝑀𝑠𝑀𝜃=𝜀cos𝜃𝑁𝜀sin𝜃𝐵.(3.5)

As 𝜆>0, 𝜀 is the sign of 𝜎 such that if 𝜎<0, then 𝜀=1 and if 𝜎>0, then 𝜀=1. From this, the components of the second fundamental form of 𝑀 are given by𝑒=𝜀𝜆𝜏2𝜀𝜅cos𝜃𝜎,𝑓=𝜀𝜆𝜏,𝑔=𝜀𝜆.(3.6)

If the second fundamental form is nondegenerate, 𝑒𝑔𝑓20, that is, 𝜅, 𝜎 and cos𝜃 are nowhere vanishing. In this case, we can define formally the second Gaussian curvature 𝐾II and the second mean curvature 𝐻II on 𝑀.On the other hand, the Gauss curvature 𝐾, the mean curvature 𝐻, the second Gaussian curvature 𝐾II and the second mean curvature 𝐻II are obtained by using (2.5), (2.6) and (2.7) as follows:𝐾=𝜅cos𝜃,𝜆𝜎𝐻=𝜀(12𝜆𝜅cos𝜃),𝐾2𝜆𝜎II=𝜀𝜅cos2𝜃6𝜅𝜆cos3𝜃+4𝜅2𝜆2cos4𝜃+1,𝐻4cos𝜃𝜎II=18𝜀𝜆𝜅3cos3𝜃𝜎36𝑖=0𝑔𝑖cos𝑖𝜃,(3.7) and where the coefficients 𝑔𝑖 are𝑔0=3𝜆2𝜅2𝜏2,𝑔1𝜅=2𝜆𝜅𝑠𝜏𝜅𝜏𝑠sin𝜃1+6𝜆2𝜏2𝜅3,𝑔2=2𝜆2𝜅2𝜅𝜏𝑠4𝜅𝑠𝜏3𝜅sin𝜃+𝜆𝑠2+3𝜅42𝜅𝜅𝑠𝑠𝜅2𝜏2,𝑔3=2𝜆2𝜅2𝜅2𝜏2𝜅3+𝜅𝜅𝑠𝑠𝜅3𝑠2𝜅3,𝑔4=16𝜆𝜅4,𝑔5=20𝜆2𝜅5,𝑔6=8𝜆3𝜅6.(3.8)

Differentiating 𝐾, 𝐾II, 𝐻, and 𝐻II with respect to 𝑠 and 𝜃, after straightforward calculations, we get, 𝐾𝑠𝜅=𝑠cos𝜃𝜆𝜎2,𝐾𝜃=𝜅sin𝜃𝜆𝜎2,(3.9)𝐻𝑠=𝜀𝜅𝑠cos𝜃2𝜎2,𝐻𝜃=𝜀𝜅sin𝜃2𝜎2,(3.10)𝐾II𝑠=𝜀𝜅𝑠8𝜆3𝜅3cos5𝜃18𝜆2𝜅2cos4𝜃+12𝜆𝜅cos3𝜃cos2𝜃14cos𝜃𝜎2,(3.11)𝐾II𝜃=𝜀𝜅sin𝜃8𝜆3𝜅3cos5𝜃18𝜆2𝜅2cos4𝜃+12𝜆𝜅cos3𝜃+sin2𝜃2𝜆𝜅cos𝜃4cos2𝜃𝜎2,(3.12)𝐻II𝑠=18𝜀𝜅4cos3𝜃𝜎46𝑖=0𝑓𝑖cos𝑖𝜃,(3.13) and where 𝑓𝑖 are 𝑓0=3𝜅2𝜏𝜅𝑠𝜏2𝜅𝜏𝑠,𝑓1=2𝜅2𝜅𝑠𝜅𝑠𝜏𝜅𝜏𝑠𝜅𝜅𝑠𝑠𝜏sin𝜃+3𝜅𝜏𝑠2𝜅𝑠𝜏6𝜆𝜅3𝑓𝜏,2=2𝜆𝜅29𝜅𝑠𝜅𝜏𝑠𝜅𝑠𝜏+2𝜅𝜅𝑠𝑠𝜏sin𝜃+6𝜆2𝜅4𝜏3𝜅𝑠𝜏2𝜅𝜏𝑠+𝜅𝑠9𝜅𝑠210𝜅𝜅𝑠𝑠+𝜅2𝜏2𝜅𝜏𝑠𝜅𝑠𝜏,𝑓3=2𝜆2𝜅3𝜅𝑠16𝜅𝑠𝜏7𝜅𝜏𝑠4𝜅𝜏𝜅𝑠𝑠sin𝜃+2𝜆𝜅15𝜅𝜅𝑠𝜅𝑠𝑠𝜅15𝑠2+𝜅4𝜅𝑠+𝜅2𝜏2𝜏𝜅𝑠5𝜅𝜏𝑠,𝑓4=2𝜆2𝜅25𝜅𝑠3𝜅𝑠22𝜅𝜅𝑠𝑠+2𝜅2𝜏2𝜅𝜏𝑠3𝜏𝜅𝑠+𝜅4𝜅𝑠2𝜅4𝜅𝑠,𝑓5=6𝜆𝜅5𝜅𝑠,𝑓6=4𝜆2𝜅6𝜅𝑠,(3.14)𝐻II𝜃=18𝜀𝜆𝜅3cos4𝜃𝜎46𝑖=0𝑖cos𝑖𝜃,(3.15) and where the coefficients 𝑖 are 0=9𝜆𝜅2𝜏2sin𝜃,1=2𝜅31+15𝜆2𝜏2sin𝜃+4𝜆𝜅𝜅𝜏𝑠𝜅𝑠𝜏,2=𝜆2𝜅𝜅𝑠𝑠8𝜅4+𝜅2𝜏2130𝜆2𝜅2𝜅3𝑠2sin𝜃+6𝜆2𝜅23𝜅𝑠𝜏2𝜅𝜏𝑠,3=4𝜆2𝜅2𝜅4𝜅2𝜏22𝜅𝜅𝑠𝑠𝜅+3𝑠2𝜅sin𝜃+2𝜆𝜅𝑠𝜏𝜅𝜏𝑠+4𝜆2𝜅2𝜅𝜏𝑠4𝜅𝑠𝜏,4=2𝜆𝜅33𝜆22𝜅𝜏2𝜅3+𝜅𝑠𝑠+𝜅sin𝜃+2𝜆2𝜅24𝜅𝜏𝑠𝜏𝜅𝑠𝜅9𝜆𝑠3,5=6𝜆2𝜅3𝜆4𝜅𝑠𝜏𝜅𝜏𝑠𝜅2,sin𝜃6=4𝜆3𝜅6sin𝜃.(3.16)

Now, we consider a tubular surface 𝑀 in E3 satisfying the Jacobi equation Φ(𝐾,𝐻II)=0. By using (3.9), (3.13), and (3.15), we obtain Φ(𝐾,𝐻II) in the following form:𝐾𝑠𝐻II𝜃𝐾𝜃𝐻II𝑠=𝜀4𝜆2𝜅3𝜎5cos3𝜃4𝑖=0𝑢𝑖cos𝑖𝜃,(3.17) with respect to the Gaussian curvature 𝐾 and the second mean curvature 𝐻II, where𝑢0=3𝜆𝜏𝜅2𝜅𝑠𝜏+𝜅𝜏𝑠𝑢sin𝜃,1=𝜅36𝜆2𝜏2𝜅+1𝑠+6𝜆2𝜅𝜏𝜏𝑠sin𝜃𝜆𝜅2𝜅𝑠𝑠𝜏+𝜆𝜅3𝜏𝑠𝑠,𝑢2𝜅=𝜆2𝜅𝑠𝑠𝑠4𝜅𝜅𝑠𝜅𝑠𝑠3𝜅4𝜅𝑠𝜅+3𝑠3+𝜅3𝜏𝜏𝑠sin𝜃+𝜆2𝜅33𝜅𝑠𝜏𝑠+4𝜅𝑠𝑠𝜏𝜅𝜏𝑠𝑠,𝑢3=𝜆𝜅7𝜆𝜅𝜅𝑠𝜅𝑠𝑠𝜆𝜅2𝜅𝑠𝑠𝑠𝜅6𝜆𝑠3+2𝜆𝜅4𝜅𝑠4𝜆𝜅3𝜏𝜏𝑠+sin𝜃𝜅𝜅𝑠𝑠𝜏+𝜅𝜅𝑠𝜏𝑠𝜅𝑠2𝜏𝜅2𝜏𝑠𝑠,𝑢4=𝜆2𝜅24𝜅𝜅𝑠𝜏𝑠𝜅4𝜏𝑠2𝜅2𝜏𝑠𝑠+𝜅𝜅𝑠𝑠𝜏.(3.18) Then, by Φ(𝐾,𝐻II)=0, (3.17) becomes4𝑖=0𝑢𝑖cos𝑖𝜃=0.(3.19)

Hence, we have the following theorem.

Theorem 3.2. Let 𝑀be a tubular surface defined by (3.1) with nondegenerate second fundamental form. 𝑀 is a (𝐾,𝐻II)-Weingarten surface if and only if 𝑀 is a tubular surface around a circle or a helix.

Proof. Let us assume that 𝑀 is a (𝐾,𝐻II)-Weingarten surface, then the Jacobi equation (3.19) is satisfied. Since polynomial in (3.19) is equal to zero for every 𝜃, all its coefficients must be zero. Therefore, the solutions of 𝑢0=𝑢1=𝑢2=𝑢3=𝑢4=0 are 𝜅𝑠=0, 𝜏=0 and 𝜅𝑠=0, 𝜏𝑠=0 that is, 𝑀 is a tubular surface around a circle or a helix, respectively.
Conversely, suppose that 𝑀 is a tubular surface around a circle or a helix, then it is easily to see that Φ(𝐾,𝐻II)=0 is satisfied for the cases both 𝜅𝑠=0, 𝜏=0 and 𝜅𝑠=0, 𝜏𝑠=0. Thus M is a (𝐾,𝐻II)-Weingarten surface.
We suppose that a tubular surface 𝑀 with nondegenerate second fundamental form in E3 is (𝐻,𝐻II)-Weingarten surface. From (3.10), (3.13), and (3.15), Φ(𝐻,𝐻II) is 𝐻𝑠𝐻II𝜃𝐻𝜃𝐻II𝑠=18𝜆𝜅3𝜎5cos3𝜃4𝑖=0𝑣𝑖cos𝑖𝜃,(3.20) with respect to the variable cos𝜃, where 𝑣0=3𝜆𝜏𝜅2𝜅𝜏𝑠+𝜅𝑠𝜏𝑣sin𝜃,1=𝜅3𝜅𝑠+6𝜆2𝜏𝜅𝑠𝜏+𝜅𝜏𝑠sin𝜃+𝜆𝜅2𝜅𝑠𝑠𝜏𝜅𝜏𝑠𝑠,𝑣2=𝜆3𝜅4𝜅𝑠𝜅3𝑠3+4𝜅𝜅𝑠𝜅𝑠𝑠𝜅3𝜏𝜏𝑠𝜅2𝜅𝑠𝑠𝑠sin𝜃+𝜆2𝜅3𝜅𝜏𝑠𝑠3𝜅𝑠𝜏𝑠4𝜅𝑠𝑠𝜏,𝑣3=𝜆2𝜅6𝜅𝑠3+𝜅2𝜅𝑠𝑠𝑠7𝜅𝜅𝑠𝜅𝑠𝑠2𝜅4𝜅𝑠+4𝜅3𝜏𝜏𝑠𝜅sin𝜃+𝜆𝜅2𝜏𝑠𝑠+𝜅𝑠2𝜏𝜅𝜅𝑠𝑠𝜏𝜅𝜅𝑠𝜏𝑠,𝑣4=𝜆2𝜅2𝜅2𝜏𝑠𝑠4𝜅𝜅𝑠𝑠𝜏4𝜅𝜅𝑠𝜏𝑠𝜅+4𝑠2𝜏.(3.21)
Then, by Φ(𝐻,𝐻II)=0, (3.22) becomes in following form: 4𝑖=0𝑣𝑖cos𝑖𝜃=0.(3.22)
Thus, we state the following theorem.

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

Proof. Considering Φ(𝐻,𝐻II)=0and by using (3.13), one can obtaine the solutions 𝜅𝑠=0, 𝜏=0, and 𝜅𝑠=0, 𝜏𝑠=0 of the equations 𝑣0=𝑣1=𝑣2=𝑣3=𝑣4=0 for all 𝜃. Thus, it is easly proved that 𝑀 is a (𝐻,𝐻II)-Weingarten surface if and only if M is a tubular surface around a circle or a helix.

We consider a tubular surface𝑀 is (𝐾II,𝐻II)-Weingarten surface with nondegenerate second fundamental form in E3. By using (3.11), (3.12), (3.13), and (3.15), Φ(𝐾II,𝐻II) is 𝐾II𝑠𝐻II𝜃𝐾II𝜃𝐻II𝑠=116𝜆𝜅3𝜎5cos5𝜃9𝑖=0𝑤𝑖cos𝑖𝜃,(3.23) where 𝑤0=3𝜆𝜏𝜅2𝜅𝜏𝑠2𝜅𝑠𝜏𝑤sin𝜃,1=𝜅3𝜅𝑠+18𝜆2𝜏𝜅𝑠𝜏2𝜅𝜏𝑠sin𝜃+𝜆𝜅4𝜅𝑠𝜅𝜏𝑠𝜅𝑠𝜏+𝜅𝜅𝑠𝑠𝜏𝜅2𝜏𝑠𝑠,𝑤2=6𝜅𝜅𝑠𝑠18𝜆2𝜅4𝜏23𝜅4𝜅6𝑠22𝜅2𝜏2𝜆𝜅𝑠+43𝜆2𝜅21𝜆𝜅3𝜏𝜏𝑠𝜆𝜅2𝜅𝑠𝑠𝑠sin𝜃+3𝜆2𝜅2𝜅𝑠6𝜅𝑠𝜏5𝜅𝜏𝑠2𝜅𝜅𝑠𝑠𝜏+𝜅2𝜏𝑠𝑠,𝑤3=𝜅2+38𝜆2𝜅2𝜏2+4𝜆2𝜅423𝜆2𝜅𝜅𝑠𝑠+24𝜆2𝜅𝑠2𝜅𝜅𝑠+48𝜆2𝜅4𝜏𝜏𝑠+3𝜆2𝜅3𝜅𝑠𝑠𝑠2𝜆sin𝜃𝜆𝜅2𝜅2𝜅12𝜏𝑠𝑠+32𝜆2𝜅2𝜅3𝑠2𝜏𝑠14𝜆2𝜅23𝜅𝜅𝑠𝜏𝑠+214𝜆2𝜅2𝜅𝜅𝑠𝑠𝜏,𝑤4=𝜆2𝜆2𝜅2𝜅12𝜅𝑠𝑠+415𝜆2𝜅2𝜅𝜅𝑠𝜅𝑠𝑠+134𝜆2𝜅2𝜅13𝜏𝜏𝑠+310𝜆2𝜅2𝜅1𝑠2+257𝜏2+𝜅2𝜆2𝜅+134𝜅𝑠sin𝜃+𝜆2𝜅222𝜅𝜅𝑠𝑠𝜏+17𝜅𝜅𝑠𝜏𝑠𝜅14𝑠2𝜏16𝜅2𝜏𝑠𝑠,𝑤5=𝜆2𝜅55𝜅𝜅𝑠𝜅𝑠𝑠+433𝜆2𝜅2𝜅43𝜏𝜏𝑠+266𝜆2𝜏2𝜅+254𝜅𝑠13𝜅2𝜅+42𝑠𝑠𝑠sin𝜃𝜆𝜅50𝜆2𝜅21𝜅𝜅𝑠𝜏𝑠+132𝜆2𝜅2𝜅𝑠2𝜏+132𝜆2𝜅2𝜅2𝜏𝑠𝑠+74𝜆2𝜅21𝜅𝜅𝑠𝑠𝜏,𝑤6=2𝜆3𝜅2𝜅63𝑠324𝜆2𝜅4𝜅𝑠𝜏241𝜅4𝜅𝑠+33𝜅3𝜏𝜏𝑠78𝜅𝜅𝑠𝜅𝑠𝑠24𝜆2𝜅5𝜏𝜏𝑠+15𝜅2𝜅𝑠𝑠𝑠sin𝜃+𝜆2𝜅2𝜅16𝑠2𝜏26𝜆2𝜅4𝜏𝑠𝑠16𝜅𝜅𝑠𝑠𝜏+13𝜅2𝜏𝑠𝑠16𝜅𝜅𝑠𝜏𝑠+54𝜆2𝜅3𝜅𝑠𝜏𝑠+80𝜆2𝜅3𝜅𝑠𝑠𝜏,𝑤7=2𝜆4𝜅330𝜅4𝜅𝑠13𝜅2𝜅𝑠𝑠𝑠40𝜅3𝜏𝜏𝑠+79𝜅𝜅𝑠𝜅𝑠𝑠𝜅60𝑠3sin𝜃+2𝜆3𝜅3𝜅33𝑠2𝜏𝑠33𝜅𝜅𝑠𝜏𝑠4𝜆2𝜅4𝜏𝑠𝑠+12𝜆2𝜅3𝜅𝑠𝜏𝑠+15𝜅2𝜏𝑠𝑠+16𝜆2𝜅3𝜅𝑠𝑠𝜏33𝜅𝜅𝑠𝑠𝜏,𝑤8=8𝜆5𝜅4𝜅6𝑠3+2𝜅4𝜅𝑠4𝜅3𝜏𝜏𝑠+7𝜅𝜅𝑠𝜅𝑠𝑠𝜅2𝜅𝑠𝑠𝑠sin𝜃+2𝜆4𝜅413𝜅2𝜏𝑠𝑠𝜅+4𝑠2𝜅𝜏40𝜅𝑠𝑠𝜏+𝜅𝑠𝜏𝑠,𝑤9=8𝜆5𝜅54𝜅𝜅𝑠𝜏𝑠𝜅4𝑠2𝜏𝜅2𝜏𝑠𝑠+4𝜅𝜅𝑠𝑠𝜏.(3.24)

Since Φ(𝐾II,𝐻II)=0, then (3.23) becomes in following form:9𝑖=0𝑤𝑖cos𝑖𝜃=0.(3.25)

Hence, we have the following theorem.

Theorem 3.4. Let 𝑀be a tubular surface defined by (3.1) with nondegenerate second fundamental form. 𝑀 is a (𝐾II,𝐻II)-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 (𝑋,𝑌){(𝐾,𝐻II),(𝐻,𝐾II),(𝐻II,𝐾II)}, and let 𝑀 be a tubular surface defined by (3.1) with nondegenerate second fundamental form. 𝑀 is a (𝑋,𝑌)-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 (𝐾,𝐻II), (𝐻,𝐻II), (𝐻II,𝐾II), (𝐾,𝐻,𝐻II), (𝐾,𝐻,𝐾II), (𝐻,𝐾II,𝐻II), (𝐾,𝐾II,𝐻II), and (𝐾,𝐻,𝐾II,𝐻II) linear Weingarten tubular surfaces in E3.  (𝐾,𝐻), (𝐾,𝐾II), and (𝐻,𝐾II) linear Weingarten tubes are studied in [1].

Let 𝑎1, 𝑎2,𝑎3, 𝑎4, and 𝑏 be constants. In general, a linear combination of 𝐾, 𝐻, 𝐾II and 𝐻II can be constructed as𝑎1𝐾+𝑎2𝐻+𝑎3𝐾II+𝑎4𝐻II=𝑏.(4.1)

By the straightforward calculations, we obtained the reduced form of (4.1) 8𝑏𝜅3𝜀𝜎3cos3𝜃+8𝑖=0𝑝𝑖cos𝑖𝜃=0,(4.2) where the coefficients are 𝑝0=3𝑎4𝜆𝜅2𝜏2,𝑝1=𝑎4𝜅𝜅2𝜆𝑠𝜏𝜅𝜏𝑠sin𝜃𝜅26𝜆2𝜏2,𝑝+12=𝑎4𝜆2𝜆𝜅2𝜅𝜏𝑠4𝜅𝑠𝜏sin𝜃+𝜅23𝜅2𝜏22𝜅𝜅𝑠𝑠+3𝜅𝑠2+2𝑎3𝜆𝜅4,𝑝3=𝑎4𝜅2𝜆2𝜅𝜅𝑠𝑠𝜅4+2𝜅2𝜏23𝜅𝑠25𝜅24𝑎2𝜅34𝑎3𝜆2𝜅5,𝑝4=8𝑎1𝜀𝜅4+16𝑎2𝜆𝜅4+2𝑎3𝜆𝜅41+𝜆2𝜅2+17𝑎4𝜆𝜅4,𝑝5=16𝑎1𝜀𝜆𝜅520𝑎2𝜆2𝜅516𝑎3𝜆2𝜅520𝑎4𝜆2𝜅5,𝑝6=8𝑎1𝜀𝜆2𝜅6+8𝑎2𝜆3𝜅6+34𝑎3𝜆3𝜅6,𝑝7=28𝑎3𝜆4𝜅7,𝑝8=8𝑎3𝜆5𝜅8.(4.3)

Then, 𝑝0, 𝑝1, 𝑝2, 𝑝7, and 𝑝8 are zero for any 𝑏𝐼𝑅. If 𝑎40 or 𝑎30, from 𝑝0=𝑝1=𝑝7=𝑝8=0, one has 𝜅=0. Hence, we can give the following theorems.

Theorem 4.1. Let (𝑋,𝑌){(𝐾,𝐻III),(𝐻,𝐻II),(𝐾II,𝐻II)}. Then, there are no (𝑋,𝑌)-linear Weingarten tubular surfaces 𝑀in Euclidean 3-space defined by (3.1) with nondegenerate second fundamental form.

Theorem 4.2. Let (𝑋,𝑌,𝑍){(𝐻,𝐾II,𝐻II),(𝐾,𝐾II,𝐻II),(𝐾,𝐻,𝐻II),(𝐾,𝐻,𝐾II)}. Then, there are no (𝑋,𝑌,𝑍)-linear Weingarten tubular surfaces 𝑀in Euclidean 3-space defined by (3.1) with nondegenerate second fundamental form.

Theorem 4.3. Let 𝑀be a tubular surface defined by (3.1) with nondegenerate second fundamental form. Then, there are no (𝐾,𝐻,𝐾II,𝐻II)-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.

References

  1. J. S. Ro and D. W. Yoon, “Tubes of weingarten types in a euclidean 3-space,” Journal of the Chungcheong Mathematical Society, vol. 22, no. 3, pp. 359–366, 2009. View at Google Scholar
  2. F. Dillen and W. Kühnel, “Ruled Weingarten surfaces in Minkowski 3-space,” Manuscripta Mathematica, vol. 98, no. 3, pp. 307–320, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  3. F. Dillen and W. Sodsiri, “Ruled surfaces of Weingarten type in Minkowski 3-space,” Journal of Geometry, vol. 83, no. 1-2, pp. 10–21, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. F. Dillen and W. Sodsiri, “Ruled surfaces of Weingarten type in Minkowski 3-space. II,” Journal of Geometry, vol. 84, no. 1-2, pp. 37–44, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. W. Kühnel, “Ruled W-surfaces,” Archiv der Mathematik, vol. 62, no. 5, pp. 475–480, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  6. R. López, “Special Weingarten surfaces foliated by circles,” Monatshefte für Mathematik, vol. 154, no. 4, pp. 289–302, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. G. Stamou, “Regelflächen vom Weingarten-Typ,” Colloquium Mathematicum, vol. 79, no. 1, pp. 77–84, 1999. View at Google Scholar · View at Zentralblatt MATH
  8. D. W. Yoon, “Some properties of the helicoid as ruled surfaces,” JP Journal of Geometry and Topology, vol. 2, no. 2, pp. 141–147, 2002. View at Google Scholar · View at Zentralblatt MATH
  9. W. Kühnel and M. Steller, “On closed Weingarten surfaces,” Monatshefte für Mathematik, vol. 146, no. 2, pp. 113–126, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. C. Baikoussis and T. Koufogiorgos, “On the inner curvature of the second fundamental form of helicoidal surfaces,” Archiv der Mathematik, vol. 68, no. 2, pp. 169–176, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. T. Koufogiorgos and T. Hasanis, “A characteristic property of the sphere,” Proceedings of the American Mathematical Society, vol. 67, no. 2, pp. 303–305, 1977. View at Publisher · View at Google Scholar
  12. D. Koutroufiotis, “Two characteristic properties of the sphere,” Proceedings of the American Mathematical Society, vol. 44, pp. 176–178, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. D. E. Blair and Th. Koufogiorgos, “Ruled surfaces with vanishing second Gaussian curvature,” Monatshefte für Mathematik, vol. 113, no. 3, pp. 177–181, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. D. W. Yoon, “On non-developable ruled surfaces in Euclidean 3-spaces,” Indian Journal of Pure and Applied Mathematics, vol. 38, no. 4, pp. 281–290, 2007. View at Google Scholar · View at Zentralblatt MATH