# Weingarten and Linear Weingarten Type Tubular Surfaces inE3

Authors: Yılmaz Tunçer; Dae Won Yoon; Murat Kemal Karacan
Journal: Mathematical Problems in Engineering
 (2011)

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## 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.


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## Body

## 1. Introduction

Letf 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 by(1.1)Φ(f,g)=det⁡(fsftgsgt),
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,c∈IR, then it is said to be a linear Weingarten surface or a LW-surface [1].When the constantb=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 formII 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), X≠Y,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 studiedW-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,KII)-Weingarten surfaces [10]. Dillen, and sodsiri, and Kühnel, gave a classification of ruled (X,Y)-Weingarten surfaces in Minkowski 3-space E13, where (X,Y)∈{K,H,KII} [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,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 curvatureK, 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 conditions(1.2)Φ(X,Y)=0,aX+bY=c,
where X,Y∈{K,H,KII,HII}, 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-spaceE3.


## 2. Preliminaries

We denote a surfaceM in E3 by (2.1)M(s,t)=(m1(s,t),m2(s,t),m3(s,t)).
Let Ube the standard unit normal vector field on a surface M defined by(2.2)U=Ms∧Mt‖Ms∧Mt‖,
where Ms=∂M(s,t)/∂s. Then, the first fundamental form Iand the second fundamental form II of a surface Mare defined by, respectively,(2.3)I=Eds2+2Fdsdt+Gdt2,II=eds2+2fdsdt+gdt2,
where(2.4)E=〈Ms,Ms〉,F=〈Ms,Mt〉,G=〈Mt,Mt〉,e=-〈Ms,Us〉=〈Mss,U〉,f=-〈Ms,Ut〉=〈Mst,U〉,g=-〈Mt,Ut〉=〈Mtt,U〉,
[14]. On the other hand, the Gaussian curvature K and the mean curvature Hare (2.5)K=eg-f2EG-F2,H=Eg-2Ff+Ge2(EG-F2),
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 by(2.6)KII=1(|eg|-f2)2{|-12ett+fst-12gss12esfs-12etft-12gsef12gtfg|-|012et12gs12etef12gsfg|},
and the second mean curvature HII of a surface is defined by(2.7)HII=H-12|det⁡II|∑i,j∂∂ui(|det⁡II|Lij∂∂uj(ln⁡|K|)),
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 ..
Letα:[a,b]→E3 be a unit-speed curve. A tubular surface of radius λ>0 about α is the surface with parametrization(3.1)M(s,θ)=α(s)+λ[N(s)cos⁡θ+B(s)sinθ],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 curveα are denoted by κ, τ. Then, Frenet formula of α(s) is defined by(3.2)[T′N′B′]=[0κ0-κ0τ0-τ0][TNB],
[1]. Furthermore, we have the natural frame {MS,Mθ} given by(3.3)Ms=(1-λκcos⁡θ)T-λτsinθN+λτcos⁡θB,Mθ=-λsinθN+λcos⁡θB.The components of the first fundamental form are(3.4)E=λ2τ2+σ2,F=λ2τ,G=λ2,
where σ=1-λκcosθ.On the other hand, the unit normal vector fieldU is obtained by (3.5)U=Ms∧Mθ‖Ms∧Mθ‖=-εcos⁡θN-εsinθB.Asλ>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 by(3.6)e=ελτ2-εκcos⁡θσ,f=ελτ,g=ελ.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 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:(3.7)K=-κcos⁡θλσ,H=ε(1-2λκcos⁡θ)2λσ,KII=-εκ(cos⁡2θ-6κλcos⁡3θ+4κ2λ2cos⁡4θ+1)4cos⁡θσ,HII=1-8ελκ3cos⁡3θσ3(∑i=06gicos⁡iθ),
and where the coefficients gi are(3.8)g0=3λ2κ2τ2,g1=2λκ(κsτ-κτs)sinθ-(1+6λ2τ2)κ3,g2=2λ2κ2(κτs-4κsτ)sinθ+λ(3(κs)2+3κ4-2κκss-κ2τ2),g3=2λ2κ(2κ2τ2-κ3+κκss-3(κs)2)-κ3,g4=16λκ4,g5=-20λ2κ5,g6=8λ3κ6.DifferentiatingK, KII, H, and HII with respect to s and θ, after straightforward calculations, we get,
(3.9)Ks=-κscos⁡θλσ2,Kθ=κsinθλσ2,(3.10)Hs=-εκscos⁡θ2σ2,Hθ=εκsinθ2σ2,(3.11)(KII)s=εκs(8λ3κ3cos⁡5θ-18λ2κ2cos⁡4θ+12λκcos⁡3θ-cos⁡2θ-1)4cos⁡θσ2,(3.12)(KII)θ=-εκsinθ(8λ3κ3cos⁡5θ-18λ2κ2cos⁡4θ+12λκcos⁡3θ+sin2θ-2λκcos⁡θ)4cos⁡2θσ2,(3.13)(HII)s=18εκ4cos⁡3θσ4(∑i=06ficos⁡iθ),
and where fi are
(3.14)f0=3κ2τ(κsτ-2κτs),f1=2κ(2κs(κsτ-κτs)-κκssτ)sinθ+(3κτs-2κsτ)6λκ3τ,f2=2λκ2(9κs(κτs-κsτ)+2κκssτ)sinθ+6λ2κ4τ(3κsτ-2κτs)+κs(9(κs)2-10κκss)+κ2τ(2κτs-κsτ),f3=2λ2κ3(κs(16κsτ-7κτs)-4κτκss)sinθ+2λκ(15κκsκss-(15(κs)2+κ4)κs+κ2τ(2τκs-5κτs)),f4=2λ2κ2(5κs(3(κs)2-2κκss)+2κ2τ(2κτs-3τκs)+κ4κs)-2κ4κs,f5=6λκ5κs,f6=-4λ2κ6κs,(3.15)(HII)θ=18ελκ3cos⁡4θσ4(∑i=06hicos⁡iθ),
and where the coefficients hi are
(3.16)h0=-9λκ2τ2sinθ,h1=2κ3(1+15λ2τ2)sinθ+4λκ(κτs-κsτ),h2=λ(2κκss-8κ4+κ2τ2(1-30λ2κ2)-3(κs)2)sinθ+6λ2κ2(3κsτ-2κτs),h3=4λ2κ(2κ4-κ2τ2-2κκss+3(κs)2)sinθ+2λκ(κsτ-κτs+4λ2κ2(κτs-4κsτ)),h4=2λκ3(3λ2(2κτ2-κ3+κss)+κ)sinθ+2λ2κ2(4(κτs-τκs)-9λ(κs)3),h5=6λ2κ3(λ(4κsτ-κτs)-κ2sinθ),h6=4λ3κ6sinθ.Now, we consider a tubular surfaceM 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:(3.17)Ks(HII)θ-Kθ(HII)s=-ε4λ2κ3σ5cos⁡3θ∑i=04uicos⁡iθ,
with respect to the Gaussian curvature K and the second mean curvature HII, where(3.18)u0=-3λτκ2(κsτ+κτs)sinθ,u1=κ3((6λ2τ2+1)κs+6λ2κττs)sinθ-λκ2κssτ+λκ3τss,u2=λ(κ2κsss-4κκsκss-3κ4κs+3(κs)3+κ3ττs)sinθ+λ2κ3(3κsτs+4κssτ-κτss),u3=λκ{(7λκκsκss-λκ2κsss-6λ(κs)3+2λκ4κs-4λκ3ττs)sinθ+(κκssτ+κκsτs-(κs)2τ-κ2τss)},u4=-λ2κ2{4κκsτs-4τ(κs)2-κ2τss+κκssτ}.
Then, by Φ(K,HII)=0, (3.17) becomes(3.19)∑i=04uicos⁡iθ=0.Hence, we have the following theorem.Theorem 3.2.
LetMbe 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 thatM 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 thatM 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 surfaceM with nondegenerate second fundamental form in E3 is (H,HII)-Weingarten surface. From (3.10), (3.13), and (3.15), Φ(H,HII) is
(3.20)Hs(HII)θ-Hθ(HII)s=18λκ3σ5cos⁡3θ∑i=04vicos⁡iθ,
with respect to the variable cosθ, where
(3.21)v0=3λτκ2(κτs+κsτ)sinθ,v1=-κ3(κs+6λ2τ(κsτ+κτs))sinθ+λκ2(κssτ-κτss),v2=λ(3κ4κs-3(κs)3+4κκsκss-κ3ττs-κ2κsss)sinθ+λ2κ3(κτss-3κsτs-4κssτ),v3=λ2κ(6(κs)3+κ2κsss-7κκsκss-2κ4κs+4κ3ττs)sinθ+λκ(κ2τss+(κs)2τ-κκssτ-κκsτs),v4=-λ2κ2(κ2τss-4κκssτ-4κκsτs+4(κs)2τ).
Then, byΦ(H,HII)=0, (3.22) becomes in following form:
(3.22)∑i=04vicos⁡iθ=0.
Thus, we state the following theorem.Theorem 3.3.
LetMbe 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.
ConsideringΦ(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
(3.23)(KII)s(HII)θ-(KII)θ(HII)s=-116λκ3σ5cos⁡5θ∑i=09wicos⁡iθ,
where
(3.24)w0=3λτκ2(κτs-2κsτ)sinθ,w1=κ3(κs+18λ2τ(κsτ-2κτs))sinθ+λκ(4κs(κτs-κsτ)+κκssτ-κ2τss),w2={(6κκss-18λ2κ4τ2-3κ4-6(κs)2-2κ2τ2)λκs+4(3λ2κ2-1)λκ3ττs-λκ2κsss}sinθ+3λ2κ2(κs(6κsτ-5κτs)-2κκssτ+κ2τss),w3={(κ2+38λ2κ2τ2+4λ2κ4-23λ2κκss+24λ2(κs)2)κκs+48λ2κ4ττs+3λ2κ3κsss}sinθ-λκ{2(λ2κ2-1)κ2τss+(32λ2κ2-3)(κs)2τs-(14λ2κ2-3)κκsτs+2(1-4λ2κ2)κκssτ},w4=-λ{(2λ2κ2-1)κ2κss+4(1-5λ2κ2)κκsκss+(134λ2κ2-1)κ3ττs+(3(10λ2κ2-1)(κs)2+(2(57τ2+κ2)λ2+13)κ4)κs}sinθ+λ2κ2(22κκssτ+17κκsτs-14(κs)2τ-16κ2τss),w5=λ2κ{55κκsκss+4(33λ2κ2-4)κ3ττs+2(66λ2τ2+25)κ4κs-(13κ2+42)κsss}sinθ-λκ{(50λ2κ2-1)κκsτs+(1-32λ2κ2)(κs)2τ+(1-32λ2κ2)κ2τss+(74λ2κ2-1)κκssτ},w6=2λ3κ2(63(κs)3-24λ2κ4κsτ2-41κ4κs+33κ3ττs-78κκsκss-24λ2κ5ττs+15κ2κsss)sinθ+λ2κ2(16(κs)2τ-26λ2κ4τss-16κκssτ+13κ2τss-16κκsτs+54λ2κ3κsτs+80λ2κ3κssτ),w7=2λ4κ3(30κ4κs-13κ2κsss-40κ3ττs+79κκsκss-60(κs)3)sinθ+-2λ3κ3(33(κs)2τs-33κκsτs-4λ2κ4τss+12λ2κ3κsτs+15κ2τss+16λ2κ3κssτ-33κκssτ),w8=-8λ5κ4(-6(κs)3+2κ4κs-4κ3ττs+7κκsκss-κ2κsss)sinθ+2λ4κ4(13κ2τss+4(κs)2τ-40κ(κssτ+κsτs)),w9=8λ5κ5{4κκsτs-4(κs)2τ-κ2τss+4κκssτ}.SinceΦ(KII,HII)=0, then (3.23) becomes in following form:(3.25)∑i=09wicos⁡iθ=0.Hence, we have the following theorem.Theorem 3.4.
LetMbe 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 Theorems3.2 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].Leta1, a2,a3, a4, and b be constants. In general, a linear combination of K, H, KII and HII can be constructed as(4.1)a1K+a2H+a3KII+a4HII=b.By the straightforward calculations, we obtained the reduced form of (4.1) (4.2)8bκ3εσ3cos⁡3θ+∑i=08picos⁡iθ=0,
where the coefficients are (4.3)p0=3a4λκ2τ2,p1=a4κ(2λ(κsτ-κτs)sinθ-κ2(6λ2τ2+1)),p2=a4λ(2λκ2(κτs-4κsτ)sinθ+κ2(3κ2-τ2)-2κκss+3κs2)+2a3λκ4,p3=a4κ(2λ2(κκss-κ4+2κ2τ2-3κs2)-5κ2)-4a2κ3-4a3λ2κ5,p4=8a1εκ4+16a2λκ4+2a3λκ4(1+λ2κ2)+17a4λκ4,p5=-16a1ελκ5-20a2λ2κ5-16a3λ2κ5-20a4λ2κ5,p6=8a1ελ2κ6+8a2λ3κ6+34a3λ3κ6,p7=-28a3λ4κ7,p8=8a3λ5κ8.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,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.
LetMbe 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].



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