Weingarten and Linear Weingarten Type Tubular Surfaces in E³

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 byΦ(𝑓,𝑔)=det𝑓^𝑠𝑓_𝑡𝑔_𝑠𝑔_𝑡,(1.1) where 𝑓_𝑠=𝜕𝑓/𝜕𝑠 and 𝑓_𝑡=𝜕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,𝑐∈𝐼𝑅, 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 𝐾_II is the second Gaussian curvature on 𝑀[1].

For a pair (𝑋,𝑌), 𝑋≠𝑌,of the curvatures 𝐾, 𝐻, 𝐾_II and 𝐻_II 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 [1–9]. 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,𝐾_II)-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,𝐾_II} [2–4]. 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,𝐾_II), (𝐻,𝐾_II)-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 𝐾_II, and the second mean curvature 𝐻_II, an interesting geometric question is raised: classify all surfaces in Euclidean 3-space satisfying the conditionsΦ𝑎𝑋(𝑋+,𝑌𝑏𝑌)==𝑐0,,(1.2) where 𝑋,𝑌∈{𝐾,𝐻,𝐾_II,𝐻_II}, 𝑋≠𝑌 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 𝑀(𝑠,𝑡)=𝑚₁(𝑠,𝑡),𝑚₂(𝑠,𝑡),𝑚₃(𝑠,𝑡).(2.1) Let 𝑈be the standard unit normal vector field on a surface 𝑀 defined by𝑈=𝑀^𝑠∧𝑀_𝑡‖‖𝑀_𝑠∧𝑀_𝑡‖‖,(2.2) where 𝑀_𝑠=𝜕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=−⟨𝑀_𝑠𝐸=⟨𝑀_𝑠,𝑀_𝑠⟩,𝐹=⟨𝑀_𝑠,𝑀_𝑡⟩,𝐺=⟨𝑀_𝑡,𝑀_𝑡⟩,,𝑈_𝑠⟩=⟨𝑀_𝑠𝑠,𝑈⟩,𝑓=−⟨𝑀_𝑠,𝑈_𝑡⟩=⟨𝑀_𝑠𝑡,𝑈⟩,𝑔=−⟨𝑀_𝑡,𝑈_𝑡⟩=⟨𝑀_𝑡𝑡,𝑈⟩,(2.4) [14]. On the other hand, the Gaussian curvature 𝐾 and the mean curvature 𝐻are 𝐾=𝑒𝑔−𝑓²𝐸𝐺−𝐹²𝐻=𝐸𝑔−2𝐹𝑓+,𝐺𝑒2𝐸𝐺−𝐹²,(2.5) respectively. From Brioschi's formula in a Euclidean 3-space, we are able to compute 𝐾_II and 𝐻_II 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 𝐾_II of a surface is defined by𝐾_II=1||𝑒𝑔||−𝑓²²⎧⎪⎪⎪⎨⎪⎪⎪⎩||||||||||−12𝑒^𝑡𝑡+𝑓_𝑠𝑡−12𝑔^𝑠𝑠12𝑒^𝑠𝑓_𝑠−12𝑒^𝑡𝑓_𝑡−12𝑔^𝑠1𝑒𝑓2𝑔^𝑡𝑓𝑔||||||||||−||||||||||012𝑒^𝑡12𝑔^𝑠12𝑒^𝑡1𝑒𝑓2𝑔^𝑠𝑓𝑔||||||||||⎫⎪⎪⎪⎬⎪⎪⎪⎭,(2.6) and the second mean curvature 𝐻_II of a surface is defined by𝐻_II=𝐻−12||detII||^𝑖,𝑗𝜕𝜕𝑢^𝑖||detII||𝐿^𝑖𝑗𝜕𝜕𝑢^𝑗ln||𝐾||,(2.7) where 𝑢^𝑖 and 𝑢^𝑗 stand for “𝑠” and “𝜃=𝑡”, respectively, and 𝐿^𝑖𝑗=(𝐿_𝑖𝑗)⁻¹, where 𝐿_𝑖𝑗 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 parametrization𝑀(𝑠,𝜃)=𝛼(𝑠)+𝜆[𝑁(𝑠)cos𝜃+𝐵(𝑠)sin𝜃],(3.1)𝑎≤𝑠≤𝑏, 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 by⎡⎢⎢⎢⎣𝑇^𝑁^𝐵^⎤⎥⎥⎥⎦=⎡⎢⎢⎢⎣−00𝜅−𝜅0𝜏𝜏00⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣𝑁𝐵𝑇⎤⎥⎥⎥⎦,(3.2) [1]. Furthermore, we have the natural frame {𝑀_𝑆,𝑀_𝜃} given by𝑀_𝑠=(1−𝑀𝜆𝜅cos𝜃)𝑇−𝜆𝜏sin𝜃𝑁+𝜆𝜏cos𝜃𝐵,_𝜃=−𝜆sin𝜃𝑁+𝜆cos𝜃𝐵.(3.3)

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

On the other hand, the unit normal vector field 𝑈 is obtained by 𝑈=𝑀^𝑠∧𝑀_𝜃‖‖𝑀_𝑠∧𝑀_𝜃‖‖=−𝜀cos𝜃𝑁−𝜀sin𝜃𝐵.(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=𝜀𝜆𝜏²−𝜀𝜅cos𝜃𝜎,𝑓=𝜀𝜆𝜏,𝑔=𝜀𝜆.(3.6)

If the second fundamental form is nondegenerate, 𝑒𝑔−𝑓²≠0, that is, 𝜅, 𝜎 and cos𝜃 are nowhere vanishing. In this case, we can define formally the second Gaussian curvature 𝐾_II and the second mean curvature 𝐻_II on 𝑀.On the other hand, the Gauss curvature 𝐾, the mean curvature 𝐻, the second Gaussian curvature 𝐾_II and the second mean curvature 𝐻_II are obtained by using (2.5), (2.6) and (2.7) as follows:𝐾=−𝜅cos𝜃𝐻=𝜀(1−2𝜆𝜆𝜎𝜅co,s𝜃)𝐾2𝜆𝜎_II,=−𝜀𝜅cos²𝜃−6𝜅𝜆cos³𝜃+4𝜅²𝜆²cos⁴𝜃+1𝐻4cos𝜃𝜎_II,=1−8𝜀𝜆𝜅³cos³𝜃𝜎³⁶_𝑖=0𝑔_𝑖cos^𝑖𝜃,(3.7) and where the coefficients 𝑔_𝑖 are𝑔₁𝑔₀=3𝜆²𝜅²𝜏²,𝑔₂=2𝜆𝜅𝜅_𝑠𝜏−𝜅𝜏_𝑠sin𝜃−1+6𝜆²𝜏²𝜅³,=2𝜆²𝜅𝑔²₃𝜅𝜏_𝑠−4𝜅_𝑠𝜏sin𝜃+𝜆3𝜅_𝑠²+3𝜅⁴−2𝜅𝜅_𝑠𝑠−𝜅²𝜏²,=2𝜆²𝜅2𝜅²𝜏²𝑔−₄𝜅³+𝜅𝜅_𝑠𝑠−3𝜅_𝑠²−𝜅³,𝑔₅=16𝜆𝜅⁴,𝑔₆=−20𝜆²𝜅⁵,=8𝜆³𝜅⁶.(3.8)

Differentiating 𝐾, 𝐾_II, 𝐻, and 𝐻_II with respect to 𝑠 and 𝜃, after straightforward calculations, we get, 𝐾_𝑠=−𝜅^𝑠cos𝜃𝜆𝜎²,𝐾_𝜃=𝜅sin𝜃𝜆𝜎²,(3.9)𝐻_𝑠=−𝜀𝜅^𝑠cos𝜃2𝜎²,𝐻_𝜃=𝜀𝜅sin𝜃2𝜎²,(3.10)𝐾_II_𝑠=𝜀𝜅^𝑠8𝜆³𝜅³cos⁵𝜃−18𝜆²𝜅²cos⁴𝜃+12𝜆𝜅cos³𝜃−cos²𝜃−14cos𝜃𝜎²,(3.11)𝐾_II_𝜃=−𝜀𝜅sin𝜃8𝜆³𝜅³cos⁵𝜃−18𝜆²𝜅²cos⁴𝜃+12𝜆𝜅cos³𝜃+sin²𝜃−2𝜆𝜅cos𝜃4cos²𝜃𝜎²,(3.12)𝐻_II_𝑠=18𝜀𝜅⁴cos³𝜃𝜎⁴⁶_𝑖=0𝑓_𝑖cos^𝑖𝜃,(3.13) and where 𝑓_𝑖 are 𝑓𝑓₀₁=3𝜅²𝜏𝜅_𝑠𝜏−2𝜅𝜏_𝑠,𝑓=2𝜅2𝜅_𝑠𝜅_𝑠𝜏−𝜅𝜏_𝑠−𝜅𝜅_𝑠𝑠𝜏sin𝜃+3𝜅𝜏_𝑠−2𝜅_𝑠𝜏6𝜆𝜅³𝜏,₂=+2𝜆𝜅𝜅²₂9𝜅_𝑠𝜅𝜏_𝑠−𝜅_𝑠𝜏+2𝜅𝜅_𝑠𝑠𝜏sin𝜃+6𝜆²𝜅⁴𝜏3𝜅_𝑠𝜏−2𝜅𝜏_𝑠+𝜅_𝑠9𝜅^𝑠²−10𝜅𝜅_𝑠𝑠𝑓₃𝜏2𝜅𝜏_𝑠−𝜅_𝑠𝜏,=+2𝜆2²𝜆𝜅𝜅³𝜅15_𝑠𝜅1𝜅6_𝑠𝜅_𝑠𝜏−7𝜅𝜏_𝑠−4𝜅𝜏𝜅_𝑠𝑠sin𝜃𝑓₄𝜅_𝑠𝑠−15𝜅_𝑠²+𝜅⁴𝜅^𝑠+𝜅²𝜏2𝜏𝜅_𝑠−5𝜅𝜏_𝑠,𝑓₅=2𝜆²𝜅²5𝜅^𝑠3𝜅^𝑠²−2𝜅𝜅_𝑠𝑠+2𝜅²𝜏2𝜅𝜏_𝑠−3𝜏𝜅_𝑠+𝜅⁴𝜅_𝑠−2𝜅⁴𝜅_𝑠,𝑓₆=6𝜆𝜅⁵𝜅_𝑠,=−4𝜆²𝜅⁶𝜅_𝑠,(3.14)𝐻_II_𝜃=18𝜀𝜆𝜅³cos⁴𝜃𝜎⁴⁶_𝑖=0ℎ_𝑖cos^𝑖𝜃,(3.15) and where the coefficients ℎ_𝑖 are ℎℎ₀=−9𝜆𝜅²𝜏²sin𝜃,₁ℎ₂=2𝜅³1+15𝜆²𝜏²sin𝜃+4𝜆𝜅𝜅𝜏_𝑠−𝜅_𝑠𝜏,ℎ₃=𝜆2𝜅𝜅_𝑠𝑠−8𝜅⁴+𝜅²𝜏²1−30𝜆²𝜅²−3𝜅_𝑠²sin𝜃+6𝜆²𝜅²3𝜅_𝑠𝜏−2𝜅𝜏_𝑠,=ℎ4𝜆²𝜅2𝜅⁴−𝜅²𝜏²−2𝜅𝜅_𝑠𝑠+3𝜅_𝑠²sin𝜃+2𝜆𝜅𝜅^𝑠𝜏−𝜅𝜏_𝑠+4𝜆²𝜅²𝜅𝜏_𝑠−4𝜅_𝑠𝜏₄,=2𝜆𝜅³3𝜆²2𝜅𝜏ℎ²₅−𝜅³+𝜅_𝑠𝑠+𝜅sin𝜃+2𝜆²𝜅²4𝜅𝜏^𝑠−𝜏𝜅_𝑠−9𝜆𝜅_𝑠³,=6𝜆²𝜅³ℎ𝜆4𝜅_𝑠𝜏−𝜅𝜏_𝑠−𝜅²sin𝜃₆,=4𝜆³𝜅⁶sin𝜃.(3.16)

Now, we consider a tubular surface 𝑀 in E³ satisfying the Jacobi equation Φ(𝐾,𝐻_II)=0. By using (3.9), (3.13), and (3.15), we obtain Φ(𝐾,𝐻_II) in the following form:𝐾_𝑠𝐻_II_𝜃−𝐾_𝜃𝐻_II_𝑠=−𝜀4𝜆²𝜅³𝜎⁵cos³𝜃⁴_𝑖=0𝑢_𝑖cos^𝑖𝜃,(3.17) with respect to the Gaussian curvature 𝐾 and the second mean curvature 𝐻_II, where𝑢𝑢₀=−3𝜆𝜏𝜅²𝜅_𝑠𝜏+𝜅𝜏_𝑠sin𝜃,₁𝑢₂=𝜅³6𝜆²𝜏²+1𝜅_𝑠+6𝜆²𝜅𝜏𝜏_𝑠sin𝜃−𝜆𝜅²𝜅_𝑠𝑠𝜏+𝜆𝜅³𝜏_𝑠𝑠,𝑢₃=𝜆𝜅²𝜅_𝑠𝑠𝑠−4𝜅𝜅_𝑠𝜅_𝑠𝑠−3𝜅⁴𝜅_𝑠+3𝜅_𝑠³+𝜅³𝜏𝜏_𝑠sin𝜃+𝜆²𝜅³3𝜅_𝑠𝜏_𝑠+4𝜅_𝑠𝑠𝜏−𝜅𝜏_𝑠𝑠,=𝜆𝜅+7𝜆𝜅𝜅𝜅𝜅^𝑠_𝑠𝑠𝜅_𝑠𝑠−𝜆𝜅²𝜅_𝑠𝑠𝑠−6𝜆𝜅_𝑠³+2𝜆𝜅⁴𝜅_𝑠−4𝜆𝜅³𝜏𝜏_𝑠sin𝜃𝑢𝜏+𝜅𝜅_𝑠𝜏_𝑠−𝜅_𝑠²𝜏−𝜅²𝜏_𝑠𝑠₄,=−𝜆²𝜅²4𝜅𝜅^𝑠𝜏_𝑠−4𝜏𝜅_𝑠²−𝜅²𝜏_𝑠𝑠+𝜅𝜅_𝑠𝑠𝜏.(3.18) Then, by Φ(𝐾,𝐻_II)=0, (3.17) becomes⁴_𝑖=0𝑢_𝑖cos^𝑖𝜃=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,𝐻_II)-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,𝐻_II)-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 𝜅_𝑠=0, 𝜏=0 and 𝜅_𝑠=0, 𝜏_𝑠=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,𝐻_II)=0 is satisfied for the cases both 𝜅_𝑠=0, 𝜏=0 and 𝜅_𝑠=0, 𝜏_𝑠=0. Thus M is a (𝐾,𝐻_II)-Weingarten surface.
We suppose that a tubular surface 𝑀 with nondegenerate second fundamental form in E³ is (𝐻,𝐻_II)-Weingarten surface. From (3.10), (3.13), and (3.15), Φ(𝐻,𝐻_II) is 𝐻_𝑠𝐻_II_𝜃−𝐻_𝜃𝐻_II_𝑠=18𝜆𝜅³𝜎⁵cos³𝜃⁴_𝑖=0𝑣_𝑖cos^𝑖𝜃,(3.20) with respect to the variable cos𝜃, where 𝑣𝑣₀=3𝜆𝜏𝜅²𝜅𝜏_𝑠+𝜅_𝑠𝜏sin𝜃,₁𝑣₂=−𝜅³𝜅_𝑠+6𝜆²𝜏𝜅_𝑠𝜏+𝜅𝜏_𝑠sin𝜃+𝜆𝜅²𝜅_𝑠𝑠𝜏−𝜅𝜏_𝑠𝑠,=+𝜆𝜆₂3𝜅⁴𝜅_𝑠−3𝜅_𝑠³+4𝜅𝜅_𝑠𝜅_𝑠𝑠−𝜅³𝜏𝜏_𝑠−𝜅²𝜅_𝑠𝑠𝑠sin𝜃𝑣₃𝜅³𝜅𝜏_𝑠𝑠−3𝜅_𝑠𝜏_𝑠−4𝜅_𝑠𝑠𝜏,=+𝜆²𝜆𝜅𝜅6𝜅₂𝜅_𝑠³+𝜅²𝜅_𝑠𝑠𝑠−7𝜅𝜅_𝑠𝜅_𝑠𝑠−2𝜅⁴𝜅_𝑠+4𝜅³𝜏𝜏_𝑠sin𝜃𝑣₄𝜏_𝑠𝑠+𝜅_𝑠²𝜏−𝜅𝜅_𝑠𝑠𝜏−𝜅𝜅_𝑠𝜏_𝑠,=−𝜆²𝜅²𝜅²𝜏_𝑠𝑠−4𝜅𝜅_𝑠𝑠𝜏−4𝜅𝜅_𝑠𝜏_𝑠+4𝜅_𝑠²𝜏.(3.21)
Then, by Φ(𝐻,𝐻_II)=0, (3.22) becomes in following form: ⁴_𝑖=0𝑣_𝑖cos^𝑖𝜃=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,𝐻_II)-Weingarten surface if and only if M is a tubular surface around a circle or a helix.

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

We consider a tubular surface𝑀 is (𝐾_II,𝐻_II)-Weingarten surface with nondegenerate second fundamental form in E³. By using (3.11), (3.12), (3.13), and (3.15), Φ(𝐾_II,𝐻_II) is 𝐾_II_𝑠𝐻_II_𝜃−𝐾_II_𝜃𝐻_II_𝑠=−116𝜆𝜅³𝜎⁵cos⁵𝜃⁹_𝑖=0𝑤_𝑖cos^𝑖𝜃,(3.23) where 𝑤𝑤₀=3𝜆𝜏𝜅²𝜅𝜏_𝑠−2𝜅_𝑠𝜏sin𝜃,₁𝑤₂=𝜅³𝜅_𝑠+18𝜆²𝜏𝜅_𝑠𝜏−2𝜅𝜏_𝑠sin𝜃+𝜆𝜅4𝜅_𝑠𝜅𝜏_𝑠−𝜅_𝑠𝜏+𝜅𝜅_𝑠𝑠𝜏−𝜅²𝜏_𝑠𝑠,=+63𝜅𝜆𝜅_𝑠𝑠2−18𝜆²𝜅⁴𝜏²−3𝜅⁴−6𝜅_𝑠²−2𝜅²𝜏²𝜆𝜅^𝑠+43𝜆²𝜅²−1𝜆𝜅³𝜏𝜏_𝑠−𝜆𝜅²𝜅_𝑠𝑠𝑠sin𝜃𝑤₃𝜅²𝜅_𝑠6𝜅_𝑠𝜏−5𝜅𝜏_𝑠−2𝜅𝜅_𝑠𝑠𝜏+𝜅²𝜏_𝑠𝑠,=−𝜅𝜆²𝜅+328𝜆𝜆²₂𝜅²𝜏²+4𝜆²𝜅⁴−23𝜆²𝜅𝜅_𝑠𝑠+24𝜆²𝜅_𝑠²𝜅𝜅^𝑠+48𝜆²𝜅⁴𝜏𝜏_𝑠+3𝜆²𝜅³𝜅_𝑠𝑠𝑠sin𝜃𝜅²−1𝜅²𝜏_𝑠𝑠+32𝜆²𝜅²−3𝜅_𝑠²𝜏_𝑠−14𝜆²𝜅²−3𝜅𝜅_𝑠𝜏_𝑠𝑤₄+21−4𝜆²𝜅²𝜅𝜅_𝑠𝑠𝜏,=−𝜆2𝜆²𝜅²−1𝜅²𝜅_𝑠𝑠+41−5𝜆²𝜅²𝜅𝜅_𝑠𝜅_𝑠𝑠+134𝜆²𝜅²−1𝜅³𝜏𝜏_𝑠+𝜆+₂310𝜆²𝜅²−1𝜅_𝑠²+257𝜏²+𝜅²𝜆²+13𝜅⁴𝜅^𝑠sin𝜃𝑤₅𝜅²22𝜅𝜅^𝑠𝑠𝜏+17𝜅𝜅_𝑠𝜏_𝑠−14𝜅_𝑠²𝜏−16𝜅²𝜏_𝑠𝑠,=−𝜆²𝜆𝜅𝜅555𝜅0𝜅𝜆_𝑠2𝜅_𝑠𝑠+433𝜆²𝜅²−4𝜅³𝜏𝜏_𝑠+266𝜆²𝜏²+25𝜅⁴𝜅_𝑠−13𝜅²+42𝜅_𝑠𝑠𝑠sin𝜃𝜅²−1𝜅𝜅_𝑠𝜏_𝑠+1−32𝜆²𝜅²𝜅_𝑠²𝜏+1−32𝜆²𝜅²𝜅²𝜏_𝑠𝑠𝑤₆+74𝜆²𝜅²−1𝜅𝜅_𝑠𝑠𝜏,=+2𝜆𝜆³₂𝜅²63𝜅^𝑠³−24𝜆²𝜅⁴𝜅_𝑠𝜏²−41𝜅⁴𝜅_𝑠+33𝜅³𝜏𝜏_𝑠−78𝜅𝜅_𝑠𝜅_𝑠𝑠−24𝜆²𝜅⁵𝜏𝜏_𝑠+15𝜅²𝜅_𝑠𝑠𝑠sin𝜃𝑤₇𝜅²16𝜅^𝑠²𝜏−26𝜆²𝜅⁴𝜏_𝑠𝑠−16𝜅𝜅_𝑠𝑠𝜏+13𝜅²𝜏_𝑠𝑠−16𝜅𝜅_𝑠𝜏_𝑠+54𝜆²𝜅³𝜅_𝑠𝜏_𝑠+80𝜆²𝜅³𝜅_𝑠𝑠𝜏,=2𝜆⁴𝜅³30𝜅⁴𝜅_𝑠−13𝜅²𝜅_𝑠𝑠𝑠−40𝜅³𝜏𝜏_𝑠+79𝜅𝜅_𝑠𝜅_𝑠𝑠−60𝜅_𝑠³sin𝜃+𝑤₈−2𝜆³𝜅³33𝜅^𝑠²𝜏_𝑠−33𝜅𝜅_𝑠𝜏_𝑠−4𝜆²𝜅⁴𝜏_𝑠𝑠+12𝜆²𝜅³𝜅_𝑠𝜏_𝑠+15𝜅²𝜏_𝑠𝑠+16𝜆²𝜅³𝜅_𝑠𝑠𝜏−33𝜅𝜅_𝑠𝑠𝜏,=+−82𝜆𝜆⁵₄𝜅⁴−6𝜅^𝑠³+2𝜅⁴𝜅_𝑠−4𝜅³𝜏𝜏_𝑠+7𝜅𝜅_𝑠𝜅_𝑠𝑠−𝜅²𝜅_𝑠𝑠𝑠sin𝜃𝑤₉𝜅⁴13𝜅²𝜏_𝑠𝑠+4𝜅_𝑠²𝜏−40𝜅𝜅_𝑠𝑠𝜏+𝜅_𝑠𝜏_𝑠,=8𝜆⁵𝜅⁵4𝜅𝜅^𝑠𝜏_𝑠−4𝜅_𝑠²𝜏−𝜅²𝜏_𝑠𝑠+4𝜅𝜅_𝑠𝑠𝜏.(3.24)

Since Φ(𝐾_II,𝐻_II)=0, then (3.23) becomes in following form:⁹_𝑖=0𝑤_𝑖cos^𝑖𝜃=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 (𝐾_II,𝐻_II)-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,𝐻_II),(𝐻,𝐾_II),(𝐻_II,𝐾_II)}, 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,𝐻_II), (𝐻,𝐻_II), (𝐻_II,𝐾_II), (𝐾,𝐻,𝐻_II), (𝐾,𝐻,𝐾_II), (𝐻,𝐾_II,𝐻_II), (𝐾,𝐾_II,𝐻_II), and (𝐾,𝐻,𝐾_II,𝐻_II) linear Weingarten tubular surfaces in E³.  (𝐾,𝐻), (𝐾,𝐾_II), and (𝐻,𝐾_II) linear Weingarten tubes are studied in [1].

Let 𝑎₁, 𝑎₂,𝑎₃, 𝑎₄, and 𝑏 be constants. In general, a linear combination of 𝐾, 𝐻, 𝐾_II and 𝐻_II can be constructed as𝑎₁𝐾+𝑎₂𝐻+𝑎₃𝐾_II+𝑎₄𝐻_II=𝑏.(4.1)

By the straightforward calculations, we obtained the reduced form of (4.1) 8𝑏𝜅³𝜀𝜎³cos³𝜃+⁸_𝑖=0𝑝_𝑖cos^𝑖𝜃=0,(4.2) where the coefficients are 𝑝₁𝑝₀=3𝑎₄𝜆𝜅²𝜏²,𝑝=𝑎₄𝜅2𝜆𝜅_𝑠𝜏−𝜅𝜏_𝑠sin𝜃−𝜅²6𝜆²𝜏²+1₂,=𝑝𝑎₄₃𝜆2𝜆𝜅²𝜅𝜏_𝑠−4𝜅_𝑠𝜏sin𝜃+𝜅²3𝜅²−𝜏²−2𝜅𝜅_𝑠𝑠+3𝜅_𝑠2+2𝑎₃𝜆𝜅⁴,=𝑎₄𝑝𝜅₄2𝜆²𝜅𝜅_𝑠𝑠−𝜅⁴+2𝜅²𝜏²−3𝜅_𝑠2−5𝜅²−4𝑎₂𝜅³−4𝑎₃𝜆²𝜅⁵,𝑝₅=8𝑎₁𝜀𝜅⁴+16𝑎₂𝜆𝜅⁴+2𝑎₃𝜆𝜅⁴1+𝜆²𝜅²+17𝑎₄𝜆𝜅⁴,=−1𝑝6₆𝑎₁𝜀𝜆𝜅⁵−20𝑎₂𝜆²𝜅⁵−16𝑎₃𝜆²𝜅⁵−20𝑎₄𝜆²𝜅⁵,=8𝑎₁𝜀𝜆𝑝²₇𝜅⁶+8𝑎₂𝜆³𝜅⁶+34𝑎₃𝜆³𝜅⁶,𝑝₈=−28𝑎₃𝜆⁴𝜅⁷,=8𝑎₃𝜆⁵𝜅⁸.(4.3)

Then, 𝑝₀, 𝑝₁, 𝑝₂, 𝑝₇, and 𝑝₈ are zero for any 𝑏∈𝐼𝑅. If 𝑎₄≠0 or 𝑎₃≠0, from 𝑝₀=𝑝₁=𝑝₇=𝑝₈=0, one has 𝜅=0. Hence, we can give the following theorems.

Theorem 4.1. Let (𝑋,𝑌)∈{(𝐾,𝐻_III),(𝐻,𝐻_II),(𝐾_II,𝐻_II)}. 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,𝐾_II,𝐻_II),(𝐾,𝐾_II,𝐻_II),(𝐾,𝐻,𝐻_II),(𝐾,𝐻,𝐾_II)}. 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,𝐾_II,𝐻_II)-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.