Abstract In this paper, we consider the idea of Bertrand curves for curves lying on surfaces in

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1 Bertrad Parter D -Curves i Miowsi -space Mustafa Kazaz a, H. Hüseyi Uğurlu b, Mehmet Öder a, Seda Oral a a Celal Bayar Uiversity, Departmet of Mathematics, Faculty of Arts ad Scieces,, Maisa, Turey. s:mustafa.azaz@bayar.edu.tr, mehmet.oder@bayar.edu.tr b Gazi Uiversity, Gazi Faculty of Educatio, Departmet of Secodary Educatio Sciece ad Mathematics Teachi, Mathematics Teachi Proram, Aara, Turey. huurlu@azi.edu.tr Abstract I this paper, we cosider the idea of Bertrad curves for curves lyi o surfaces i Miowsi -space E. By cosideri the Darboux frame, we defie these curves as Bertrad D -curves ad ive the characterizatios for those curves. We also fid the relatios betwee the eodesic curvatures, the ormal curvatures ad the eodesic torsios of these associated curves. Furthermore, we show that i Miowsi -space E, the defiitio ad the characterizatios of Bertrad D -curves iclude those of Bertrad curves i some special cases. MSC: 5A5, 5B0, 5C50. Key wor: Bertrad D -curves, Darboux frame, Miowsi -space.. Itroductio I the theory of space curves i differetial eometry, the associated curves, the curves for which at the correspodi poits of them oe of the Freet vectors of a curve coicides with the oe of the Freet vectors of the other curve have a importat role for the characterizatios of space curves. The well-ow examples of such curves are Bertrad curves. These special curves are very iteresti ad characterized as a id of correspodi relatio betwee two curves such that the curves have the commo pricipal ormal i.e., the Bertrad curve is a curve which shares the ormal lie with aother curve. These curves have a importat role i the theory of curves. Hereby, from the past to today, a lot of mathematicias have studied o Bertrad curves i differet areas [,,,4,4,5,9]. Also these curves have a importat role i the theory of ruled surface ad i the characterizatios of some other special curves. I [7], Izumiya ad Taeuchi have studied cylidrical helices ad Bertrad curves from the view poit as curves o ruled surfaces. They have show that cylidrical helices ca be costructed from plae curves ad Bertrad curves ca be costructed from spherical curves. Also, they have studied eeric properties of cylidrical helices ad Bertrad curves as applicatios of siularity theory for plae curves ad spherical curves[8]. I [], Gluc has ivestiated the Bertrad curves i -dimesioal Euclidea space E. The correspodi characterizatios of the Bertrad curves i - dimesioal Loretzia space E have bee ive by Tosu ad Ozür[5]. Furthermore, by cosideri the Freet frame of the ruled surfaces, Ravai ad Ku exteded the otio of Bertrad curve to the ruled surfaces ad amed as Bertrad offsets[]. The correspodi characterizatios of the Bertrad offsets of timelie ruled surface i Miowsi -space E were ive by Kuraz [0]. The differetial eometry of the curves fully lyi o a surface i Miowsi -space E has bee ive by Uurlu, Kocayiit ad Topal[9,6,7,8]. They have ive the Darboux frame of the curves accordi to the Loretzia characters of surfaces ad the curves. I this paper, we cosider the otio of the Bertrad curve for the curves lyi o the surfaces i Miowsi -space E. We call these ew associated curves as Bertrad D - curves ad by usi the Darboux frame of the curves we ive the defiitio ad the characterizatios of these special curves. E

2 . Prelimiaries The Miowsi -space E is the real vector space IR provided with the stadart flat metric ive by, = dx + dx + dx where ( x, x, x ) is a rectaular coordiate system of E. A arbitrary vector v = ( v, v, v) i E ca have oe of three Loretzia causal characters; it ca be spacelie if v, v > 0 or v = 0, timelie if v, v < 0 ad ull (lihtlie) if v, v = 0 ad v 0. Similarly, a arbitrary curve α = α ( s) ca locally be spacelie, timelie or ull (lihtlie), if all of its velocity vectors α (s) are respectively spacelie, timelie or ull (lihtlie). We say that a timelie vector is future poiti or past poiti if the first compoud of the vector is positive or eative, respectively. For ay vectors x = ( x) ad y = ( y, y, y) i E, i the meai of Loretz vector product of x ad y is defied by where e e e x y = x x x = ( x y x y, x y x y, x y x y ) y y y δ i = j, ij = ei = ( i, i, i) 0 i j, δ δ δ ad e e = e, e e = e, e e = e. T, N, B the movi Freet frame alo the curve α (s) i the Miowsi Deote by { } space E. For a arbitrary spacelie curve α (s) i the space formulae are ive, T 0 0T N ε 0 = N, B 0 0 B where T, T =, N, N = ε = ±, B, B = ε, T, N = T, B = N, B = 0 E, the followi Freet ad ad are curvature ad torsio of the spacelie curve α (s) respectively. Here, ε determies the id of spacelie curve α (s). If ε =, the α (s) is a spacelie curve with spacelie first pricipal ormal N ad timelie biormal B. If ε =, the α (s) is a spacelie curve with timelie pricipal ormal N ad spacelie biormal B. Furthermore, for a timelie curve α (s) i the space E, the followi Freet formulae are ive i as follows, T 0 0T N 0 = N. B 0 0 B where T, T =, N, N = B, B =, T, N = T, B = N, B = 0 ad ad are curvature ad torsio of the timelie curve α (s) respectively[6,7]. Defiitio.. i) Hyperbolic ale: Let x ad y be future poiti (or past poiti) timelie vectors i IR. The there is a uique real umber 0 such that < y >= x y cosh. This umber is called the hyperbolic ale betwee the vectors x ad y.

3 ii) Cetral ale: Let x ad y be spacelie vectors i IR that spa a timelie vector subspace. The there is a uique real umber 0 such that < y >= x y cosh. This umber is called the cetral ale betwee the vectors x ad y. iii) Spacelie ale: Let x ad y be spacelie vectors i IR that spa a spacelie vector subspace. The there is a uique real umber 0 such that < y >= x y cos. This umber is called the spacelie ale betwee the vectors x ad y. iv) Loretzia timelie ale: Let x be a spacelie vector ad y be a timelie vector i IR. The there is a uique real umber 0 such that < y >= x y sih. This umber is called the Loretzia timelie ale betwee the vectors x ad y []. Defiitio.. A surface i the Miowsi -space IR is called a timelie surface if the iduced metric o the surface is a Loretz metric ad is called a spacelie surface if the iduced metric o the surface is a positive defiite Riemaia metric, i.e., the ormal vector o the spacelie (timelie) surface is a timelie (spacelie) vector, []. Lemma.. I the Miowsi -space IR, the followi properties are satisfied: (i) Two timelie vectors are ever orthooal. (ii) Two ull vectors are orthooal if ad oly if they are liearly depedet. (iii) A timelie vector is ever orthooal to a ull (lihtlie) vector [].. Darboux Frame of a Curve Lyi o a Surface i Miowsi -space Let S be a orieted surface i three-dimesioal Miowsi space E ad let cosider a o-ull curve x( s ) lyi o S fully. Sice the curve x( s ) is also i space, there exists T, N, B at each poits of the curve where T is uit taet vector, N is Freet frame { } pricipal ormal vector ad B is biormal vector, respectively. Sice the curve x( s ) lies o the surface S there exists aother frame of the curve ( ) T,,. I this frame T is the uit x s which is called Darboux frame ad deoted by { } taet of the curve, is the uit ormal of the surface S ad is a uit vector ive by = T. Sice the uit taet T is commo i both Freet frame ad Darboux frame, the vectors N, B, ad lie o the same plae. The, if the surface S is a orieted timelie surface, the relatios betwee these frames ca be ive as follows If the curve x( s ) is timelie. If the curve x( s ) is spacelie. T 0 0 T T 0 0 T = 0 cosϕ siϕ N, 0 coshϕ sihϕ = N. 0 siϕ cosϕ B 0 sihϕ coshϕ B If the surface S is a orieted spacelie surface, the the curve x( s) lyi o S is a spacelie curve. So, the relatios betwee the frames ca be ive as follows T 0 0 T 0 coshϕ sihϕ = N. 0 sihϕ coshϕ B I all cases, ϕ is the ale betwee the vectors ad N. Accordi to the Loretzia causal characters of the surface S ad the curve x( s ) lyi o S, the derivative formulae of the Darboux frame ca be chaed as follows: E

4 i) If the surface S is a timelie surface, the the curve x( s ) lyi o S ca be a spacelie or a timelie curve. Thus, the derivative formulae of the Darboux frame of x( s ) is ive by T 0 ε T = 0 ε, T, T = ε = ±,, = ε,, =. ( a ) 0 ii) If the surface S is a spacelie surface, the the curve x( s ) lyi o S is a spacelie curve. Thus, the derivative formulae of the Darboux frame of x( s ) is ive by T 0 T = 0, T, T =,, =,, =. ( b ) 0 I these formulae, ad are called the eodesic curvature, the ormal curvature ad the eodesic torsio, respectively. Here ad i the followi, we use dot to deote the derivative with respect to the arc leth parameter of a curve. The relatios betwee eodesic curvature, ormal curvature, eodesic torsio ad κ, are ive as follows dϕ = κ cosϕ, = κ siϕ, = +, if both S ad x( s ) are timelie or spacelie, ( a ) dϕ = κ coshϕ, = κ sihϕ, = +, if S is timelie ad x( s ) is spacelie. ( b ) (See [9,8,9]). Furthermore, the eodesic curvature ad eodesic torsio of the curve x( s ) ca be calculated as follows dx d x =,, dx, d = () I the differetial eometry of surfaces, for a curve x( s) lyi o a surface S the followis are well-ow i) x( s) is a eodesic curve = 0, ii) x( s) is a asymptotic lie = 0, iii) x( s) is a pricipal lie = 0 []. 4. Bertrad D -Curves i Miowsi -space E I this sectio, by cosideri the Darboux frame, we defie Bertrad D -curves ad ive the characterizatios of these curves i Miowsi -space E. Defiitio 4.. Let S ad S be orieted surfaces i Miowsi -space E ad let cosider the arc-leth parameter curves x( s ) ad x ( s ) lyi fully o S ad S, respectively. Deote the Darboux frames of x( s ) ad ( ) T,, T,,, respectively. If x s by { } ad { } there exists a correspodi relatioship betwee the curves x ad x such that, at the correspodi poits of the curves, the Darboux frame elemet of x coicides with the Darboux frame elemet of x, the x is called a Bertrad D -curve, ad x is a Bertrad x is said to be a Bertrad D -pair. If there exist such parter D -curve of x. The, the pair { } curves lyi o the orieted surfaces S ad as Bertrad pair surfaces (Fi. ). S, respectively, we call the surface pair { S, S } 4

5 Fi. Bertrad parter D -curves By cosideri the Loretzia casual characters of the surfaces ad the curves, from Defiitio 4., it is easily see that there are five differet types of the Bertrad D -curves i x be a Bertrad D -pair. The accordi to the Miowsi -space. Let the pair { } character of the surface S we have the followis: If both the surface S ad the curve x( s) lyi o S are spacelie the, there are two cases; first oe is that both the surface S ad the curve x ( s ) fully lyi o S are spacelie. x is a Bertrad D -pair of the type. The secod case is I this case we say that the pair { } that both the surface S ad the curve x ( s ) fully lyi o S are timelie. The the pair x is called a Bertrad D -pair of the type. If both the surface S ad the curve { } x( s) lyi o S are timelie the, there are two cases; oe is that both the surface S ad the curve x ( s ) fully lyi o S are timelie. I this case we say that the pair {, } x x is a Bertrad D -pair of the type. The other case is that both the surface S ad the curve x ( s ) fully lyi o S are spacelie the the pair { x } is a Bertrad D -pair of the type 4. If the surface S is timelie ad the curve x( s) lyi o S is spacelie the the surface S is timelie ad the curve x ( s ) fully lyi o S is spacelie. I this case we say that the pair {, } x x is a Bertrad D -pair of the type 5. Theorem 4.. Let S be a orieted surface ad x( s ) be a Bertrad D -curve i E with arc leth parameter s fully lyi o S. If S is aother orieted surface ad x ( s ) is a curve with arc leth parameter s fully lyi o S, the x ( s ) is Bertrad parter D -curve of x( s ) if ad oly if the ormal curvature of x( s ) ad the eodesic curvature ormal curvature ad the eodesic torsio of x ( s ) satisfy the followi equatio i) if the pair {, } x x is a Bertrad D -pair of the type, the ( λ ) λ λ λ x is a Bertrad D -pair of the type, the = + λ ( λ ) cosh λ ii) if the pair { } ( + λ ) λ + λ λ = + + λ ( + λ ) sih λ +, the 5

6 iii) if the pair {, } x x is a Bertrad D -pair of the type, the ( + λ ) λ + λ λ x is a Bertrad D -pair of the type 4, the = + λ ( + λ ) cosh + λ iv) if the pair { } ( λ ) λ λ λ x is a Bertrad D -pair of the type 5, the = + λ ( λ ) sih λ v) if the pair { } ( + λ ) + λ + λ λ = + λ ( λ ) + cos + λ for some ozero costats λ, where is the ale betwee the taet vectors T ad T at the correspodi poits of x ad x. Proof: i) Suppose that the pair {, } frames of x( s ) ad ( ) x s by { } x x is a Bertrad D -pair of the type. Deote the Darboux T,, T,,, respectively. The by the defiitio we ad { } ca assume that x( s ) = x ( s ) + λ( s ) ( s ), (4) for some fuctio λ ( s ). By tai derivative of (4) with respect to s ad applyi the Darboux formulas () we have T = ( λ ) T + λ + λ (5) Sice the directio of coicides with the directio of, i.e., the taet vector T of the curve lies o the plae spaed by the vectors T ad, we et λ ( s ) = 0. This meas that λ is a ozero costat. Thus, the equality (5) ca be writte as follows T = ( λ ) T + λ. (6) Furthermore, we have T = cosht + sih, (7) where is the ale betwee the taet vectors T ad T at the correspodi poits of x ad x. By differetiati this last equatio with respect to s, we et ( + ) = ( + )sih T + ( cosh + sih ) + ( + ) cosh. (8) From this equatio ad the fact that = siht + cosh, (9) we et ( siht + + cosh ) = ( + ) sih T + ( cosh + sih ) (0) + ( + )cosh Sice the directio of is coicidet with we have 6

7 = +. () From (6) ad (7) ad otice that T is orthooal to we obtai λ λ = =. cosh sih () Equality () ives us λ tah = λ. () By tai the derivative of this equatio ad applyi () we et ( λ ) λ λ λ = +, (4) λ ( λ ) cosh λ that is desired. Coversely, assume that the equatio (4) hol for some ozero costats λ. The by usi () ad (), (4) ives us = λ ( λ ) + (( ) ) λ + λ λ (5) Let defie a curve x( s ) = x ( s ) + λ( s ) ( s ). (6) We will prove that x is a Bertrad D -curve ad x is the Bertrad parter D -curve of x. By tai the derivative of (6) with respect to s twice, we et T = ( λ ) T + λ, (7) ad d s ( + ) + T = ( λ + λ ) T + (( λ ) + λ ) + + (( λ ) λ ) respectively. Tai the cross product of (7) with (8) we have + = ( ) T λ λ + λ + ( λ ) ( ) λ λ λ λ ( λ ) ( ) + λ λ By substituti (5) i (9) we et + = ( λ ( ) ) λ + λ T + + ( λ ) + λ ( λ ) ( ) Tai the cross product of (7) with (0) we have 4 + = λ T ( )( ) + ( λ ) λ ( ) λ + λ () + ( λ ) (8) (9) (0) 7

8 From (0) ad () we have 4 ( ) ( ) = λ λ + λ λ T + ( ) λ λ + ( λ ) + λ ( λ ) ( λ ) Furthermore, from (7) ad (0) we et = ( λ ) λ, () = ( λ ) + λ respectively. Substituti () i () we obtai 4 ( ) ( ) = λ λ + λ λ T (4) + ( λ ) + λ ( λ ) ( λ ) sp T,. So, at the Equality (7) ad (4) shows that the vectors T ad lie o the plae { } correspodi poits of the curves, the Darboux frame elemet of x coicides with the Darboux frame elemet of x, i.e, the curves x ad x are Bertrad D -pair curves. Let ow ive the characterizatios of Bertrad parter D -curves of the type i some special cases. Assume that x( s ) be a asymptotic lie. The, from (4) we have the followi special cases: i) Cosider that x ( s ) is a eodesic curve. The x ( s ) is Bertrad parter D -curve of x( s) if ad oly if the followi equatio hol, λ = ( λ ) ii) Assume that x ( s ) is also a asymptotic lie. The x ( s ) is Bertrad parter D - curve of x( s) if ad oly if the eodesic torsio of x ( s ) satisfies the followi equatio, λ =. λ I this case, the Freet frame of the curve x ( s ) coicides with its Darboux frame. From () we have = κ ad =. So, the Bertrad parter D -curves become the Bertrad parter curves, i.e., if both x( s ) ad x ( s ) are asymptotic lies the, the defiitio ad the characterizatios of the Bertrad parter D -curves ivolve those of the Bertrad parter curves i Miowsi -space. The, a ew characterizatio of Bertrad curves ca be ive as follows () 8

9 Corollary 4.. Let x( s) be a spacelie Bertrad curve with arcleth parameter s i Miowsi -space E. The the spacelie curve x ( s ) is a Bertrad parter curve of x( s) if ad oly if the curvature κ ad the torsio of x ( s ) satisfy the followi equatio λ = λ for some ozero costats λ. iii) If x ( s ) is a pricipal lie the x ( s ) is Bertrad parter D -curve of x( s ) if ad oly if the eodesic curvature ad the eodesic torsio of x ( s ) satisfy the followi equality, λ = ( λ ). The proofs of the statemet (ii), (iii), (iv) ad (v) of Theorem 4. ad the particular cases ive above ca be ive by the same way of the proof of statemet (i). Theorem 4.. Let the pair {, } x x is a Bertrad D -pair of the type. The the relatio betwee eodesic curvature, eodesic torsio of x( s) ad the eodesic curvature the eodesic torsio of x ( s ) is ive as follows i) if the pair {, } = λ( + ) ii) if the pair {, } + = λ( + ) iii) if the pair {, } = λ( ) iv) if the pair {, } + = λ( + ) v) if the pair {, } = λ( ). Proof: i) Suppose that the pair {, } from (6) we ca write x ( s ) = x( s ) λ( s ) ( s ) x x is a Bertrad D -pair of the type, the x x is a Bertrad D -pair of the type, the x x is a Bertrad D -pair of the type, the x x is a Bertrad D -pair of the type 4, the x x is a Bertrad D -pair of the type 4, the, x x is a Bertrad D -pair of the type. The by defiitio for some costats λ. By differetiati () with respect to s we have T = ( + λ ) T λ (4) By the defiitio we have T = cosht sih (5) From (4) ad (5) we obtai cosh = ( + λ ), sih = λ (6) Usi () ad (6) it is easily see that = λ( + ). () 9

10 From Theorem 4., we obtai the followi special cases. x be a Bertrad D -pair of the type. The, Let the pair { } i) if oe of the curves x ad x is a pricipal lie, the the relatio betwee the eodesic curvatures ad = λ is ii) if x is a eodesic curve, the the eodesic curvature of the curve x is ive by = λ iii) if x is a eodesic curve, the the eodesic curvature of the curve x is ive by = λ Theorem 4.. Let {, } i) d = x x be Bertrad D -pair of the type. The the followi relatios hold: ii) = sih cosh iii) = cosh + sih iv) = ( sih cosh ) + Proof: i) Sice the pair {, } x x is a Bertrad D -pair of the type, we have T T By differetiati this equatio with respect to s we have d ( + ), T + T, + = sih. Usi the fact that the directio of coicides with the directio of ad T = cosht sih, = siht + cosh we easily et that d =., = cosh. ii) By defiitio we have, = 0. Differetiati this equatio with respect to s we have ( T + ), +, T + = 0. By (7) we obtai = sih cosh iii) By differetiati the equatio T, = 0 ( + ), + T,( T + = 0. From (7) it follows that = cosh + sih. with respect to s we et (7) 0

11 iv) By differetiati the equatio, = 0 ( +, +,( + ) = 0, T T with respect to s we obtai ad usi the fact that directio of coicides with the directio of ad T = cosht + sih, = siht + cosh we et = ( sih cosh ) +. The statemets of Theorem 4. for the pairs {, } as follows ad the proofs ca be easily doe by the same way of the case the pairs {, } of the type. x x of the type,, 4, ad 5 ca be ive x x is For the pair { x } of the type For the pair {, } i) d = + ii) = cosh sih iii) = sih + cosh iv) = ( cosh sih ) d i) = + x x of the type ii) = sih cosh iii) = cosh + sih iv) = ( sih cosh ) + For the pair { x } of the type 4 For the pair {, } x x of the type 5 i) d = ii) = cosh + sih iii) = sih + cosh iv) = ( cosh sih ) d i) = + ii) = si + cos iii) = cos + si iv) = ( si cos ) + Let ow x be a Bertrad D -curve ad x be a Bertrad parter D -curve of x ad the pair { x } be of the type. From the first equatio of () ad by usi the fact that = siht + cosh we have = ( + λ ) cosh sih λ λ + λ. (8)

12 The we ca ive the followi corollary. Corollary 4.. Let x be a Bertrad D -curve ad x be a Bertrad parter D -curve of x ad the pair { x } be of the type. The the relatios betwee the eodesic curvature of x ( s ) ad the eodesic curvature ad the eodesic torsio follows. i) If x is a eodesic curve, the the eodesic curvature of x( s) are ive as of x ( s ) is = λ (cosh sih ) λ. (9) ii) If x is a pricipal lie, the the relatio betwee the eodesic curvatures ad is ive by = ( ) cosh + λ + λ. (0) If the pair {, } is ive as follows x x is of the type,, 4 or 5 the the eodesic curvature of the curve x ( s ) If the pair { x } is of the type If the pair {, } ( λ ) sih λ cosh ( ) + λ λ λ = + x x is of the type = λ cosh + λ sih + λ If the pair { x } is of the type 4 If the pair {, } ( λ ) sih λ cosh ( ) λ + λ λ = + x x is of the type 5 = λ cos + λ si + + λ ad the statemets i Corollary 4. are obtaied by the same way. Similarly, From the secod equatio of () ad by usi the fact that is coicidet with, i.e., = siht + cosh, the eodesic torsio of x is ive by = ( ) cosh ( ) sih cosh sih λ λ λ λ () From () we ca ive the followi corollary. Corollary 4.. Let x be a Bertrad D -curve ad x be a Bertrad parter D -curve of x ad the pair { x } be of the type. The the relatios betwee the eodesic torsio of x ( s ) ad the eodesic curvature ad the eodesic torsio of x( s) are ive as follows. i) If x is a eodesic curve the the eodesic torsio of x is = ( cosh sih cosh ) + λ. ()

13 ii) If x is a pricipal lie the the relatio betwee ad is = ( )sih cosh + λ. () Furthermore, by usi () ad (), from () ad () we have the followi corollary. Corollary 4.4. i) Let {, } x x be Bertrad D -pair of the type ad let x be a eodesic lie. The the eodesic torsio of x ( s ) is ive by ii) Let { } = ( ) ( ) λ λ + λ. (4) x be Bertrad D -pair of the type ad let x be a pricipal lie. The the relatio betwee the eodesic curvatures ad is ive as follows ( + λ )( λ ) = = costat. (5) λ Whe the pair {, } torsio of x ( s ) are ive as follows. x x is of the type,, 4 or 5, the the relatios which ive the eodesic For the pair { x } of the type For the pair {, } = sih λ ( ) + λ λ ) sih cosh x x of the type = ( ) cosh + ( + + ) λ si λ For the pair { x } of the type 4 For the pair {, } ( ) sih ( ) = + λ λ + + λ sih cosh + λ cosh λ λ sih cosh x x of the type 5 ( ) cos ( ) = λ + λ + λ si cos + λ si 4. Coclusios I this paper, the defiitio ad characterizatios of Bertrad parter D -curves are ive which is a ew study of associated curves lyi o surfaces. It is show that the defiitio ad the characterizatios of Bertrad parter D -curves iclude those of Bertrad parter curves i some special cases. Furthermore, the relatios betwee the eodesic curvatures, the ormal curvatures ad the eodesic torsios of these curves are ive. REFERENCES [] Blasche, W., Differetial Geometrie ad Geometrische Grudlae ve Eisteis Relativitasttheorie Dover, New Yor, (945). [] Bure J. F., Bertrad Curves Associated with a Pair of Curves, Mathematics Maazie, Vol. 4, No.. (Sep. - Oct., 960), pp

14 [] Gluc, H., Hiher curvature of Curves i Euclidea Space E, Amer. Math. Mothly, 7 (996), 699. [4] Görülü, E., Ozdamar, E., A eeralizatios of the Bertrad curves as eeral iclied curves i E, Commuicatios de la Fac. Sci. Ui. Aara, Series A, 5 (986), [5] Hacisalihoğlu, H. H., Diferasiyel Geometri, İöü Üiversitesi Fe-Edebiyat Faültesi Yayıları No:, (98). [6] Izumiya, S., Taeuchi, N., Special Curves ad Ruled surfaces, Beitr ae zur Alebra ud Geometrie Cotributios to Alebra ad Geometry, Vo. 44, No., 0-, 00. [7] Izumiya, S., Taeuchi, N., Special Curves ad Ruled surfaces, preprit Isaac Newto Istitute formathematical Scieces, Cambride Uiversity, preprit series, NI-000-SGT 000. [8] Izumiya, S., Taeuchi, N., Geeric properties of helices ad Bertrad curves, Joural of Geometry 74 (00) [9]Kocayiğit, H., Miowsi -Uzayıda Time-lie Asal Normalli Space-lie Eğrileri Freet ve Darboux vetörleri, C.B.Ü. Fe Bilimleri Estisüsü Yüse Lisas Tezi, 004. [0] Kuraz, M., Timelie Rele Yüzeyleri Bertrad Ofsetleri, C.B.Ü. Fe Bilimleri Estitüsü Yüse Lisas Tezi, 004. [] O Neill, B., Elematery Differetial Geometry Academic Press Ic. New Yor, 966. [] O Neill, B., Semi-Riemaia Geometry with Applicatios to Relativity, Academic Press, Lodo, (98). [] Ravai, B., Ku, T. S., Bertrad Offsets of ruled ad developable surfaces, Comp. Aided Geom. Desi, (), No., (99). [4] Strui, D. J., Lectures o Classical Differetial Geometry, d ed. Addiso Wesley, Dover, (988). [5] Tosu M., Özür İ., "Bertrad Curves i the -dimesioal Loretz Space L ", The joural of arts ad scieces, Saarya Uiversity, Faculty of arts ad scieces, Nr., Series A, 997. [6] Uğurlu, H. H., Kocayiğit, H., The Freet ad Darboux Istataeous Rotai Vectors of Curves o Time-Lie Surface, Mathematical & Computatioal Applicatios, Vol., No., pp.-4 (996). [7] Uğurlu, H. H., Topal, A., Relatio Betwee Daboux Istataeous Rotai Vectors of Curves o a Time-Lie Surfaces, Mathematical & Computatioal Applicatios, Vol., No., pp (996). [8]Uğurlu, H. H., O the Geometry of Timelie Surfaces Commuicatio, Aara Uiversity, Faculty of Scieces, Dept. of Math., Series Al, Vol. 46, pp. -(997). [9] Whittemore, J. K., Bertrad curves ad helices, Due Math. J. Volume 6, Number (940),