Some Characterizations of Partially Null Curves in Semi-Euclidean Space

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1 International Mathematical Forum, 3, 28, no. 32, Some Characterizations of Partially Null Curves in Semi-Euclidean Space Melih Turgut Dokuz Eylul University, Buca Educational Faculty Department of Mathematics, 3516 Buca, Izmir, Turkey Suha Yilmaz Dokuz Eylul University, Buca Educational Faculty Department of Mathematics, 3516 Buca, Izmir, Turkey Abstract In this paper, first, position vector of all partially null curves in Semi-Euclidean space E 4 2 is determined. Then, in the same space characterizations of spherical and inclined partially null curves are given. Mathematics Subject Classification: 53B3, 53C5 Keywords: Semi-Euclidean Space, Partially Null Curves 1 Introduction Suffice it to say that the many important results in the theory of the curves in E 3 were initiated by G. Monge and the moving frames idea was due to G. Darboux. E. Cartan opened door of notion of null curves (for more details see 2]). And, thereafter null curves deeply studied by W.B. Bonnor 7] in Minkowski space-time. In the same space, Frenet equations for some special null; Partially and Pseudo Null curves are given in 4]. By means of Frenet equations, in 3] authors gave characterizations of such kind null curves lying on the pseudohyperbolic space in E1 4. In 5] authors defined Frenet equations of pseudo null and partially null curves in Semi-Euclidean space E2. 4 And, inclined curves are well-known concept in the classical differential geometry 6].

2 157 M. Turgut and S. Yilmaz In this work, by means of Frenet equations defined in 5], we investigated position vector of partially null curves in E2 4. Additionally, characterizations of spherical and inclined partially null curves are given. 2 Preliminaries To meet the requirements in the next sections, here, the basic elements of the theory of curves in the space E2 4 are briefly presented.(a more complete elementary treatment can be found in 1]). Semi-Euclidean space E2 4 is an Euclidean space E 4 provided with the standart flat metric given by g = dx 2 1 dx2 2 + dx2 3 + dx2 4, (1) where (x 1,x 2,x 3,x 4 ) is a rectangular coordinate system in E2 4. Since g is an indefinite metric, recall that a vector v E2 4 can have one of the three causal characters; it can be space-like if g(v, v) > orv =, time-like if g(v, v) < and null (light-like) if g(v, v)= and v. Similary, an arbitrary curve α = α(s) ine2 4 can be locally be space-like, time-like or null (light-like), if all of its velocity vectors α (s) are respectively space-like, time-like or null. Also, recall the norm of a vector v is given by v = g(v, v). Therefore, v is a unit vector if g(v, v) =±1. Next, vectors v, w in E2 4 are said to be orthogonal if g(v, w) =. The velocity of the curve α(s) is given by α (s). Let a and b be two space-like vector in E2. 4 Then, there is unique real number δ π, called angle between a and b, such that g(a, b) = a. b cos δ. Recall that space-like curve with time-like principal normal N and a null first and second binormal is called a partially null curve in E2 4 5]. Let ϑ = ϑ(s) be a partially null curve in E2 4. If tangent vector field of this curve is forming a constant angle with a constant vector field U, then this curve is called an inclined curve. The Lorentzian hypersphere of center m =(m 1,m 2,m 3,m 4 ) and radius r R + in the space E2 4 defined by S 3 2(m, r) = { α =(α 1,α 2,α 3,α 4 ) E 4 2 : g(α m, α m) =r 2}. (2) Denote by {T (s),n(s),b 1 (s),b 2 (s)} the moving Frenet frame along the curve α(s) in the space E2 4. Then T,N,B 1,B 2 are, respectively, the tangent, the principal normal, the first binormal and the second binormal vector fields. Space-like or time-like curve α(s) is said to be parametrized by arclength function s, ifg(α (s),α (s)) = ±1. Let α(s) be a partially null curve in the space-time E2 4, parametrized by arclength function s. Then for the curve α the following Frenet equations are

3 Characterizations of partially null curves 1571 given in 5] T N B 1 B 2 = τ σ τ σ T N B 1 B 2, (3) where T,N,B 1 and B 2 are mutually orthogonal vectors satisfying equations g(t,t)=1,g(n,n) = 1,g(B 1,B 1 )=g(b 2,B 2 )=,g(b 1,B 2 )=1. And here,, τ and σ are first, second and third curvature of the curve α, respectively. In 5] authors gave a characterization about partially null curves with the following statement. Theorem 2.1 A partially null unit speed curve ϕ(s) in E2 4, τ for each s I R has σ =for each s. with curvatures 3 Position Vector of a Partially Null Curve in E 4 2 In this section, we shall investigate position vector of partially null curves in Semi-Euclidean space E 4 2. Let ψ = ψ(s) be a unit speed partially null curve in E 4 2 with curvatures,τ. Considering theorem (2.1), we can write this curve respect to frame {T,N,B 1,B 2 } as ψ = ψ(s) =m 1 T + m 2 N + m 3 B 1. (4) Differentiating both sides of (4) respect to s and using Frenet equations, we have a system of differential equation as follow: dm 1 + m ds 2 1= dm 2 + m ds 1 = dm 3 + m ds 2τ =. (5) Using (5) 1 and (5) 2 we have second order differential equation respect to u 2 as d 1 ds Using a exchange variable t = s ds in (6) we get ] dm 2 m 2 +1=. (6) ds

4 1572 M. Turgut and S. Yilmaz Solution of (7) gives us m 2 (t) = cosh t A + d 2 m 2 dt 2 m t sinh t =. (7) dt + sinh t B t cosh t where A and B are real numbers. Rewriting t = s ds we have ] m 2 (s) = (cosh s ds) A + s (sinh s ds)ds + ] (sinh s ds) B s (cosh s ds)ds dt, (8) Let us denote m 2 (s) =φ. And using (5) 1 and (5) 3 we get other components, respectively, (9) m 1 = 1 dφ ds, (1) s m 3 = (φ.τ)ds. (11) Corollary 3.1 Position vector of all partially null curves in Semi-Euclidean space E2 4 can be formed by the equations (9), (1) and (11). 4 The Inclined Partially Null Curves in E 4 2 Theorem 4.1 Let ψ = ψ(s) be a unit speed partially null curve in E 4 2. ψ is an inclined curve, if and only if τ = constant. (12) Proof. Let ψ = ψ(s) be a unit speed partially null curve in E2 4 and also be an inclined curve. From definition of inclined curves, we write that g(t,u) = cos ξ, (13) where U is a constant space-like vector and ξ is a constant angle. Differentiating (13) respect to s, we have

5 Characterizations of partially null curves 1573 as.g(n,u) =, (14) which implies that N U. And therefore we compose constant vector U U = u 1 T + u 2 B 1 + u 3 B 2. (15) Differentiating (15) and considering Frenet equations we have following equation system: Solution of (16) yields that du 1 = ds du 2 ds = du 3 ds = u 1 + u 3 τ =. (16) = constant. (17) τ Conversely, let us consider a vector given by U = { T + B + } τ E. cos ξ, (18) where ξ is a constant angle. Differentiating (18) and considering differential of (17), we have du =. (19) ds (19) implies that U is a constant vector. And then considering a partially null curve ψ = ψ(s); using inner product, we get which shows that ψ is an inclined curve in E 4 2. g(t,u) = cos ξ, (2) 5 A Characterization of Spherical Partially Null Curves in E 4 2 Theorem 5.1 Let ψ = ψ(s) be a unit speed partially null curve in E2 4 with curvatures, τ for each s. If ψ lies on S2 3 Lorentzian hypersphere, then 1 d τ ds ( 1 )=constant. (21)

6 1574 M. Turgut and S. Yilmaz Proof. Let us suppose that ψ lies on S2 3 Lorentzian hypersphere with center m and radius r. From definition, we write g(ψ m, ψ m) =r 2 (22) for each s. Differentiating (22) three times with respect to s and considering theorem (2.1), we have g(ψ m, B 1 )= 1 d τ ds ( 1 ). (23) One more differentiating of (23) gives us d 1 d ds τ ds ( 1 ] ) =, (24) which implies that References 1 d τ ds ( 1 )=constant. (25) 1] B. O Neill, Semi-Riemannian Geometry, Adacemic Press, New York, ] C.Boyer, A History of Mathematics, New York: Wiley, ] C.Camci, K. Ilarslan and E. Sucurovic, On pseudohyperbolical curves in Minkowski space-time. Turk J.Math. 27 (23) ] J. Walrave, Curves and surfaces in Minkowski space. Dissertation,K. U. Leuven, Fac. of Science, Leuven (1995) 5] M. Petrovic-Torgasev, K.Ilarslan and E. Nesovic, On partially null and pseudo null curves in the semi-euclidean space R2 4. J. of Geometry. 84 (25) ] R. S. Milman and G. D. Parker, Elements of Differential Geometry, Prentice-Hall Inc., Englewood Cliffs, New Jersey, ] W.B. Bonnor, Null curves in a Minkowski space-time, Tensor. 2 (1969) Received: January 27, 28