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(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 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), 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 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,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 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,
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-space E3.

2. Preliminaries

We denote a surface M in E3 by 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 byU=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,I=Eds2+2Fdsdt+Gdt2,II=eds2+2fdsdt+gdt2,
whereE=〈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 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 byKII=1(|eg|-f2)2{|-12ett+fst-12gss12esfs-12etft-12gsef12gtfg|-|012et12gs12etef12gsfg|},
and the second mean curvature HII of a surface is defined byHII=H-12|detII|∑i,j∂∂ui(|detII|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 ..

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],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′]=[0κ0-κ0τ0-τ0][TNB],
[1]. Furthermore, we have the natural frame {MS,Mθ} given byMs=(1-λκcosθ)T-λτsinθN+λτcosθB,Mθ=-λsinθN+λcosθB.

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

On the other hand, the unit normal vector field U is obtained by U=Ms∧Mθ‖Ms∧Mθ‖=-εcosθN-εsinθB.

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.

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:K=-κcosθλσ,H=ε(1-2λκcosθ)2λσ,KII=-εκ(cos2θ-6κλcos3θ+4κ2λ2cos4θ+1)4cosθσ,HII=1-8ελκ3cos3θσ3(∑i=06gicosiθ),
and where the coefficients gi areg0=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.

Differentiating K, KII, H, and HII with respect to s and θ, after straightforward calculations, we get,
Ks=-κscosθλσ2,Kθ=κsinθλσ2,Hs=-εκscosθ2σ2,Hθ=εκsinθ2σ2,(KII)s=εκs(8λ3κ3cos5θ-18λ2κ2cos4θ+12λκcos3θ-cos2θ-1)4cosθσ2,(KII)θ=-εκsinθ(8λ3κ3cos5θ-18λ2κ2cos4θ+12λκcos3θ+sin2θ-2λκcosθ)4cos2θσ2,(HII)s=18εκ4cos3θσ4(∑i=06ficosiθ),
and where fi are
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,(HII)θ=18ελκ3cos4θσ4(∑i=06hicosiθ),
and where the coefficients hi are
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 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:Ks(HII)θ-Kθ(HII)s=-ε4λ2κ3σ5cos3θ∑i=04uicosiθ,
with respect to the Gaussian curvature K and the second mean curvature HII, whereu0=-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∑i=04uicosiθ=0.

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
Hs(HII)θ-Hθ(HII)s=18λκ3σ5cos3θ∑i=04vicosiθ,
with respect to the variable cosθ, where
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, by Φ(H,HII)=0, (3.22) becomes in following form:
∑i=04vicosiθ=0.

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
(KII)s(HII)θ-(KII)θ(HII)s=-116λκ3σ5cos5θ∑i=09wicosiθ,
where
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}.

Since Φ(KII,HII)=0, then (3.23) becomes in following form:∑i=09wicosiθ=0.

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.

By the straightforward calculations, we obtained the reduced form of (4.1) 8bκ3εσ3cos3θ+∑i=08picosiθ=0,
where the coefficients are 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.

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.

RoJ. S.YoonD. W.Tubes of weingarten types in a euclidean 3-spaceDillenF.KühnelW.Ruled Weingarten surfaces in Minkowski 3-spaceDillenF.SodsiriW.Ruled surfaces of Weingarten type in Minkowski 3-spaceDillenF.SodsiriW.Ruled surfaces of Weingarten type in Minkowski 3-space. IIKühnelW.Ruled W-surfacesLópezR.Special Weingarten surfaces foliated by circlesStamouG.Regelflächen vom Weingarten-TypYoonD. W.Some properties of the helicoid as ruled surfacesKühnelW.StellerM.On closed Weingarten surfacesBaikoussisC.KoufogiorgosT.On the inner curvature of the second fundamental form of helicoidal surfacesKoufogiorgosT.HasanisT.A characteristic property of the sphereKoutroufiotisD.Two characteristic properties of the sphereBlairD. E.KoufogiorgosTh.Ruled surfaces with vanishing second Gaussian curvatureYoonD. W.On non-developable ruled surfaces in Euclidean 3-spaces