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On the elliptic cylindrical Tzitzeica curves in Minkowski 3-space.

[section]1. Preliminaries and Introduction

The Minkowski 3-space [E.sup.3.sub.1] is the Euclidean 3-space [E.sup.3] provided with the Lorentzian inner product

[<x, y>.sub.L] = [x.sub.1][y.sub.1] + [x.sub.2][y.sub.2] - [x.sub.3][y.sub.3],

where x = ([x.sub.1],[x.sub.2],[x.sub.3]) and y = ([y.sub.1],[y.sub.2],[y.sub.3]). An arbitrary vector x = ([x.sub.1],[x.sub.2],[x.sub.3]) in [E.sup.3.sub.1] can have one of three Lorentzian causal characters: it is spacelike if [<x,x>.sub.L] > 0 or x = 0, timelike if [<x, x>.sub.L] < 0 and null (lightlike) if [<x, x>.sub.L] = 0 and x [not equal to] 0. Similarly, an arbitrary curve [alpha] = [alpha](s) in [E.sup.3.sub.1] is locally spacelike, timelike or null (lightlike), if all of its velocity vectors (tangents) [alpha]'(s) = T(s) are respectively spacelike, timelike or null, for each s [member of] I [subset] IR. Lorentzian vectoral product of x and y is defined by

x [[conjunction].sub.L] y = ([x.sub.2][y.sub.3] - [x.sub.3][y.sub.2], [x.sub.3][y.sub.1] - [x.sub.1][y.sub.3], [x.sub.2][y.sub.1] - [x.sub.1][y.sub.2]).

Recall that the pseudo-norm of an arbitrary vector x [member of] [E.sup.3.sub.1] is given by [[parallel]x[parallel].sub.L] = [square root of [|<x, x>.sub.L|]]. If the curve [alpha] is non-unit speed, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

If the curve a is unit speed, then

[kappa](s) = [parallel][alpha]"(s) [[parallel].sub.L] , [tau](s) = [parallel]B'(s) [[parallel].sub.L]. (1.2)

[1,4].

In this paper we have interested in Tzitzeica elliptic cylindrical curves in Minkowski 3-Space, more precisely we ask in what conditions a cylindrical curve is a Tzitzeica one, namely the function t [right arrow] [tau](t)/[d.sup.2(t) is constant, where d(t) is the distance from origin to the osculating plane of curve. The Tzitzeica condition yields a third-order ODE which in our framework admits a direct integration. Therefore the final answer of main problem is given via a second order ODE which in the elliptic case is exactly the equation of a forced harmonic oscillator. In this case, the solution depends of four real constants: one defining the Tzitzeica condition and other three obtained by integration.

[section]2. Elliptic Cylindrical Tzitzeica curves in Minkowski 3-Space

Proposition 1. Let [alpha](t) be an elliptic cylindrical curve in Minkowski 3- Space.Then, the curve [alpha](t) is Tzitzeica curve if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where f(0), f'(0), K [not equal to] 0 and c are real constants.

Proof. Let in [E.sup.3.sub.1] a curve C given in vectorial form C : [alpha] = [alpha](t). This curve is called elliptic cylindrical if has the expression

[alpha](t) = (cos t, sin t, f(t)) (2.1)

and differentiation of [alpha](t), we have

[alpha]'(t) = (- sin t, cos t, f'(t))

[alpha]"(t) = (-cos t, - sin t, f"(t)) (2.2)

[alpha]"'(t) = (sin t, - cos t, f"(t))

for some f [member of] [C.sup.[infinity]] (R). From Eq.(1.1) and Eq.(2.2), the torsion function is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the distance from origin to the osculating plane is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let us suppose that the curve is Tzitzeica with the constant K [not equal to] 0, because the curve is not contained in a plane

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Integration gives

f"(t) + f(t) = - 1/Kt+c' (2.3)

where c is a real constant. Then the Laplace transform gives

[[s.sup.2]Y(s) - sf(0) - f'(0)] + Y(s) = -L {G(t)} = -g(s),

where Y(s) and G(s) denote the Laplace transform of f(t) and g(t) respectively, and f(0) and f'(0) are arbitrary constants. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and therefore

f(t) = f(0) cos t + f'(0) sin t - sin(t)*G(t),

where the function denoted by sin(t)*G(t) and defined as

sin(t)* G(t) = [[integral].sup.t.sub.0] G(u) sin(u - t)du

is called the convolution of the functions sin t and G(t) or

f(t) = f(0) cos t + f'(0) sin t + [[integral].sup.t.sub.0] G(u) sin(u - t)du.

Theorem 1. Let [alpha](t) be elliptic cylindrical Tzitzeica curve, then the curve [alpha](t) are space-like, timelike and null curve if and only if [f'.sup.2](0) < 1, [f'.sup.2](0) > 1 and [f'.sup.2](0) = 1, respectively.

Proof. Since the curve [alpha](t) = (cos t, sin t, f(t)), the tangent of the curve is T = [alpha]'(t) = (- sin t, cos t, f'(t)). The taylor series of the function f in the neighbourhood of zero is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We take into consideration satisfying

f(0) = f'(0)[not equal to]0, 0 = f"(0) = f"'(0) =... (2.4)

in the neighbourhood of zero. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From the last equation, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, for t [right arrow] 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, satisfying Eq.(2.4) as t [right arrow] 0, sin t = t and the function f is written such as

f(t) = f'(0)sin t.

If we take derivative of the last equation for t and square, we have

[f'.sup.2](t) = [f'.sup.2](0)[cos.sup.2]t, [<T(t), T(t)>.sub.L] = 1 - [f'.sup.2] > 0,

or

[(f').sup.2] (t) < 1, [(f').sup.2] (0)[cos.sup.2] t < 1.

Since |cos t| < 1, from the last equation, we have (i) The curve [alpha](t) is spacelike curve if and only if

[<T(t),T(t))>.sub.L] = 1 - [f'.sup.2] > 0.

Then, we have

[(f').sup.2] (0) < 1.

(ii) The Tzitzeica curve [alpha](t) is timelike curve if and only if

[<T(t),T(t)>.sub.L] = 1 - [f'.sup.2] < 0.

Then, we have

[f'.sup.2](0) > 1.

(iii) The Tzitzeica curve [alpha](t) is null curve if and only if

[<T(t),T(t)>.sub.L] = 1 - [f'.sup.2] =0.

Then, we have

[f'.sup.2](0) = 1.

References

[1] Akyildiz, E., Akyildiz, Y., Alpay, S., Erkip, A. and Yazici, A., Lecture Notes on Differential Equations, Metu Publications, 1981, 293.

[2] Bukcu, B., Karacan, M. K., On the involute and evolute curves of the spacelike curve with a spacelike binormal in Minkowski 3-space, Int. J.of Contemp. Math. Sci., 2(2007), No. 5-8.

[3] Crasmareanu, M., Cylindrical Tzitzeica Curves Implies Forced Harmonic, Oscillators, Balkan Journal of Geometry and Its Applications, 7(2002), No. 1, 37-42.

[4] Petrovic-Torgasev, M., Sucurovic, E., Some Characterizations of The Spacelike, The Timelike and The Null Curves on The Pseudohyperbolic Space [H.sup.2.sub.0] in [E.sup.3.sub.1], Kragujevac J.Math., 22(2000), 71-82.

[5] Ross, S.L., Differential Equation, John Wiley & Sons, Inc, New York, 1984.

Murat Kemal Karacan ([dagger]) and Bahaddin Bukcu ([double dagger])

([dagger])Department of Mathematics, Usak University, Usak, 64200, Turkey ([double dagger])Department of Mathematics, Gazi Osman Pasa University, Tokat, Turkey E-mail: murat.karacan@usak.edu.tr
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Author:Karacan, Murat Kemal; Bukcu, Bahaddin
Publication:Scientia Magna
Article Type:Report
Geographic Code:7TURK
Date:Sep 1, 2009
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