Timelike Constant Angle Surfaces in Minkowski Space IR

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 6, 0, no. 44, Timelike Constant Angle Surfaces in Minkowski Space IR Fatma Güler, Gülnur Şaffak and Emin Kasap Department of Mathematics, Arts and Science Faculty, Ondokuz Mayis Uniersity, 559 Samsun, Turkey. Abstract In this paper, the classi.cations are gien for the timelike surfaces whose the normal make a constant angle with a constant direction. By taking the direction spacelike (timelike, timelike constant angle surfaces with constant spacelike (timelike direction are deried. It is shown that the minimal timelike constant angle surfaces are planes. Finally,examples are gien to show the timelike constant angle surfaces. Mathematics Subject Classi.cation: 4J6, 5B0, 5B0 Keywords: Constant angle, Minkowski space, Ruled surface. Introduction A constant angle surfaces in Euclidean -space is a surface whose the unit normal makes a constant angle with a fixed direction. These surfaces generalize the concept of helix, that is, cures whose tangent lines make a constant angle with a fixed ector of E : Helical features characterise all screws and bolts as well as some gears. These components therefore play important roles in mechanical construction and are studied in undergraduate mechanical engineering. Their representations in an assembly would be treated in courses on mechanics and machines, and their production, to some extent, in workshop practice. Also, DNA (deoxyribonucleic acid is a double-stranded molecule that is twisted into a helix like spiral staircase. Recently, constant angle surfaces hae been the subject of some studies: Munteanu and Nistor [6] studied constant angle surfaces in Euclidean -space. They obtained classifications for all constant angle surfaces in E : Cermelli and Scala [] proed that constant angle surfaces hae some important applications to physics, it was shown how constant angle surfaces can be used to describe interfaces occurring in special equilibrium con.gurations of liquid crystals and layered.uids. Lopez and Munteanu [] studied spacelike constant angle surfaces in Minkowski -space. They gae the extension of constant angle

2 90 F. Güler, G. Şaffak and E. Kasap property for the special ruled surfaces. The objectie of the study in this paper is to classify timelike constant angle surfaces in Minkowski -space IR. By choosing the constant direction spacelike or timelike, we obtain different parametrizations for the timelike surfaces. Moreoer, we show that minimal timelike constant angle surfaces are planes.. Preliminaries Let us considerminkowski -space IR =, IR ( ++,, of X = ( x, x, x and Y = ( y, y, y IR and let the Lorentzianinner product X, Y = xy+ xy xy (. The norm of X IR is detoned by X and defines as X = X, X, []. A ector X ( x, x, x IR = is called a spacelike, timelike and null (light- like ector if X, X > 0 or X = 0, X, X < 0 and X, X = 0 for X 0, respectiely, []. A timelike ector is said to be positie (resp.negatie if and only if x > 0 (resp. x < 0, [4]. Let X and Y be spacelike ectors in IR. If the inequality X, Y > X Y is satisfied, there is a unique real number α such that XY, = X Ycoshα (. If the inequality X, Y X Y is satisfied, there is a unique real number α such that XY, = X Ycosα (. Let X be spacelike ector and Y be a positie timelike ector in IR. Then there is a unique nonnegatie real number α such that XY, = X Ysinhα,[5]. (.4 A surface M in IR is called a timelike surface if the induced metric g on the surface is a Lorentz metric. The normal ector on the timelike surface is a spacelike ector, [5]. Let M is a timelike surface in X, Y χ M, we get IR and N is unit normal of M. For all DXY D Y g( S( X, Y N = + (.5 X g is induced metric on the surface M, D and D are Lei-Ciita connections on and M, respectiely. S: χ( M χ( M, S( X = DX N is the shape operator of M, []. r = r u, is gien by The mean curature H of te timelike surface H = ( FM GL En( F EG (.6 E = g( r, r, F = g r, r, G = g r, r, L= g r, N, M = g r, N, n= g r, N u u u uu u IR

3 Timelike constant angle surfaces 9 are coefficients of the Minkowski first and second fundamental forms, [7]. A timelike surface with anishing mean cuırature is called a minimal surface, [8]. Timelike Constant Angle Surfaces with Constant Spacelike Direction Let M be a timelike surface and α be constant angle between the unit normal N = ( n, n, n and the fixed spacelike direction k. Without loss of generality, the fixed direction is taken to be the.rst real axis. There are two cases for angle α : a if g N, k = coshα n >, from (., we get n, from (., we get g( N k b if, = cosα Now we will examine these cases : a Let n >. Since k is a spacelike unit ector, for an unitary timelike ector field e on M, we get k = sinhαe + coshα N (. Lemma. Let e be an unitary ector field on M and orthogonal to e. For the, χ M, we get orthonormal basis { e e } of e e D N = λe, D e = λcoth αe, λ = λ u, D N D e D e e = e = e = 0 Proof. By applying D e to the equality (., we hae Dek = sinhαdee + coshαden (. Since e g( N, e = 0, we get g De N, e + g N, De e = 0 (. ( ( As e g( N, N = 0. This clearly implies that D N χ ( M Therefore e. De N = λe+ λe (.4 From (. and (.4, we get Dee = cothα ( λe+ λe (.5 we will inestigate the case α = 0 later. It is easy to see, from (., that DeN = λe and Dee = λcothαe Next, by applying D e for the equality (., we obtain Dek = sinhαdee + coshαden (.6 Since e g( N, N = 0, we get De N = μe + μ e, μ, μ IR (.7

4 9 F. Güler, G. Şaffak and E. Kasap, = 0,, we hae De e = μcothαe (.8 Because the shape operator S of M is symmetric, we see that, g S e, e = g e, S e, From (.6, (.7 and e g( e e ( = λ = 0 μ Hence, we get De N = Dee = 0 With respect to the basis { e, e, } N of De e = ae + a e + a N, a, a, a IR Simply calculations gie De e = 0 Now, we can gie the following results: IR, we can write Corollary. De= De= 0, De= λ coth αe, De= λcothαe. e e e e The timelike ruled surface M can be expressed as r= r u, = xu,, y u,, z u, We may assume that, β (, r = e r = u e β : M IR is a smooth function. By u corollary., we get ruu = 0, ru = βu r, ru = λcothαr β and r = λβ cothαru + βr λβ N β Since r u = r u and Nu = Nu, we obtain the following diferential equations: βu + λβ cothα = 0, λu λ cothα = 0 (.9 soling the equations (.9, we hae tanhα λ( u, =, β( u, = ϕ ( u+γ (.0 u+γ or ( u, 0, ( u, λ = β = β (. Now we can gie the classifications for M. Let (.0 be a solution for equations (.9. Since g( ru, k = sinh α and g( r, k = 0, we get (, = sinh α, (, r u u h u

5 Timelike constant angle surfaces 9 hu (, IR. As ( u g( h, h = y z = cosh α. u u u u Therefore, we obtain h = cosh α f,coshα f g r, r =, we hae that u ( f = ( f, f IR and f ( =. Thus, we get r( u, = usinh α, ucosh α f + γ, ucoshα f + γ As r u ( = r, it is easy to see that u dγ df = coshα Γ d d dγ df = cosh α Γ. d d Without loss of generality, if we get f ( cosh,sinh, r( u, usinh α, ucoshαcosh γ, ucoshαsinh γ = this implies that ( = + + (. γ ( γ, γ coshα = = sinh τγ( τ dτ, coshτγ( τ dτ 0 0 Let (. be a solution for equations (.9. Since β u = 0, we hae r u = 0. As r uu = 0and r u = 0, we get h uu = 0 and h u = 0. This implies that h u is constant ector in IR. We may assume that hu = ( coshα cosh μ, coshαsinh μ such that α is a constant. Therefore, we can write hu, = ucoshαcosh μ+ γ, ucoshαsinh μ+ γ ( Because r u and r are orthogonal, we obtain (, ( sinh, cosh γ = γ γ = μγ μγ The last equation implies that r( u, = ( usinh α, ucoshαcosh μ+ sinh μγ, ucoshαsinh μ cosh μγ By applying a Lorentz transformation of the form 0 coshμ sinhμ 0 sinhμ coshμ 0 0 We obtain r( u, = ( ucosh α, Γ, usinhα This is a parametrization for the Lorentz plane sinhα x+ coshα z = 0 (.

6 94 F. Güler, G. Şaffak and E. Kasap Special Case: If α = 0 then k = N. It follows that the unit normal of M is a constant. Since, 0, 0 yz, -plane. k =, M is a Lorentz plane which is parallel to b Let n. Since k is a spacelike unit ector, for an unitary spacelike ector field e on M, we get k = sinαe + cosα N (.4 Lemma. Let e be an unitary ector field on M and orthogonal to e. For the orthonormal e, e of χ M, we get basis { } e e D N = λe, D e = λcot αe, λ = λ u, e = e = e = 0 D N D e D e Proof. It can be proed as Lemma.. Corollary. De e = De e = 0, De e = λ cot αe, De e = λcotαe. By similar arguments, for the surface r= r u, = xu,, y u,, z u, such that r e, r β ( u, e u = = we obtain that ruu = 0, ru = βu r, ru = λcotαr β and r = λβ cotαru βr + λβ N β Since r u = r u and Nu = Nu, we hae the following diferential equations: βu + λβcotα = 0, λu λ cotα = 0 (.5 soling the equations (.5, we hae tanα λ( u, =, β( u, = ϕ ( u+γ (.6 u+γ or λ( u, = 0, β( u, = β (.7 By similar arguments if (.6 is a solution for equations (.5, we hae the following parametrization of M : r( u, = ( usin α, ucosαcos + γ, ucosαsin + γ (.8 γ ( γ, γ cosα = = sinh τγ( τ dτ, coshτγ( τ dτ 0 0 If (.7 is a solution for equations (.5, we obtain that r u, = usin α, ucosαcosh μ+ sinh μγ, ucosαsinh μ cosh μγ

7 Timelike constant angle surfaces 95 By applying a Lorentz transformation of the form 0 coshμ sinhμ 0 sinhμ cosh μ. 0 0 We obtain r( u, = ( ucos α, Γ, usinα This is a parametrization for the Lorentz plane sinα x cosα z = 0 (.9 Special Cases: If α = 0 then k N. yz, plane. If = Thus, M is a Lorentz plane which is parallel to π α = then k is tangent to M. It is follows that r u, = u, γ, γ (.0 ( Where γ γ, γ ( = IR. In this case, M is a part of the cylindrical surface. Finally, from (., (., (.8, (.9 and (.0 we can gie the following theorem: Theorem. Eery timelike constant angle surface M with constant spacelike direction is a congruent to the following surfaces: (i r( u, = ( usinh α, ucoshαcosh + γ, ucoshαsinh + γ γ ( γ, γ coshα = = sinh τγ( τ dτ, coshτγ( τ dτ 0 0 ( (ii r( u, = usin α, ucosαcosh + γ, ucosαsinh + γ γ ( γ, γ cosα = = sinh τγ( τ dτ, coshτγ( τ dτ 0 0 (iii A Lorentz plane which hae the equation sinhα x+ coshα z = 0 or sinα x cosα z = 0. (i A Lorentz plane which is parallel to ( yz, plane ( A part of the cylindrical surface. Example. a If we take η = + and α = in (., we obtain the following parametrization for M :

8 96 F. Güler, G. Şaffak and E. Kasap r( u, = usinh (,cosh ucosh + sinh,cosh usinh + sinh +, (Fig. π b If we take η = and α = in (.8, we obtain the following parametrization 4 for M : r( u, = u, (( u+ cosh, ( u+ sinh, (Fig. Figure.Timelike constant angle surface with spacelike direction Figure.Timelike constant angle surface with spacelike direction 4 Timelike Constant Angle Surfaces with Constant Timelike Direction Let M be a timelike surface and α be constant angle between the unit normal N and the fixed timelike direction k. Without loss of generality, the fixed direction is taken to be the third real axis. Since k is a positie timelike ector, for an unitary timelike ector field e on M, we get k = coshαe + sinhα N (4. Lemma 4. Let e be an unitary ector field on M and orthogonal to e. For the orthonormal e, e of χ M, we get basis { } e e D N = λe, D e = λtanh αe, λ = λ u,

9 Timelike constant angle surfaces 97 DeN = Dee = Dee = 0 Proof. It can be proed as Lemma.. Corollary 4. De= De= 0, De= λ tanh αe, De= λtanhαe. e e e e By similar arguments, for the surface r= r u, = xu,, y u,, z u, such that r e, r β ( u, e u = = we obtain that ruu = 0, ru = βu r, ru = λtanhαr β and r = λβ tanhαru + βr λβ N β Since r u = r u and Nu = Nu, we hae the following diferential equations: βu + λβ tanhα = 0, λu λ tanhα = 0 (4. soling the equations (4., we hae cothα λ( u, =, β( u, = ϕ ( u+γ (4. u+γ or ( u, 0, ( u, λ = β = β (4.4 Now we can gie the classifications for M. Let (4. be a solution for equations (4.. Since g( r, u k = cosh α and g( r, k = 0, we get: r( u, = h( u,, ucoshα hu (, IR. As g( ru r u g( h, h = x + y = sinh α. u u u u Therefore, we obtain h = sinh α f,sinhα f, =,we hae that u ( f = ( f, f IR and. = + + f = Thus, we get ( α γ α γ α r u, usinh f, usinh f, ucosh r As u = r, it is easy to see that u dγ df = sinhα Γ d d dγ df = sinh α Γ. d d Without loss of generality, if we get f ( = cos,sin, this implies that

10 98 F. Güler, G. Şaffak and E. Kasap ( α γ α γ α r u, = usinh cos +, usinh sin +, ucosh (4.5 γ = ( γ, γ = sinhα sin τγ( τ dτ, cosτγ( τ dτ 0 0 Let (4.4 be a solution for equations (4.. By similar arguments, we can obtain that r( u, = ( usinhα cos μ sin μγ, usinhαsin μ+ cos μγ, ucoshα γ = γ, γ = sin μγ,cos μγ. ( ( In the spacelike ( x, y plane. By applying a transformation of the form cos μ sin μ 0 sin μ cos μ We obtain r( u, = ( usinh α, Γ, ucoshα This is a parametrization for the Lorentz plane coshα x sinhα z = 0. (4.6 Special Case: If α = 0 then k is tangent to M. It ifollows that r u, = γ, γ, u (4.7 ( γ γ γ ( =, IR. In this case, M is a part of the cylindrical surface. Finally, we can gie the following theorem: Theorem 4. Eery timelike constant angle surface M with constant timelike direction is a congruent to the following surfaces: (i r( u, = ( usinhα cos + γ, usinhαsin + γ, ucoshα γ = ( γ, γ = sinhα sin τγ( τ dτ, cosτγ( τ dτ 0 0 (ii A Lorentz plane which hae the equation coshα x sinhα z = 0. (iii A part of the cylindrical surface. From (.6, the mean curature of the timelike constant angle surface M is gien by H = ελ ε = g k, k. Thus we can gie the following result: Corollary 4. Minimal timelike constant angle surface are the planes.

11 Timelike constant angle surfaces 99 Example 4. If we take for M : η = and α = in (4.5, we obtain the following parametrization ( ( r u, = sinh u+ cos sin,sinh u+ sin + cos, ucosh,(fig. Figure.Timelike constant angle surface with timelike direction References [] Beem, J. K. and Ehrlich, P. E. Global Lorentzian Geometry. Marcel Dekker. Inc. New York, 98. [] Cermelli, P. and Di Scala, A. J.: Constant-angle surfaces in liquid crystals, Philosophical Magazine, 87(, (007. [] Lopez and Munteanu Constant-angle surfaces in Minkowski space, arxi: [math.dg], 009. [4] M.I.Munteanu, A.I.Nistor, A New Approach on Constant Angle Surface in Math. (009., -0 [5] O.Neill, B. Semi-Riemannian Geometry. Academic Press. New York, 98. [6] Ratcliffe, J. G. Foundations of Hyperbolic Manifolds. Springer 005. [7] Weinstein, T. Lorentz Surfaces. Walter de Gruyter, New York 996. E, Turkish J.

12 00 F. Güler, G. Şaffak and E. Kasap [8] Woestijne, V. D. I. Minimal surfaces of the -dimensional Minkowski space, World Scientific Publishing Singapore, 44-69, 990. Receied: March, 0