Research Article Weingarten and Linear Weingarten Type Tubular Surfaces in E 3

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1 Mathematical Problems in Engineering Volume 20, Article ID 9849, pages doi:0.55/20/9849 Research Article Weingarten and Linear Weingarten Type Tubular Surfaces in E 3 Yılmaz Tunçer, Dae Won Yoon, 2 and Murat Kemal Karacan Department of Mathematics, Arts and Science Faculty, Usak University, Usak, Turkey 2 Department of Mathematic Education and RINS, Gyeongsang National University, Jinju 66070, Republic of Korea Correspondence should be addressed to Yılmaz Tunçer, yilmaz.tuncer@usak.edu.tr Received 5 January 20; Revised 30 March 20; Accepted 26 April 20 Academic Editor: Victoria Vampa Copyright q 20 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. 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.. Introduction Let f and g be smooth functions on a surface M in Euclidean 3-space E 3. The Jacobi function Φ f, g formed with f, g is defined by Φ f, g fs f t det,. g s g t 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 H on 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, thatis,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. 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.

2 2 Mathematical Problems in Engineering If the second fundamental form II of a surface M in E 3 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. For a pair X, Y, X / Y, of the curvatures K, H, K II and H II of M in E 3,ifM satisfies Φ X, Y 0byaX by c, then it said to be a X, Y -Weingarten surface or X, Y -linear Weingarten surface, respectively. Several geometers have studied W-surfaces and LW-surfaces and obtained many interesting results 9. For the study of these surfaces, Kühnel and Stamou investigated ruled X, Y -Weingarten surfaces in Euclidean 3-space E 3 7, 9. Also, Baikoussis and Koufogiorgos studied helicoidal H, K II -Weingarten surfaces 0. Dillen, and sodsiri, and Kühnel, gave a classification of ruled X, Y -Weingarten surfaces in Minkowski 3-space E 3, where X, Y {K, H, K II } 2 4. Koufogiorgos, Hasanis, and Koutroufiotis investigated closed ovaloid X, Y -linear Weingarten surfaces in E 3, 2. Yoon, Blair and Koufogiorgos classified ruled X, Y -linear Weingarten surfaces in E 3 8, 3, 4. 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. 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 conditions Φ X, Y 0, ax by c,.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 E Preliminaries We denote a surface M in E 3 by M s, t m s, t,m 2 s, t,m 3 s, t. 2. Let U be the standard unit normal vector field on a surface M defined by U M s M t M s M t, 2.2 where M s M s, t / s. Then, the first fundamental form I and the second fundamental form II of a surface M are defined by, respectively, I Eds 2 2Fdsdt Gdt 2, II eds 2 2fdsdt gdt 2, 2.3

3 Mathematical Problems in Engineering 3 where E M s,m s, F M s,m t, G M t,m t, e M s,u s M ss,u, f M s,u t M st,u, g M t,u t M tt,u, On the other hand, the Gaussian curvature K and the mean curvature H are H K eg f 2 EG F 2, Eg 2Ff Ge, 2 EG F 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,andG by the components of the second fundamental form e, f,andg, respectively 4. Consequently, the second Gaussian curvature K II of a surface is defined by 2 e tt f st 2 g ss 2 e s f s 2 e t 0 2 e t 2 g s K II eg f 2 2 f t 2 g s e f 2 e t e f, g t f g 2 g s f g and the second mean curvature H II of a surface is defined by H II H 2 det II L ij det II u i i,j u j ln K, 2.7 where u i and u j stand for s and θ t, respectively, and L ij L ij, where L ij are the coefficients of the second fundamental form 3, 4. Remark 2.. 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 Weingarten Tubular Surfaces Definition 3.. Let α : a, b E 3 be a unit-speed curve. A tubular surface of radius λ>0 about α is the surface with parametrization M s, θ α s λ N s cos θ B s sin θ, 3.

4 4 Mathematical Problems in Engineering a s b, where N s, B s are the principal normal and the binormal vectors of α, respectively. 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 0 κ 0 T κ 0 τ N, 0 τ 0 B 3.2. Furthermore, we have the natural frame {M S,M θ } given by M s λκ cos θ T λτ sin θn λτ cos θb, M θ λ sin θn λ cos θb. 3.3 The components of the first fundamental form are E λ 2 τ 2 σ 2, F λ 2 τ, G λ 2, 3.4 where σ λκ cos θ. On the other hand, the unit normal vector field U is obtained by U M s M θ ε cos θn ε sin θb. M s M θ 3.5 As λ>0, ε is the sign of σ such that if σ<0, then ε andifσ>0, then ε. From this, the components of the second fundamental form of M are given by e ελτ 2 εκ cos θσ, f ελτ, g ελ. 3.6 If the second fundamental form is nondegenerate, eg f 2 / 0, that is, κ, σ and cos θ 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

5 Mathematical Problems in Engineering 5 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 κ cos θ λσ, ε 2λκ cos θ H, 2λσ K II εκ cos 2 θ 6κλcos 3 θ 4κ 2 λ 2 cos 4 θ, 4 cos θσ 6 H II g 8ελκ 3 cos 3 θσ 3 i cos i θ, 3.7 and where the coefficients g i are g 0 3λ 2 κ 2 τ 2, g 2λκ κ s τ κτ s sin θ 6λ 2 τ 2 κ 3, g 2 2λ 2 κ 2 κτ s 4κ s τ sin θ λ 3 κ s 2 3κ 4 2κκ ss κ 2 τ 2, g 3 2λ 2 κ 2κ 2 τ 2 κ 3 κκ ss 3 κ s 2 κ 3, 3.8 g 4 6λκ 4, g 5 20λ 2 κ 5, g 6 8λ 3 κ 6. Differentiating K, K II, H, andh II calculations, we get, with respect to s and θ, after straightforward K s κ s cos θ λσ 2, K θ κ sin θ λσ 2, 3.9 H s εκ s cos θ εκ sin θ, H 2σ 2 θ, 2σ K II s εκ s 8λ 3 κ 3 cos 5 θ 8λ 2 κ 2 cos 4 θ 2λκcos 3 θ cos 2 θ, 4 cos θσ 2 3. K II θ εκ sin θ 8λ 3 κ 3 cos 5 θ 8λ 2 κ 2 cos 4 θ 2λκcos 3 θ sin 2 θ 2λκ cos θ, 3.2 4cos 2 θσ 2 6 H II s f 8εκ 4 cos 3 θσ 4 i cos i θ, 3.3

6 6 Mathematical Problems in Engineering and where f i are f 0 3κ 2 τ κ s τ 2κτ s, f 2κ 2κ s κ s τ κτ s κκ ss τ sin θ 3κτ s 2κ s τ 6λκ 3 τ, f 2 2λκ 2 9κ s κτ s κ s τ 2κκ ss τ sin θ 6λ 2 κ 4 τ 3κ s τ 2κτ s κ s 9 κ s 2 0κκ ss κ 2 τ 2κτ s κ s τ, f 3 2λ 2 κ 3 κ s 6κ s τ 7κτ s 4κτκ ss sin θ 2λκ 5κκ s κ ss 5 κ s 2 κ 4 κ s κ 2 τ 2τκ s 5κτ s, f 4 2λ 2 κ 2 5κ s 3 κ s 2 2κκ ss 2κ 2 τ 2κτ s 3τκ s κ 4 κ s 2κ 4 κ s, f 5 6λκ 5 κ s, f 6 4λ 2 κ 6 κ s, H II θ h 8ελκ 3 cos 4 θσ 4 i cos i θ, 3.5 and where the coefficients h i are h 0 9λκ 2 τ 2 sin θ, h 2κ 3 5λ 2 τ 2 sin θ 4λκ κτ s κ s τ, h 2 λ 2κκ ss 8κ 4 κ 2 τ 2 30λ 2 κ 2 3 κ s 2 sin θ 6λ 2 κ 2 3κ s τ 2κτ s, h 3 4λ 2 κ 2κ 4 κ 2 τ 2 2κκ ss 3 κ s 2 sin θ 2λκ κ s τ κτ s 4λ 2 κ 2 κτ s 4κ s τ, h 4 2λκ 3 3λ 2 2κτ 2 κ 3 κ ss κ sin θ 2λ 2 κ 2 4 κτ s τκ s 9λ κ s 3, h 5 6λ 2 κ 3 λ 4κ s τ κτ s κ 2 sin θ, h 6 4λ 3 κ 6 sin θ. 3.6 Now, we consider a tubular surface M in E 3 satisfying the Jacobi equation Φ K, H II 0. By using 3.9, 3.3, and 3.5, weobtainφ K, H II in the following form: K s H II θ K θ H II s ε 4λ 2 κ 3 σ 5 cos 3 θ 4 u i cos i θ, 3.7

7 Mathematical Problems in Engineering 7 with respect to the Gaussian curvature K and the second mean curvature H II, where u 0 3λτκ 2 κ s τ κτ s sin θ, u κ 3 6λ 2 τ 2 κ s 6λ 2 κττ s sin θ λκ 2 κ ss τ λκ 3 τ ss, u 2 λ κ 2 κ sss 4κκ s κ ss 3κ 4 κ s 3 κ s 3 κ 3 ττ s sin θ λ 2 κ 3 3κ s τ s 4κ ss τ κτ ss, u 3 λκ {7λκκ s κ ss λκ 2 κ sss 6λ κ s 3 2λκ 4 κ s 4λκ 3 ττ s sin θ } κκ ss τ κκ s τ s κ s 2 τ κ 2 τ ss, u 4 λ 2 κ 2{ } 4κκ s τ s 4τ κ s 2 κ 2 τ ss κκ ss τ. 3.8 Then, by Φ K, H II 0, 3.7 becomes 4 u i cos i θ Hence, we have the following theorem. Theorem 3.2. Let M be a tubular surface defined by 3. 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 3.9 is satisfied. Since polynomial in 3.9 is equal to zero for every θ, all its coefficients must be zero. Therefore, the solutions of u 0 u u 2 u 3 u 4 0areκ s 0, τ 0andκ 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 0andκ 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 E 3 is H, H II -Weingarten surface. From 3.0, 3.3,and 3.5, Φ H, H II is H s H II θ H θ H II s 8λκ 3 σ 5 cos 3 θ 4 v i cos i θ, 3.20 with respect to the variable cos θ, where v 0 3λτκ 2 κτ s κ s τ sin θ, v κ 3 κ s 6λ 2 τ κ s τ κτ s sin θ λκ 2 κ ss τ κτ ss,

8 8 Mathematical Problems in Engineering v 2 λ 3κ 4 κ s 3 κ s 3 4κκ s κ ss κ 3 ττ s κ 2 κ sss sin θ λ 2 κ 3 κτ ss 3κ s τ s 4κ ss τ, v 3 λ 2 κ 6 κ s 3 κ 2 κ sss 7κκ s κ ss 2κ 4 κ s 4κ 3 ττ s sin θ λκ κ 2 τ ss κ s 2 τ κκ ss τ κκ s τ s, v 4 λ 2 κ 2 κ 2 τ ss 4κκ ss τ 4κκ s τ s 4 κ s 2 τ. 3.2 Then, by Φ H, H II 0, 3.22 becomes in following form: 4 v i cos i θ Thus, we state the following theorem. Theorem 3.3. Let M be a tubular surface defined by 3. 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 0 and by using 3.3, one can obtaine the solutions κ s 0, τ 0, and κ s 0, τ s 0 of the equations v 0 v v 2 v 3 v 4 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 surface M is K II,H II -Weingarten surface with nondegenerate second fundamental form in E 3.Byusing 3., 3.2, 3.3, and 3.5, Φ K II,H II is K II s H II θ K II θ H II s 6λκ 3 σ 5 cos 5 θ 9 w i cos i θ, 3.23 where w 0 3λτκ 2 κτ s 2κ s τ sin θ, w κ 3 κ s 8λ 2 τ κ s τ 2κτ s sin θ λκ 4κ s κτ s κ s τ κκ ss τ κ 2 τ ss, w 2 { 6κκ ss 8λ 2 κ 4 τ 2 3κ 4 6 κ s 2 2κ 2 τ 2 } λκ s 4 3λ 2 κ 2 λκ 3 ττ s λκ 2 κ sss sin θ 3λ 2 κ 2 κ s 6κ s τ 5κτ s 2κκ ss τ κ 2 τ ss,

9 Mathematical Problems in Engineering 9 w 3 {κ 2 38λ 2 κ 2 τ 2 4λ 2 κ 4 23λ 2 κκ ss 24λ 2 κ s 2 } κκ s 48λ 2 κ 4 ττ s 3λ 2 κ 3 κ sss sin θ { λκ 2 λ 2 κ 2 κ 2 τ ss 32λ 2 κ 2 3 κ s 2 τ s 4λ 2 κ 2 3 κκ s τ s 2 4λ 2 κ 2 } κκ ss τ, { w 4 λ 2λ 2 κ 2 κ 2 κ ss 4 5λ 2 κ 2 κκ s κ ss 34λ 2 κ 2 κ 3 ττ s 3 0λ 2 κ 2 κ s τ 2 κ 2 λ 2 3 κ 4 } κ s sin θ λ 2 κ 2 22κκ ss τ 7κκ s τ s 4 κ s 2 τ 6κ 2 τ ss, { } w 5 λ 2 κ 55κκ s κ ss 4 33λ 2 κ 2 4 κ 3 ττ s 2 66λ 2 τ 2 25 κ 4 κ s 3κ 2 42 κ sss sin θ { λκ 50λ 2 κ 2 κκ s τ s 32λ 2 κ 2 κ s 2 τ 32λ 2 κ 2 κ 2 τ ss } 74λ 2 κ 2 κκ ss τ, w 6 2λ 3 κ 2 63 κ s 3 24λ 2 κ 4 κ s τ 2 4κ 4 κ s 33κ 3 ττ s 78κκ s κ ss 24λ 2 κ 5 ττ s 5κ 2 κ sss sin θ λ 2 κ 2 6 κ s 2 τ 26λ 2 κ 4 τ ss 6κκ ss τ 3κ 2 τ ss 6κκ s τ s 54λ 2 κ 3 κ s τ s 80λ 2 κ 3 κ ss τ, w 7 2λ 4 κ 3 30κ 4 κ s 3κ 2 κ sss 40κ 3 ττ s 79κκ s κ ss 60 κ s 3 sin θ 2λ 3 κ 3 33 κ s 2 τ s 33κκ s τ s 4λ 2 κ 4 τ ss 2λ 2 κ 3 κ s τ s 5κ 2 τ ss 6λ 2 κ 3 κ ss τ 33κκ ss τ, w 8 8λ 5 κ 4 6 κ s 3 2κ 4 κ s 4κ 3 ττ s 7κκ s κ ss κ 2 κ sss sin θ 2λ 4 κ 4 3κ 2 τ ss 4 κ s 2 τ 40κ κ ss τ κ s τ s, w 9 8λ 5 κ 5{ } 4κκ s τ s 4 κ s 2 τ κ 2 τ ss 4κκ ss τ. Since Φ K II,H II 0, then 3.23 becomes in following form: w i cos i θ Hence, we have the following theorem. Theorem 3.4. Let M be a tubular surface defined by 3. 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.

10 0 Mathematical Problems in Engineering 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. 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. 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 E 3. K, H, K, K II,and H, K II linear Weingarten tubes are studied in. Let a, a 2, a 3, a 4,andb be constants. In general, a linear combination of K, H, K II and H II can be constructed as a K a 2 H a 3 K II a 4 H II b. 4. By the straightforward calculations, we obtained the reduced form of bκ 3 εσ 3 cos 3 θ p i cos i θ 0, 4.2 where the coefficients are p 0 3a 4 λκ 2 τ 2, p a 4 κ 2λ κ s τ κτ s sin θ κ 2 6λ 2 τ 2, p 2 a 4 λ 2λκ 2 κτ s 4κ s τ sin θ κ 2 3κ 2 τ 2 2 2κκ ss 3κ s 2a 3 λκ 4, p 3 a 4 κ 2λ 2 κκ ss κ 4 2κ 2 τ 2 2 3κ s 5κ 2 4a 2 κ 3 4a 3 λ 2 κ 5, p 4 8a εκ 4 6a 2 λκ 4 2a 3 λκ 4 λ 2 κ 2 7a 4 λκ 4, 4.3 p 5 6a ελκ 5 20a 2 λ 2 κ 5 6a 3 λ 2 κ 5 20a 4 λ 2 κ 5, p 6 8a ελ 2 κ 6 8a 2 λ 3 κ 6 34a 3 λ 3 κ 6, p 7 28a 3 λ 4 κ 7, p 8 8a 3 λ 5 κ 8. Then, p 0, p, p 2, p 7,andp 8 are zero for any b IR.Ifa 4 / 0ora 3 / 0, from p 0 p p 7 p 8 0, one has κ 0. Hence, we can give the following theorems.

11 Mathematical Problems in Engineering Theorem 4.. Let X, Y { K, H III, H, H II, K II,H II }. Then, there are no X, Y -linear Weingarten tubular surfaces M in Euclidean 3-space defined by 3. 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 M in Euclidean 3-space defined by 3. with nondegenerate second fundamental form. Theorem 4.3. Let M be a tubular surface defined by 3. 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. Acknowledgments The authors would like to thank the referees for the helpful and valuable suggestions. References J. S. Ro and D. W. Yoon, Tubes of weingarten types in a euclidean 3-space, the Chungcheong Mathematical Society, vol. 22, no. 3, pp , F. Dillen and W. Kühnel, Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Mathematica, vol. 98, no. 3, pp , F. Dillen and W. Sodsiri, Ruled surfaces of Weingarten type in Minkowski 3-space, Geometry, vol. 83, no. -2, pp. 0 2, F. Dillen and W. Sodsiri, Ruled surfaces of Weingarten type in Minkowski 3-space. II, Geometry, vol. 84, no. -2, pp , W. Kühnel, Ruled W-surfaces, Archiv der Mathematik, vol. 62, no. 5, pp , R. López, Special Weingarten surfaces foliated by circles, Monatshefte für Mathematik, vol. 54, no. 4, pp , G. Stamou, Regelflächen vom Weingarten-Typ, Colloquium Mathematicum, vol. 79, no., pp , D. W. Yoon, Some properties of the helicoid as ruled surfaces, JP Geometry and Topology, vol. 2, no. 2, pp. 4 47, W. Kühnel and M. Steller, On closed Weingarten surfaces, Monatshefte für Mathematik, vol. 46, no. 2, pp. 3 26, 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 , 997. T. Koufogiorgos and T. Hasanis, A characteristic property of the sphere, Proceedings of the American Mathematical Society, vol. 67, no. 2, pp , D. Koutroufiotis, Two characteristic properties of the sphere, Proceedings of the American Mathematical Society, vol. 44, pp , D. E. Blair and Th. Koufogiorgos, Ruled surfaces with vanishing second Gaussian curvature, Monatshefte für Mathematik, vol. 3, no. 3, pp. 77 8, D. W. Yoon, On non-developable ruled surfaces in Euclidean 3-spaces, Indian Pure and Applied Mathematics, vol. 38, no. 4, pp , 2007.

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