# 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

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

 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.

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

 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.

 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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