The differential geometry of regular curves on a regular time-like surface.

1. INTRODUCTION

It is well-known that, if a curve differentiate on an open interval, at each point, a set of mutually orthogonal unit vectors can be constructed. And these vectors are called Frenet frame or moving frame vectors. The set, whose elements are frame vectors and curvatures of a curve, is called Frenet apparatus of the curves. In recent years, the theory of degenerate submanifolds has been treated by researchers and some classical differential geometry topics have been extended to Lorentz manifolds. For instance, in , the authors extended and studied spacelike involute- evolute curves in Minkowski space. Classical differential geometry of the curves may be surrounded by the topics which are general helices, involute-evolute curve couples, spherical curves and Bertrand curves. Such special curves are investigated and used in some of real world problems like mechanical design or robotics by well-known Frenet- Serret equations. Because, we think of curves as the path of a moving particle in the Euclidean space.

At the beginning of the twentieth century Einstein's theory opened a door to new geometries such as Minkowski space-time, which is simultaneously the geometry of special relativity and the geometry induced on each fixed tangent space of an arbitrary Lorentzian manifold. Some authors have aimed to determine Frenet-Serret invariants in higher dimensions. There exists a vast literature on this subject, for instance [1-4, 6, 7, 8]. In the light of the available literature, in  the author extended spherical images of curves to a four-dimensional Lorentzian space and studied such curves in the case where the base curve is a space-like curve according to the signature (+++-). By using the Darboux vector, various well-known formulas of differential geometry had been produced by . Then, in , authors had been given these formulas in Minkowski 3-space.

In this work, we study to investigate the formulae between the Darboux vectors of the curve (c), the parameter curves ([c.sub.1]) and ([c.sub.2]) which are not intersecting perpendicularly. Thus, we will find an opportunity to investigate regular time-like surface by taking the parameter curves which are intersect under the angle [theta].

2. PRELIMINARIES

To meet the requirements in the next sections, here, the basic elements of the theory of curves in the space [R.sup.3.sub.1] are briefly presented. (A more complete elementary treatment can be found in .) The Minkowski 3-space provided with the standard flat metric given by

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

where ([x.sub.1], [x.sub.2], [x.sub.3]) is rectangular coordinate system in [R.sup.3.sub.1]. Recall that, the norm of an arbitrary vector a [member of] [R.sup.3.sub.1] is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [PHI] = [PHI] (s) be a regular curve in [R.sup.3.sub.1]. [PHI] is called an unit speed curve if the velocity vector [??] of [PHI] satisfies [parallel][??][parallel] = 1. For the vectors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] it is said to be orthogonal if and only if <[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]> = 0.

On the other hand, the vector [??] is called angular velocity vector of motion. If we consider any orthogonal trihedron as {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]}, we can write their derivative formulas as follows:

(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [LAMBDA] is Lorentzian vectorial product, .

Let us take a time-like surface as [??] = [??](u,v). Denote by {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} the moving Frenet-Serret frame along the time-like curve (c) on [??] = [??](u,v). Another orthogonal frame on [??] = [??](u, v) is the Darboux trihedron as {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]}. For an arbitrary time-like curve (c) on time-like surface, the orientation of the Darboux trihedron is written as

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the Darboux vector of this trihedron is written as

(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and 1/[T.sub.g], 1/[R.sub.n] and 1/[R.sub.g] are geodesic torsion, normal curvature and geodesic curvature, respectively. Also, the Darboux derivative formulae can be written as follows:

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] .

3. THE DARBOUX VECTOR FOR THE DARBOUX TRIHEDRON OF A TIME-LIKE CURVE

Let us express the parameter curves u = const. as ([c.sub.1]) and v = const. as ([c.sub.2]) which are constant on a time-like surface y = y(u,v). But, these curves are intersect under the angle [theta] (not perpendicular). Let any time-like curve that is passing through a point P on the surface be (c). Let us take time-like curves which are passing through the same point P as ([c.sub.1]) and ([c.sub.2]). Let the unit tangent vectors of curves (c), ([c.sub.1]) and ([c.sub.2] ) and at the point P be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[??].sub.2], respectively. From , the edges of the Darboux trihedrons of parameter curves are

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, three Darboux trihedrons are written as below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let s, [s.sub.1] and [s.sub.2] be the arc-elements of the curves (c), ([c.sub.1]) and ([c.sub.2]), respectively. Thus, we can write

(3.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Moreover, because of the parameter curves are intersect under the angle [theta] we have

(3.3) [[??].sub.1].[[??].sub.2] = -ch[theta]

Then, the normal vector of time-like surface is

(3.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Other than, considering the first two formulae of (3.2) in the third term,

(3.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is written, .

On the other hand, let us consider the hyperbolic angle between [??] and [[??].sub.1] as [alpha], and if we take inner product both sides of (3.5) with [[??].sub.1] and [[??].sub.2] then

(3.6) [??].[[??].sub.1] = ch[alpha] = - [square root of (E)] du/ds - ch[theta][square root of (G)]dv/ds

(3.7) [??].[[??].sub.1] = -ch([theta] - [alpha]) = -ch[theta][square root of (E)]du/ds - [square root of (G)]dv/ds are obtained. Thus, from (3.6) and (3.7)

(3.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are written. Finally, if we put (3.8) into (3.5), we have the following equation between the tangent vectors of the curves (c), ([c.sub.1]) and ([c.sub.2]) as

(3.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, we shall denote the arc elements ds, d[s.sub.1] and d[s.sub.2] of the parameter curves which are belongs to time-like surface y = y(u,v), and then we express as follows:

(3.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, considering (3.8) and (3.10), we have

(3.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Corollary 3.1. The third elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the Darboux trihedrons [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} and {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} are linear dependent.

Proof. If we substitute the equation (3.9) in the first equality of (2.3) and consider the Darboux trihedrons of ([c.sub.1]) and ([c.sub.2]) we have

(3.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, we get the expression.

Theorem 3.1. The Darboux trihedrons {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} and {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} of the parameters curves ([c.sub.1]) and ([c.sub.2]) of the time-like surface are written by Darboux instantaneous vectors as follows:

(3.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. If we consider the Darboux trihedrons {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]}, we see that the normal vector [??] is coincide. Then, considering (3.4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are obtained. From (2.2), we write

(3.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If (3.14) is substituted in the third equality (3.16), we get

(3.18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, from (3.18) and (3.19), we have

(3.20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, the derivative of [[??].sub.2] with respect to [s.sub.1] is written by the Lorentzian vectorial product of [[??].sub.1] and [[??].sub.2]. Similarly, it is easy to see that the other vectors can be written by the same method.

Corollary 3.2. By using the vectors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we can express [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as follows:

(3.21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 1/[T.sub.g], 1/[R.sub.n] and 1/[R.sub.g] are geodesic torsion, normal curvature and geodesic curvature, respectively.

From (2.4), we can write the Darboux vectors of the {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} and {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]}

(3.24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, if we consider the equations (3.9), (3.14) and (3.15) according to the vectors [[??].sub.1] and [[??].sub.2], and substitute in (3.24), we get (3.21), (3.22) and (3.23).

Theorem 3.2. If we consider the tangent vectors [[??].sub.1] and [[??].sub.2] of the parameter curves ([c.sub.1]) and ([c.sub.2]) on the time-like surface, then we obtain the following relations:

(3.25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(3.26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof, i) From (3.2), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are written. And also, we know that

(3.27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(3.28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(3.29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By taking differential from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, we write

(3.30) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(3.31) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

On the other hand, we have

(3.32) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(3.33) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, taking inner product of (3.32) and (3.33) by the vector [[??].sub.2] and the vector [[??].sub.1], and considering (3.30) and (3.31), we have (3.26) and (3.25). The other cases can be seen easily.

Result 3.1. If we take differential from [[??].sub.1] [[??].sub.2] = - ch[theta] with respect to u and v, we get

(3.34) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(3.35) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, we have

(3.36) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 3.3. The following geodesic curvature equalities are satisfied for the parameter curves ([c.sub.1]) and ([c.sub.2])

(3.37) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.38) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. i) From (3.25) and (3.20), we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From here, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, from (3.24), if we take inner product both of side [[??].sub.1] with -sh[theta][??] we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is obtained. Similarly, (ii) can be proofed.

Theorem 3.4. Let us consider any curve (c) on the time-like surface and the arc elements of curves (c), ([c.sub.1]) and ([c.sub.2]) as s, [s.sub.1] and [s.sub.2], respectively. Let the Darboux instantaneous rotation vectors of ([c.sub.1]) and ([c.sub.2]) be [[??].sub.1] and [[??].sub.2], and if the hyperbolic angle between the tangent [??] of curve (c) and [[??].sub.1] is [alpha], then

(3.39) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.40) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are satisfied.

Proof. If we consider (3.11) and (3.13), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

is obtained. Similarly, the others are satisfied.

Result 3.2. The following equality

(3.41) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is valid.

Theorem 3.5. Let us consider the curves (c), ([c.sub.1]) and ([c.sub.2]) which are intersect a point P on time-like surface. Let the Darboux instantaneous rotation vectors of these curves at the point P be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively. Then the following equality is satisfied

(3.42) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. From (3.9)

(3.43) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

can be written. Then, by taking derivatives with respect to s from equation (3.43), we obtain

(3.44) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, considering the Darboux trihedrons {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} and {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]}, we write

(3.45) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From (3.14) and (3.15), if [[??].sub.1] and [[??].sub.2] are substituted in (3.45) we obtain

(3.46) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

And then, substituting the equations (3.46) in (3.44), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

According to the Theorem 3.4,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are known. And, by using the trigonometric expression, we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus,

(3.47) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.48) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are written. After that,

(3.49) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is obtained. By writing (3.49) in the third expression of (3.40) we obtain

(3.50) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [??] is Darboux vector, we have

(3.51) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, considering (2.5), (3.47), (3.50) and (3.51)

(3.52) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(3.53) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

are written. At the end, if we make equal (3.52) to (3.53), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be written. Thus, we get the theorem.

Corollary 3.3 (General Form of Euler Formula). Taking dot product both of the (3.42) with [??], we have following equation among the timelike curves (c), ([c.sub.1]) and ([c.sub.2]):

(3.54) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (i = 1, 2) and 1/[R.sub.n] are geodesic torsion, normal curvatures of the parameter curves and the normal curve, respectively.

Corollary 3.4 (General Form of O. Bonnet). Taking dot product both of the (3.42) with [??], we have following equation among the timelike curves (c), ([c.sub.1]) and ([c.sub.2]):

(3.55) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] {i = 1,2) and 1/[T.sub.g] are geodesic torsions, normal curvatures of the parameter curves and the normal torsion of normal curve, respectively.

Corollary 3.5 (Liouville Formula). Taking dot product both of the (3.42) with [??], we have following equation among the timelike curves (c), ([c.sub.1]) and ([c.sub.2]):

(3.56) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and 1/[R.sub.g] (i = 1,2) are geodesic curvatures of the parameter curves and the normal curve, respectively. Now, we give some special cases of the formulae (3.54) and (3.55).

Corollary 3.6. If we take [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (i.e. the parameter curves are curvature lines) in (3.55) we have

(3.57) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 1/[T.sub.g] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (i = 1,2) are geodesic torsion of normal curve and normal curvature of parameter curves, respectively.

Result 3.3: If we take [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (i.e. the parameter curves are curvature lines) in (3.54) we have

(3.58) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (i = 1,2) and 1/[R.sub.n] are normal curvatures of parameter curves and normal curve, respectively.

Result 3.4: If we take [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (i.e. the parameter curves are asymptotic) in (3.54) we have

(3.59) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (i = 1,2) and 1/[R.sub.n] are geodesic torsions of parameter curves and normal curvature of normal curve, respectively.

EMIN OZYILMAZ AND YUSUF YAYLI

Department of Mathematics, Ege University, Bornova, Izmir, 35100, Turkey

Department of Mathematics, Ankara University, Dogol cad. Ankara, 06100, Turkey

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 Kiziltug, S. and Yayli, Y. "Space-Like Curve on Space-like Parallel Surface in Minkowski 3-space, "Int. J. Math. Comput., 19 (2013), No. 2, 94-105

 Ali, A. T.; Lopez, R. and Turgut, M. "k-type Partially Null and Nseudo Null Slant Helices in Minkowski 4-space", Math. Commun., 17, (2012), No. 1, 93-103.

 Turgut, M. and Yilmaz, S. "On The Frenet Frame and A Characterization of Space-Like Involute-Evolute Curve Couple in Minkowski Space-Time", Int. Math. Forum, Vol.3, No. 16 (2008), 793-801.