Generalized timelike Mannheim curves in Minkowski space-time [E.sup.4.sub.1].(Research Article)(Report)

1. Introduction

The geometry of curves has long captivated the interests of mathematicians, from the ancient Greeks through to the era of Isaac Newton (1643-1727) and the invention of the calculus. It is a branch of geometry that deals with smooth curves in the plane and in the space by methods of differential and integral calculus. The theory of curves is the simpler and narrower in scope because a regular curve in a Euclidean space has no intrinsic geometry. One of the most important tools used to analyze curve is the Frenet frame, a moving frame that provides a coordinate system at each point of curve that is "best adopted" to the curve near that point. Every person of classical differential geometry meets early in his course the subject of Bertrand curves, discovered in 1850 by J. Bertrand. A Bertrand curve is a curve such that its principal normals are the principal normals of a second curve. There are many works related with Bertrand curves in the Euclidean space and Minkowski space, [1-3].

Another kind of associated curve is called Mannheim curve and Mannheim partner curve. The notion of Mannheim curves was discovered by A. Mannheim in 1878. These curves in Euclidean 3-space are characterized in terms of the curvature and torsion as follows: a space curve is a Mannheim curve if and only if its curvature k1 and torsion k2 satisfy the relation

[k.sub.1] = [beta]([k.sup.2.sub.1] + [k.sup.2.sub.2]) (1.1)

for some constant [beta]. The articles concerning Mannheim curves are rather few. In [4], a remarkable class of Mannheim curves is studied. General Mannheim curves in the Euclidean 3-space are obtained in [5-7] . Recently, Mannheim curves are generalized and some characterizations and examples of generalized Mannheim curves are given in Euclidean 4-space [E.sup.4] by [8].

In this paper, we study the generalized timelike Mannheim partner curves in 4-dimensional Minkowski space-time. We will give the necessary and sufficient conditions for the generalized timelike Mannheim partner curves.

2. Preliminaries

To meet the requirements in the next sections, the basic elements of the theory of curves in Minkowski space-time [E.sup.4.sub.1] are briefly presented in this section. A more complete elementary treatment can be found in [9].

Minkowski space-time [E.sup.4.sub.1] is a usual vector space provided with the standard flat metric given by

<,> = -[dx.sup.2.sub.1] + [dx.sup.2.sub.2] + [dx.sup.2.sub.3] + [dx.sup.2.sub.4], (2.1)

where ([x.sub.1],[x.sub.2],[x.sub.3],[x.sub.4]) is a rectangular coordinate system in [E.sup.4.sub.1].

Since <,> is an indefinite metric, recall that a v [member of] [E.sup.4.sub.1] can have one of the three causal characters; it can be spacelike if <v, v> > 0 or v = 0, timelike if <v, v> < 0, and null(ligthlike) if <v, v> = 0 and v [not equal to] 0. Similarly, an arbitrary curve c = c(t) in [E.sup.4.sub.1] can locally be spacelike, timelike, or null (lightlike) if all of its velocity vectors c'(t) are, respectively, spacelike, timelike, or null. The norm of v [member of] [E.sup.4.sub.1] is given by [parallel]v[parallel] = [square root [absolute value of <v, v>]. If [parallel]c'(f)[parallel] = [square root of [absolute value of (c'(t),c'(t))] [not equal to] 0 for all t [member of] I, then C is a regular curve in [E.sup.4.sub.1]. A timelike (spacelike) regular curve C is parameterized by arc-length parameter t which is given by c : I [right arrow] [E.sup.4.sub.1], then the tangent vector c'(t) along C has unit length, that is, <c'(t), c'(t)> = -1, (<c'(t), c'(t)> = 1) for all t [member of] I.

Hereafter, curves considered are timelike and regular [C.sup.[infinity]] curves in [E.sup.4.sub.1]. Let T(t) = c'(t) for all t [member of] I; then the vector field T(t) is timelike and it is called …