NULL HELIX AND k-type NULL SLANT HELICES IN E 4 1

REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 57, No. 1, 2016, Page 71 83 Publihed online: March 3, 2016 NULL HELIX AND k-type NULL SLANT HELICES IN E 4 1 JINHUA QIAN AND YOUNG HO KIM Abtract. We tudy the null helix and k-type null lant helice in 4-dimenional Minkowki pace E 4 1 and characterize them in term of the null curvature, null torion and tructure function. Some null helice and k-type null lant helice are preented by olving pecial differential equation. 1. Introduction In recent year, with the development of the theory of relativity, geometer have extended ome topic in claical differential geometry of Riemannian manifold to that of Lorentzian manifold. For intance, in the Euclidean pace E 3, a curve whoe tangent vector make a contant angle with a fixed direction i called a helix, and the curve with non-vanihing curvature i called a lant helix if the principal normal line make a contant angle with a fixed direction of the ambient pace. Some work ha been done extending the definition of helix and lant helix to thoe in Minkowki pace (ee [1, 2, 6, 12]). Many problem in Euclidean pace can find their counterpart in Minkowki pace. However, due to the caual character of the vector in Minkowki pace, ome problem become different and new method are needed in order to dicu them, epecially for the problem related to null curve. In 2011, the author of [10] introduced a new method to decribe cone curve in which the tructure function of the cone curve on Q 3 are preented, where Q 3 = {v E 4 1 v, v = 0}. Naturally, a null curve a the integration curve of a cone curve alo can be characterized by thoe tructure function. In [2] the author defined k-type lant helice and dicued them for the partially null and peudo null curve in E 4 1. Motivated by thee idea, we introduce the definition of null helix and k-type null lant helice in E 4 1. Then we dicu their propertie by uing the Cartan Frenet frame in 2010 Mathematic Subject Claification. 53A04, 53B30, 53C40. Key word and phrae. Minkowki pace, null helix, k-type null lant helix, null curvature, null torion, tructure function. The firt author wa upported by NSFC (No. 11371080) and the Baic Science Reearch Program funded by the Minitry of Education (No. 150504007). The econd author wa upported by the Baic Science Reearch Program through the National Reearch Foundation of Korea (NRF) funded by the Minitry of Education, Science and Technology (2012R1A1A2042298). 71

72 JINHUA QIAN AND YOUNG HO KIM [10] and characterize them in term of the null curvature, null torion and tructure function. The curve under conideration in thi paper are regular unle otherwie tated. 2. Preliminarie and definition In thi ection we review ome baic fact of null curve in E 4 1 and introduce the definition of null helix and k-type null lant helice. 2.1. Minkowki 4-pace E 4 1. Minkowki 4-pace E 4 1 i the real 4-dimenional vector pace R 4 equipped with the tandard flat metric given by g = dx 2 1 + dx 2 2 + dx 2 3 dx 2 4 in term of rectangular coordinate (x 1, x 2, x 3, x 4 ). Recall that a non-zero vector υ E 4 1 i pace-like if g(v, v) > 0, time-like if g(v, v) < 0, and null (light-like) if g(v, v) = 0. In particular, the zero vector i regarded a a pace-like vector. A curve r() in E 4 1 i called pace-like, time-like or null (light-like) if all of it velocity vector r () are pace-like, time-like or null, repectively. For null curve, it i not poible to normalize the tangent vector in the uual way becaue the arc-length vanihe. A method of proceeding i to denote a new parameter called the null arc-length parameter which normalize the derivative of the tangent vector uch that g(r (), r ()) = 1 (ee [10]). Propoition 2.1. Let r() : I E 4 1 be a null curve parametrized by null arclength. Then r() can be framed by a unique Cartan Frenet frame {x, α, y, β} uch that r () = x(), x () = α(), α () = κ()x() y(), (2.1) y () = κ()α() τ()β(), β () = τ()x(), where g(x, x) = g(y, y) = g(x, α) = g(y, α) = g(x, β) = g(y, β) = g(α, β) = 0, g(x, y) = g(α, α) = g(β, β) = det(x, α, β, y) = 1. In equence, x, α, y, β i called the tangent, principal normal, firt binormal, and econd binormal vector field of r(). The function κ() and τ() are called the null curvature and null torion function, repectively. 2.2. Repreentation formula of null curve in E 4 1. In [10] the author decribed cone curve with the tructure function f() and g() in Q 3 E 4 1. A null curve in E 4 1 a the integration curve of a cone curve in Q 3 can be characterized in the ene of [10] a follow: Propoition 2.2. Let r() : I E 4 1 be a null curve with null arc-length parameter. Then r() can be written a r() = ρ(2f, 2g, 1 f 2 g 2, 1 + f 2 + g 2 ) d, (2.2)

NULL HELIX AND k-type NULL SLANT HELICES IN E 4 1 73 where f() and g() are called the tructure function of r(). The null curvature κ(), null torion τ(), tructure function f(), and g() atify where κ() = 1 2 [(log ρ) ] 2 + (log ρ) 1 2 θ2, τ 2 () = (θ (log ρ) + θ ) 2, 4ρ 2 ()[f 2 () + g 2 ()] = 1, θ = ( 1 + f 2 g 2 2.3. Null helix and k-type null lant helice in E 4 1. ) 1 ( ) f g. (2.3) Definition. Let r = r() be a null curve parametrized by null arc-length with the Cartan Frenet frame {x, α, y, β} in E 4 1. If there exit a non-zero contant vector field V in E 4 1 uch that g(x, V ) 0 i a contant for all I, then r i aid to be a null helix and V i called the axi of r. Definition. Let r = r() be a null curve parametrized by null arc-length with the Cartan Frenet frame {x, α, y, β} in E 4 1. If there exit a non-zero contant vector field V in E 4 1 uch that g(α, V ) 0 (repectively, g(y, V ) 0, g(β, V ) 0) i a contant for all I, then r i aid to be a k-type (k = 1, 2, 3, repectively) null lant helix and V i called the lope axi of r. At the end of thi ection, we make ome notation and calculation for later ue. Let V be an axi (or a lope axi) of a null helix (or a k-type null lant helix) r(). Then V can be decompoed a V = v 1 x() + v 2 α() + v 3 β() + v 4 y(), (2.4) where {x, α, y, β} i the Cartan Frenet frame of r(), and v i = v i () (i = 1, 2, 3, 4) are differentiable function on null arc-length. Thu v 1 = g(y, V ), v 2 = g(α, V ), v 3 = g(β, V ), v 4 = g(x, V ). By taking derivative on both ide of (2.4), we get (v 1 + v 2 κ + v 3 τ)x + (v 1 + v 2 v 4 κ)α + (v 3 v 4 τ)β + (v 4 v 2 )y = 0, which implie v 1 + v 2 κ + v 3 τ = 0, v 1 + v 2 v 4 κ = 0, v 3 v 4 τ = 0, v 4 v 2 = 0. 3. Null helix (2.5) Theorem 3.1. Let r = r() be a null curve in E 4 1. Then r i a null helix if and only if κ = τ τd.

74 JINHUA QIAN AND YOUNG HO KIM Proof. According to the definition of null helix, we have v 4 = g(x, V ) = C 0, where C 0 i a non-zero contant. Subtituting it into (2.5), we get v 1 + v 3 τ = 0, v 1 = C 0 κ, v 3 = C 0 τ, v 2 = 0. By (3.1) we find that C 0 κ = C 0 τ τd. (3.1) Becaue C 0 0, we have κ = τ τd. (3.2) Converely, uppoe (3.2) hold for a null curve r(). By chooing the vector field V a ( ) V = cκx + c τd β + cy, (c R {0}), we have V = 0 and g(x, V ) = c. So V i a contant vector field and r() i a null helix. From Theorem 3.1, we have: Corollary 3.2. Let r = r() be a null helix in E 4 1. Then the axe of r are obtained by ( ) V = cκx + c τd β + cy, (c R {0}), and the axe V lie fully in the rectifying pace of r(). Remark 3.3. The rectifying pace of r() i the orthogonal complement of it principal normal vector field α in E 4 1. Corollary 3.4. Let r = r() be a null helix with vanihing null torion in E 4 1. Then r can be written a follow: (1) r = (, 1 2 2, 1 6 3, 1 6 3 + ) for κ = 0; ( ) (2) r = a, b, (a 2 + b 2 ) coh( ), a2 +b 2 (a2 + b 2 ) inh( ) a2 for κ > 0; +b 2 ( ) (3) r = (b 2 a 2 ) co( ), a 2 b 2 (a2 b 2 ) in( ), b, a a for κ < 0, 2 b 2 where a, b R, a 2 + b 2 0 for κ > 0, and a 2 b 2 > 0 for κ < 0. Proof. When the null torion τ 0, from Theorem 3.1 the null curvature κ i a contant. From (2.3), the tructure function can be written a follow:

NULL HELIX AND k-type NULL SLANT HELICES IN E 4 1 75 Cae 1: If κ = 0, then we have f() = 1 g() = 2 +1, 2 +1, 2ρ() = 2 + 1. Cae 2: If κ > 0, then we get f() = a 2ρ, g() = b 2ρ, 2ρ() = a 2 + b (inh( 2 ) + coh( a 2 +b 2 where a, b R and a 2 + b 2 0. Cae 3: If κ < 0, then we obtain f() = a 2 b 2 a+b g() = a 2 b 2 a+b 2ρ() = a + b, ( ) in( ) a, ( 2 b 2 ) co( ) a2, b 2 ) ) a 2 +b 2, (3.3) (3.4) (3.5) where a, b R and a 2 b 2 > 0. From (2.2), (3.3), (3.4) and (3.5), the expreion form of r() can be eaily achieved (ee [10]). Theorem 3.5. Let r = r() be a null helix with non-zero contant null curvature and vanihing null torion in E 4 1. Then (1) r i a Bertrand curve; (2) r i a null oculating curve of the firt kind; (3) r i a 2-type and 3-type null lant helix. Proof. In E 4 1, a null curve i a Bertrand curve if and only if κ i a non-zero contant and τ 0 (ee [4]), o r i a Bertrand curve. In [8], a null curve i an oculating curve of the firt kind if and only if τ i equal to zero, therefore we get the econd concluion. At the ame time, from the axe of null helice denoted in Corollary 3.2, we have g(y, V ) = cκ, g(β, V ) = c τd. Thu, g(y, V ) and g(β, V ) are all non-zero contant when τ 0 and κ i a non-zero contant, where τd 0 i alway aumed. The proof i thu completed. 4. k-type null lant helice In thi ection we dicu three kind of null lant helice.

76 JINHUA QIAN AND YOUNG HO KIM 4.1. 1-type null lant helix. Theorem 4.1. Let r = r() be a null curve in E 4 1. Then r i a 1-type null lant helix if and only if 2κ + κ = τ τd. Proof. Baed on the definition of 1-type null lant helix, we have v 2 = g(α, V ) = C 0, where C 0 i a non-zero contant. Subtituting it into (2.5), we get v 1 + C 0 κ + v 3 τ = 0, v 1 v 4 κ = 0, v 3 v 4 τ = 0, v 4 = C 0. (4.1) By the lat equation in (4.1), we have v 4 = C 0 (putting the integration contant to be zero by tranformation). Then, the coefficient of V are expreed by v 1 = C 0 κ, v 2 = C 0, v 3 = C 0 ( 2κ+κ v 4 = C 0. τ ) = C 0 τd, (4.2) Becaue C 0 0, from the third equation in (4.2) we get 2κ + κ = τ τd. (4.3) Converely, aume (4.3) hold for a null curve r(). We can define a vector field V a follow: ( ) V = cκx + cα + c τd β + cy, (c R {0}). Then, we have V = 0 and g(α, V ) = c. Thi complete the proof. A a conequence of Theorem 4.1, we have Corollary 4.2. Let r = r() be a 1-type null lant helix in E 4 1. Then the lope axe of r are written a ( ) V = cκx + cα + c τd β + cy, (c R {0}). Corollary 4.3. Let r = r() be a 1-type null lant helix with vanihing null torion in E 4 1. Then the null curvature i given by k = c 2, where c i a contant, and r() can be written a follow: (1) r() = C 1 2 + C 2 2 log + C 3 2 log 2 + C 4, for 2c = 1;

NULL HELIX AND k-type NULL SLANT HELICES IN E 4 1 77 (2) r() = C 1 2 + C 2 (2+ 1+2c) + C 3 (2 1+2c) + C 4, for 2c > 1; (3) r() = C 1 2 + C 2 2 in[( 1 2c) log ] + C 3 2 co[( 1 2c) log ] + C 4, for 2c < 1, where C i (i = 1, 2, 3, 4) E 4 1. Proof. When the null torion τ = 0, from Theorem 4.1 we have from which we get k = c 2, By the Frenet formula (2.1) we have 2κ + κ = 0, (c R). Solving the above Euler equation (4.4), we get 3 x 2cx + 2cx = 0. (4.4) (1) x() = B 1 + B 2 log + B 3 log 2, for 2c = 1; (2) x() = B 1 + B 2 (1+ 1+2c) + B 3 (1 1+2c), for 2c > 1; (3) x() = B 1 + B 2 in[( 1 2c) log ] + B 3 co[( 1 2c) log ], for 2c < 1, where B i (i = 1, 2, 3) E 4 1. Integrating x() with repect to, the expreion form of r() can be obtained. Theorem 4.4. Let r = r() be a 1-type null lant helix with vanihing null torion in E 4 1. Then (1) r i a null oculating curve of the firt kind; (2) r i a 3-type null lant helix. Proof. A denoted in the proof of Theorem 3.5, r i a null oculating curve of the firt kind. At the ame time, from the lope axe of 1-type null lant helice, we have g(v, β) = c τd. Thu, g(v, β) i a non-zero contant when τ = 0 and τd 0. Thi complete the proof. 4.2. 2-type null lant helix. Theorem 4.5. Let r = r() be a null curve in E 4 1. Then r i a 2-type null lant helix if and only if where v 4 atifie (4.6). τ κ = ± v 4 2C0 v 4 v 2 4 + a, (C 0 0, a R),

78 JINHUA QIAN AND YOUNG HO KIM Proof. From the definition of 2-type null lant helix, we have v 1 = g(y, V ) = C 0, where C 0 i a non-zero contant. Subtituting it into (2.5), we get v 2 κ + v 3 τ = 0, C 0 + v 2 v 4 κ = 0, v 3 v 4 τ = 0, v 4 = v 2, (4.5) from which we have v 4 κv 4 + C 0 = 0. (4.6) Firt, in order to olve equation (4.6), we conider Cae 1: If κ < 0, then we have v 4 = c exp v 4 κv 4 = 0. (4.7) ( ) κ co θ in θ d, where c i a contant and the newly defined function θ() atifie θ = κ + κ in θ co θ. 2κ Cae 2: If κ > 0, then we get ( ) κ coh θ v 4 = c exp inh θ d, where c i a contant and θ() atifie θ = κ + κ inh θ coh θ. 2κ Let y 1, y 2 be two non-trivial linearly independent olution of (4.7). Then the general olution of (4.6) can be obtained in the form d v 4 = C 1 y 1 + C 2 y 2 C 0 y 2 y 1 W + C d 0y 1 y 2 W, where C 1, C 2 R and W = y 1 y 2 y 2 y 1 (ee [11, p. 142]). At the ame time, from (4.5) we have By (4.6), we have Therefore, Thu, we obtain from (4.5) v 3 v 3 + v 4 v 4κ = 0. (v 2 3) = 2C 0 v 4 2v 4v 4. v 3 = ± 2C 0 v 4 (v 4 )2 + a, (C 0 0, a R). τ κ = ± v 4 (4.8) 2C0 v 4 v 4 2 + a.

NULL HELIX AND k-type NULL SLANT HELICES IN E 4 1 79 Converely, aume (4.8) hold for a null curve r(). By chooing the vector field V a follow: V = cx + v 4α κv 4 τ β + v 4y, (0 c R) we have V = 0 and g(y, V ) = c. A a conequence of the above theorem, we have Corollary 4.6. Let r = r() be a 2-type null lant helix. Then the lope axe of r are obtained by V = cx + v 4α κv 4 τ β + v 4y, where v 4 atifie (4.6). (0 c R), Theorem 4.7. Let r = r() be a 2-type null lant helix. Suppoe the ratio of the null curvature and null torion i a non-zero contant. (1) If κ i a contant, then we have r() = ( ( a 2 + b 2 ) 1 1 a co a, 1 a in a, 1 b coh b, 1 ) b inh b. (2) If κ i not a contant, then the null curvature and null torion can be expreed by 2c 2 C 0 κ = C 0 2 + 2a(1 + c 2 ) + 2b(1 + c 2 ), 2cC 0 τ = C 0 2 + 2a(1 + c 2 ) + 2b(1 + c 2 ), where a, b, c, C 0 R and c 0, C 0 0. Proof. Aume κ = cτ (0 c R). Then we have the following two cae: Cae 1: κ i a contant. Thi mean that τ i alo a contant. In uch cae, the tructure function can be obtained by (2.3) a follow: in a f() = inh b+coh b, co a g() = inh b+coh b, (4.9) inh b+coh b 2ρ() = a2 +b 2. By (2.2) and (4.9), r() can be written a r() = ( a 2 + b 2 ) 1 ( 1 a co a, 1 a in a, 1 b coh b, 1 b inh b ), where a, b R and ab 0. Cae 2: κ i not a contant. In uch cae, from κ = cτ and (4.5) we get Subtituting it into (4.5), we have v 3 = cv 2. v 4 = C 0 1 + c 2,

80 JINHUA QIAN AND YOUNG HO KIM from which we have Furthermore, by (4.5) we get C 0 v 4 = 1 2 1 + c 2 2 + a + b, (a, b R). where a, b, c, C 0 R and c 0, C 0 0. 2c 2 C 0 κ = C 0 2 + 2a(1 + c 2 ) + 2b(1 + c 2 ), 2cC 0 τ = C 0 2 + 2a(1 + c 2 ) + 2b(1 + c 2 ), 4.3. 3-type null lant helix. Theorem 4.8. Let r = r() be a null curve in E 4 1. Then r i a 3-type null lant helix if and only if the null torion vanihe and where v 4 = g(x, V ). v 4 = 2v 4κ + v 4 κ, Proof. From the definition of 3-type null lant helix, we have v 3 = g(β, V ) = C 0, where C 0 i a non-zero contant. Subtituting it into (2.5), we get v 1 + v 2 κ + C 0 τ = 0, v 1 + v 2 v 4 κ = 0, v 4 τ = 0, v 4 = v 2. (4.10) By the third equation in (4.10), we conider the open ubet O = {τ() 0} of r. Firt, we aume O. Then, v 4 = 0 on O by (4.10). However, v 4 = 0 implie v i = 0 (i = 1, 2, 3, 4). Thi i a contradiction. So O = and τ 0 by continuity. Thu, (4.10) implie v 1 + v 2 κ = 0, v 1 + v 2 v 4 κ = 0, v 4 = v 2. (4.11) From (4.11) we have v 4 = 2v 4κ + v 4 κ. (4.12) Solving the above equation (4.12) we get v 4 = c 1 ω 2 1 + c 2 ω 1 ω 2 + c 3 ω 2 2, (4.13) where c 1, c 2, c 3 R and ω 1, ω 2 are linearly independent olution of the following equation (ee [11, p. 141 and p. 538]): The olution of (4.14) are a follow: 2ω κω = 0. (4.14)

NULL HELIX AND k-type NULL SLANT HELICES IN E 4 1 81 Cae 1: If κ < 0, then we get ω = c exp ( ) 1 κ co θ 2 in θ d, where c i a contant and the newly defined function θ() atifie κ θ = 2 + κ in θ co θ. 2κ Cae 2: If κ > 0, then we have ω = c exp ( ) 1 κ coh θ 2 inh θ d, where c i a contant and θ() atifie κ θ = 2 + κ inh θ coh θ. 2κ Converely, aume (4.12) hold for a null curve r() with vanihing null torion. We can define the vector field V a then V = 0 and g(β, V ) = c. V = (v 4 κ v 4 )x + v 4α + cβ + v 4 y, (0 c R); From Theorem 4.8, we have: Corollary 4.9. Let r = r() be a 3-type null lant helix. Then the lope axe of r are obtained by where V 4 atifie (4.12). V = (v 4 κ v 4 )x + v 4α + cβ + v 4 y, (c R {0}), For 3-type null lant helice, becaue τ = 0, β i a contant vector, cβ i a contant function. Then, we have: Corollary 4.10. The lope axe of 3-type null lant helice lie fully in the firt kind oculating pace of r(). Remark 4.11. The firt kind oculating pace of r() i the orthogonal complement of it econd binormal vector field β in E 4 1. In [8], a null curve i a null oculating curve of the firt kind if and only if the null torion i equal to zero. Thu, we have: Theorem 4.12. Let r = r() be a 3-type null lant helix. Then r i congruent to a null oculating curve of the firt kind. Next, we ue an example to how the proce to obtain the lope axe of ome kind of 3-type null lant helice.

82 JINHUA QIAN AND YOUNG HO KIM Example. Let r = r() be a 3-type null lant helix with vanihing null torion and the null curvature κ = 2a 4, (a R {0}). By olving the equation we get ω a 4 ω = 0, (1) ω = C 1 in( a ) + C 2 co( a ), for a < 0; (2) ω = C 1 inh( a ) + C 2 coh( a ), for a > 0, where C 1, C 2 R (ee [11, p. 142 and p. 171]). Taking the firt cae (a < 0) a an example, we chooe a a ω 1 = in( ), ω 2 = co( ) a two linearly independent olution. Then, from (4.13), we get a a a a v 4 = C 1 2 in 2 ( ) + C 2 2 in( ) co( ) + C 3 2 co 2 ( ), where C 1, C 2, C 3 R. From Corollary 4.9, the lope axe of uch kind of 3-type null lant helice can be obtained. Reference [1] A. Ali and R. López, Slant helice in Minkowki pace E 3 1, J. Korean Math. Soc. 48 (2011), 159 167. MR 2778006. [2] A. Ali, R. López and M. Turgut, k-type partially null and peudo null lant helice in Minkowki 4-pace, Math. Commun. 17 (2012), 93 103. MR 2946135. [3] W. B. Bonnor, Null curve in a Minkowki pace-time, Tenor (N.S.) 20 (1969), 229 242. MR 0248658. [4] A. C. Çöken and Ü. Çiftçi, On the Cartan curvature of a null curve in Minkowki pacetime, Geom. Dedicata 114 (2005), 71 78. MR 2174094. [5] J. H. Choi and Y. H. Kim, Aociated curve of a Frenet curve and their application, Appl. Math. Comput. 218 (2012), 9116 9124. MR 2923010. [6] A. Ferrández, A. Giménez and P. Luca, Null generalized helice in Lorentz-Minkowki pace, J. Phy. A: Math. Gen. 35 (2002), 8243 8251. MR 1946488. [7] J. Inoguchi and S. Lee, Null curve in Minkowki 3-pace, Int. Electron. J. Geom. 1 (2008), 40 83. MR 2443735. [8] K. İlarlan and E. Nešović, Some characterization of null oculating curve in the Minkowki pace-time, Proc. Et. Acad. Sci. 61 (2012), 1 8. MR 2952460. [9] H. Liu, Curve in the lightlike cone, Beiträge Algebra Geom. 45 (2004), 291 303. MR 2070651. [10] H. Liu and Q. Meng, Repreentation formula of curve in a two- and three-dimenional lightlike cone, Reult Math. 59 (2011), 437 451. MR 2793466.

NULL HELIX AND k-type NULL SLANT HELICES IN E 4 1 83 [11] A. D. Polyanin and V. F. Zaitev, Handbook of exact olution for ordinary differential equation, 2nd edition, Chapman and Hall/CRC, 2003. MR 2001201. [12] M. Turgut and S. Yilmaz, Some characterization of type-3 lant helice in Minkowki pacetime, Involve 2 (2009), 115 120. MR 2501349. J. Qian Department of Mathematic, Northeatern Univerity Shenyang 110004, People Republic of China ruoyunqian@163.com Y. H. Kim Department of Mathematic, Kyungpook National Univerity Taegu 702-701, Korea yhkim@knu.ac.kr Received: September 15, 2014 Accepted: December 29, 2015