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PARALLEL SURFACES TO TRANSLATION SURFACES IN EUCLIDEAN 3-SPACE
MUHAMMED ÇETIN AND YILMAZ TUNÇER
Abstract. In this paper, we investigate geometric properties of surfaces that are parallel to translation surfaces in 3-dimensional Euclidean space which are constructed by generator curves with constant curvatures and torsions are given.
1. Introduction
The theory of translation surfaces is always one of interesting topics in Euclidean space. Translation surfaces have been investigated by some di¤erential geometers. Verstraelen et al. investigated the minimal translation surfaces in n-dimensional Euclidean spaces [5]. Liu gave the classi…cation of the translation surfaces with constant mean curvature or constant Gauss curvature in 3-dimensional Euclidean space and 3-dimensional Minkowski space [3]. Yoon studied translation surfaces in the 3-dimensional Minkowski space whose Gauss map satis…es the condition where denotes the Laplacian of the surface with respect to the induced metric and the set of real metrics [6]. Munteanu and Nistor studied the second fundamental form of translation surfaces in [4]. They gave a non-existence result for polynomial trans-lation surfaces in with vanishing second Gauss curvature. They classi…ed those translation surfaces for which and are proportional. Çetin et al. investigated the translation surfaces in 3-dimensional Euclidean space by using non-planar space curves and they gave the di¤erential geometric properties for both translation sur-faces and minimal translation sursur-faces [1].
In this paper, we investigate geometric properties of surfaces that are parallel to translation surfaces in 3-dimensional Euclidean space which are constructed by generator curves with constant curvatures and torsions are given.
Received by the editors: December 26, 2014; Accepted: March 12, 2015. 2010 Mathematics Subject Classi…cation. 53A05.
Key words and phrases. Translation surfaces, Parallel surfaces, Euclidean 3-space.
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2. Parallel Surfaces to Translation Surfaces in Euclidean 3-Space Let (s) be a space curve with Frenet frame fT ; N ; B g and ; be cur-vature and torsion of the curve respectively, and let (t) be a space curve with Frenet frame fT ; N ; B g and ; be curvature and torsion of the curve re-spectively. Let M (s; t) = (s) + (t) be a translation surface in Euclidean 3-Space that is generated by two space curves with constant curvatures and torsions. Then we can write the equation of parallel surface to translation surface M as follow.
M (s; t) = M (s; t) + U (s; t) (2.1)
where 2 R ( 6= 0) and U(s; t) is the unit normal vector of M, and U(s; t) can be written as follow.
U = T ^ T
sin ' (2.2)
where ' (s) is the angle between tangent vector …elds of and . Here, because of being easy the angle of ' has been constant.
Let and are space curves with non-zero curvature and torsion. Di¤erentiating (2.2) with respect to s, we get
Us= sin '(N ^ T ) ; and Uss= 2 sin '(T ^ T ) + sin ' (B ^ T ) : Similarly, di¤erentiating (2.2) with respect to t, we get
Ut= sin '(T ^ N ) ; and Utt= 2 sin '(T ^ T ) + sin '(T ^ B ) : Also, Ust= Uts= sin '(N ^ N ) : Di¤erentiating (2.1) with respect to s, we get,
Ms= T +
sin '(N ^ T )
Mss= N
2
sin '(T ^ T ) + sin ' (B ^ T ) : And di¤erentiating (2.1) with respect to t, we get
Mt= T +
Mtt= N 2 sin '(T ^ T ) + sin ' (T ^ B ) : Also, Mst= Mts = sin ' (N ^ N ) : We can give following equalities.
hT ; T i = A1 hT ; N i = A2 hT ; B i = A3 hN ; T i = A4 hN ; N i = A5 hN ; B i = A6 hB ; T i = A7 hB ; N i = A8 hB ; B i = A9 : Also, U = cos N + sin B (2.3) and U = cos N + sin B : (2.4)
can be written. From (2.4), we get
cos A2+ sin A3= 0: (2.5)
By using (2.3), we obtain
cos A4+ sin A7= 0: (2.6)
From (2.4), we get
U ^ T = sin N cos B : (2.7)
From (2.3) and (2.7), we get
cos sin A5 cos cos A6+ sin sin A8 sin cos A9= 0: (2.8)
By using (2.3), we get
U ^ T = sin N cos B : (2.9)
From (2.4) and (2.9), we get
sin cos A5+ sin sin A6 cos cos A8 cos sin A9= 0: (2.10)
Taking the inner product of the U and U , we get
cos cos A5+ cos sin A6+ sin cos A8+ sin sin A9= 1: (2.11)
From (2.7), we obtain
sin A2 cos A3= sin ': (2.12)
By using (2.9), we get
hU ^ T ; T i = sin A4 cos A7= sin ': (2.13)
From (2.7) and (2.9), we get
Also, hT ; T i = cos '; and thus we write
A1= cos ': (2.15)
If the equation system consisting of (2.5), (2.6), (2.8), (2.10), (2.11), (2.12), (2.13), (2.14) and (2.15) is solved we will get following equalities.
A1 = cos '; A2= sin sin '; A3= cos sin '
A4 = sin sin '; A5= cos cos + sin sin cos '
A6 = cos sin sin cos cos '; A7= cos sin '
A8 = sin cos cos sin cos '
A9 = sin sin + cos cos cos '
The …rst fundamental form I of the surface M is I = Eds2+ 2F dsdt + Gdt2 where E; F and G are the coe¢ cients of the I and
E = 1 2 n+ 2( n)2+ 2(k1)2cot2' F = cos ' 2 n ncot ' sin ' 2 g gcot2' G = 1 2 n+ 2 n 2 + 2 k1 2 cot2'
Let U be the unit normal of the surface M . Then, it can be written as follow.
U = 0 B B B B @ gT + 2k 1 n sin ' k1 sin ' N + g 2 g n sin2' + 2 n gcot ' sin ' T k1 sin 'N + (T ^ T ) 1 C C C C A where 1 = 0 B B B B B B B B @ 2 n 2 n 2 g 2 2 g 2 2 2 g gcos ' 2 3 n g 2cot2' + 2 3 g 2 n+ 2 2 n n+ sin2' + 4 ( g) 2 ( n) 2 sin4' 2 4 n g n gcot3' sin ' + 4 (k1) 2 ( n) 2 sin2' 2 3(k1) 2 n sin2' + 2(k1)2 sin2' 2 4( g) 2 ( n) 2 sin2' 2 3 n( n) 2 sin2' + 2 2 n n sin2' + 2(k 1) 2 sin2' + 4( g) 2 ( g) 2 cot2' sin2' + 2 2 g gcot ' sin ' 1 C C C C C C C C A 1 2 .
The second fundamental form II of the surface M is II = lds2+ 2mdsdt + ndt2
where l; m and n are the coe¢ cients of the II and l = 0 B B B B B @ 2 k2 g g sin2' n n( 2(k1)2+ k2cot '+1) sin ' + (k1)2( n+2 n 1) sin ' + 2(k 1)2k2cot '( n 1) sin ' 2( g) 2 n sin ' + g 2
sin ' + kg g( ncot ' + k2 cot ') + nsin ' (k1)2sin ' + k2 ncos '
1 C C C C C A m = 0 B B @ 3 nkg n g sin3' 3 cot ' ( g) 2 ( n) 2 +( n) 2 ( g) 2 sin2' 2 g g( n+ n) sin ' + 3 nkg n gcot2' sin '
+ 2 n g 2cot ' + 2 g 2 ncot ' + g gsin ' 1 C C A and n = 0 B B B B B B B B B B B @ g 2 sin ' + n 1 n n sin ' + g gcot ' + 1 n 2 k2 g g sin2' + 2 k2 n ncot ' sin ' + 2 n(k1) 2 sin ' + 2k2 g g n sin2' n gcot ' sin ' g + (k1) 2
sin ' 1 + n+ k2cot ' + nsin '
k1 2sin ' k2 ncos ' 1 C C C C C C C C C C C A :
3. Shape Operator Matrices of the Parallel Surfaces to Translation Surfaces
In [5], authors gave the shape operator matrix S, the Gauss curvature K and the mean curvature H of M are
S = 1 sin2' n cos ' n cos ' n n (3.1) K = n n sin2' (3.2) and H = n+ n 2 sin2' (3.3)
respectively. The shape operator matrix S of the parallel surface M is expressed in the form
S = S (I S) 1: (3.4)
Then, by substituting (3.1) into (3.4), we obtain
S = 1
sin2' n+ n + 2 n n
n n n ncos '
ncos ' n n n
From (3.5), we can …nd the Gauss curvature K of M as
K = n+ n
sin2' n+ n + 2 n n
; (3.6)
and the mean curvature H of M as
H = n+ n 2 n n
2 sin2' n+ n + 2 n n
: (3.7)
Also, the …rst and second principal curvature of M are
k1= n+ n 2 sin2'+ r n+ n 2 sin2' 2 n n sin2' 1 n+ n 2 sin2'+ r n+ n 2 sin2' 2 n n sin2' ! and k2= n+ n 2 sin2' r n+ n 2 sin2' 2 n n sin2' 1 n+ n 2 sin2' r n+ n 2 sin2' 2 n n sin2' ! :
respectively. In addition, we can …nd the harmonic mean curvature of M which is de…ned as K=H. Thus harmonic mean curvature is
K
H =
n+ n
n+ n 2 n n
: (3.8)
Corollary 1. i. The parallel surface M of the translation surface M , which is constructed by generator curves and , is K-‡at if and only if the curve and the curve lying on the translation surface M are asymptotic line or n = n.
ii. The parallel surface M of the translation surface M has constant Gaussian curvature if and only if the surface M has constant normal curvatures along the generator curves.
Corollary 2. i. All parallel surfaces of M , which is constructed by generator curves, is minimal if and only if the curve and the curve lying on M are asymptotic line. If the generator curves lying on M aren’t asymptotic line, then there is a minimal parallel surface in parallel surfaces for = n+ n
2 n n.
ii. The parallel surface M of the translation surface M has constant mean curvature if and only if the surface M has constant normal curvatures along the generator curves.
iii. The parallel surface M has the same Gauss and mean curvatures if and only if = KH. In this case, from (3.8) we get @ n
@s n and
@ n
@t n. Thus, both of
generator curves are asymptotic curves of M .
Let K and H be Gauss curvature and mean curvature of M . Then, Gauss curvature and mean curvature of M given by (3.6) and (3.7) can be written in terms K and H by using (3.2) and (3.3) as follow.
K = 2H
1 2 H + 2K (3.9)
and
H = H K
1 2 H + 2K: (3.10)
Both (3.9) and (3.10) give us important inequality H2 < K which is guarantee
to us that is neither HK nor H pKH2 K. Thus, on these conditions, we can give the following important remarks.
Remark 1. i. M is a minimal translation surface if and only if its parallel surface is a K ‡at.
ii. Only translation surface whose Gauss and mean curvatures satis…es H2< K
has parallel surfaces, so it can be clearly state that if a translation surface has parallel surfaces then its Gauss curvature is positive.
iii. The harmonic mean curvature of M is equal to 1 = const: if and only if
K = H.
iv. Parallel surfaces of any translation surface can not be minimal surface. In addition, we can give following theorem.
Theorem 1. Let K and H be Gauss curvature and mean curvature of the trans-lation surface M , and let K and H be Gauss curvature and mean curvature of the parallel surface M , respectively. Then, the following equality is satis…ed.
HK KK + 2HH = 0:
By using (3.9) and (3.10), we can write harmonic mean curvature of the parallel surface as K H = 2H H K: (3.11) From (3.11), we obtain H K = 1 2 2 H K: Then, we can give following theorem.
Theorem 2. A translation surface has constant harmonic mean curvature if and only if its parallel surface has constant harmonic mean curvature.
Acknowledgement 1. We thank to referees for providing valuable suggestions and careful reading.
References
[1] Çetin M., Tunçer Y. and Ekmekçi N., "Translation surfaces in Euclidean 3-space", Interna-tional Journal of Physical and Mathematical Sciences", Vol.2, pp.49-56, 2011.
[2] Gray A., "Modern Di¤erential Geometry of Curves and Surfaces with Mathematica 2nd ed.", CRC Press, Washington, 1998.
[3] Liu H., "Translation surfaces with constant mean curvature in 3-dimensional spaces" Journal of Geometry, Vol.64, pp.141-149, 1999.
[4] Munteanu M. and Nistor A. I., On the geometry of the second fundamental form of translation surfaces in E3. Houston J. Math. 37(4), pp.1087-1102, 2011.
[5] Verstraelen L., Walrave J. and Yaprak S., "The minimal translation surfaces in Euclidean space", Soochow Journal of Mathematics, Vol.20, No.1, pp.77-82, 1994.
[6] Yoon D. W., "On the Gauss map of translation surfaces in Minkowski 3-space", Taiwanese Journal of Mathematics, Vol.6, No.3, pp.389-398, 2002.
Address : Celal Bayar University, Instutition of Science and Technology, Manisa-TURKEY E-mail : mat.mcetin@hotmail.com
Address : U¸sak University, Faculty of Sciences and Arts, Department of Mathematics, U¸ sak-TURKEY
E-mail : yilmaz.tuncer@usak.edu.tr
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