On the Elliptic cylindrical Tzitzeica curves in Minkowski 3-Space by ProQuest

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```									   Scientia Magna
Vol. 5 (2009), No. 3, 44-48

On the Elliptic cylindrical Tzitzeica curves in
Minkowski 3-Space
Murat Kemal Karacan† and Bahaddin Bukcu‡
†Department of Mathematics, Usak University, Usak, 64200, Turkey
‡Department of Mathematics, Gazi Osman Pasa University, Tokat, Turkey
E-mail: murat.karacan@usak.edu.tr

Abstract Elliptic cylindrical curves satisfying Tzitzeica condition are obtained via the
solution of the forced harmonic equation in Minkowski 3-Space. In addition, we have given
the conditions to be of spacelike, timelike and null curve of the elliptic cylindrical Tzitzeica
curve.

Keywords Tzitzeica curve, Elliptic cylindrical curve, Minkowski 3-Space.

§1. Preliminaries and Introduction
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The Minkowski 3-space E1 is the Euclidean 3-space E 3 provided with the Lorentzian inner
product
x, y L = x1 y1 + x2 y2 − x3 y3 ,
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where x = (x1 , x2 , x3 ) and y = (y1 , y2 , y3 ). An arbitrary vector x = (x1 , x2 , x3 ) in E1 can have
one of three Lorentzian causal characters: it is spacelike if x, x L > 0 or x = 0, timelike if
x, x L < 0 and null (lightlike) if x, x L = 0 and x = 0. Similarly, an arbitrary curve α = α(s)
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in E1 is locally spacelike, timelike or null (lightlike), if all of its velocity vectors (tangents)
α (s) = T (s) are respectively spacelike, timelike or null, for each s ∈ I ⊂ IR. Lorentzian
vectoral product of x and y is deﬁned by

x ∧L y = (x2 y3 − x3 y2 , x3 y1 − x1 y3 , x2 y1 − x1 y2 ) .
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Recall that the pseudo-norm of an arbitrary vector x ∈ E1 is given by x                 L   =   | x, x L |. If
the curve α is non-unit speed, then

α (t) ∧L α (t)                   det α (t), α (t), α (t)
L
κ(t) =
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