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Tangent and trinormal spherical images of a time-like curve on the pseudohyperbolic space [H.sup.3.sub.0]/ Pseudohuperboolse ruumi [H.sup.3.sub.0] ajasarnase joone puutuja ja trinormaali sfaarilised kujutised.

1. INTRODUCTION

At each point of a differentiable curve a tetrad of mutually orthogonal unit vectors (called tangent, normal, binormal, and trinormal) was defined and constructed. The rates of changes of these vectors along the curve define the curvatures of the curve in the space [E.sup.4.sub.1]. Spherical images (indicatrices) are a well-known concept in classical differential geometry of curves [3].

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. In recent years the theory of degenerate submanifolds has been treated by researchers and some of the classical differential geometry topics have been extended to Lorentzian manifolds. Some authors have aimed to determine Frenet-Serret invariants in higher dimensions. There exists a vast literature on this subject, for instance [2,8-10]. In the light of the available literature, in [8] 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 (+, +, +,-).

In this work we study spherical images of a time-like curve lying on the pseudohyperbolic space [H.sup.3.sub.0] in Minkowski space-time. We investigate relations between Frenet-Serret invariants of spherical images and the base curve. Additionally, some characterizations of spherical images to be general helices if the base curve is a ccr-curve are presented.

2. PRELIMINARIES

To meet the requirements in the next sections, here the basic elements of the theory of curves in the space [E.sup.4.sub.1] are briefly presented. (A more complete elementary treatment can be found in [7].)

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

g = -d[x.sup.2.sub.1] + d[x.sup.2.sub.2] + d[x.sup.2.sub.3] + d[x.sup.2.sub.4],

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

Since g is an indefinite metric, recall that a vector v [member of] [E.sup.4.sub.1] can have one of the three causal characters: it can be space-like if g(v,v) > 0 or v = 0, time-like if g(v,v) < 0, and null (light-like) if g(v,v) = 0 and v [not equal to] 0. Similary, an arbitrary curve [alpha] = [alpha](s) in [E.sup.4.sub.1] can be locally space-like, time-like, or null (light-like), if all of its velocity vectors [alpha]'(s) are respectively space-like, time-like, or null. Also, recall that the norm of a vector v is given by [parallel]v[parallel] = [square root of ([absolute value of g(v,v))]. Therefore, v is a unit vector if g(v,v) = [+ or -]1. Next, vectors v,w in [E.sup.4.sub.1] are said to be orthogonal if g(v,w) = 0. The velocity of the curve [alpha](s) is given by [parallel][alpha]'(s)[parallel]. Let a and b be two space-like vectors in [E.sup.4.sub.1]; then there is a unique real number 0 [less than or equal to] [delta] [less than or equal to] [pi], called the angle between a and b, such that g(a,b) = [parallel]a[parallel]*[parallel]b[parallel]cos [delta]. Let v = v(s) be a curve in [E.sup.4.sub.1]. If the tangent vector field of this curve forms a constant angle with a constant vector field U, this curve is called a general helix.

Denote by {T(s), N(s), [B.sub.1](s), [B.sub.2](s)} the moving Frenet-Serret frame along the curve [alpha](s) in the space [E.sup.4.sub.1]. Then T, N, [B.sub.1], [B.sub.2] are, respectively, the tangent, the principal normal, the binormal (the first binormal), and the trinormal (the second binormal) vector fields. A space-like or time-like curve [alpha](s) is said to be parametrized by arclength function s, if g([alpha]'(s), [alpha]'(s)) = [+ or -]1.

Let [alpha](s) be a time-like curve in the space-time [E.sup.4.sub.1], parametrized by arclength function s. In [7] the following Frenet-Serret equations are given:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where T, N, [B.sub.1], and [B.sub.2] are mutually orthogonal vectors satisfying the equations

g(T,T) = -1,g(N,N) = g([B.sub.1], [B.sub.1]) = g([B.sub.2], [B.sub.2]) = 1,

and where k, [tau], and [sigma] are the first, second, and third curvatures of the curve [alpha], respectively.

In the same space, in [1], the authors express a characterization of space-like curves lying on [H.sup.3.sub.0] by the following theorem:

Theorem 2.1. Let [alpha](s) be a unit speed space-like curve in [E.sup.4.sub.1], with space-like N, [B.sub.1] and curvatures k [not equal to] 0, [tau] [not equal to] 0, [sigma] [not equal to] 0 for each s [member of] I [subset] R. Then a lies on a pseudohyperbolic space if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

where [rho] = 1/k.

Recall that if a regular curve has constant Frenet-Serret curvature ratios (i.e., [tau]/k and [sigma]/[tau] are constants), it is called a ccr-curve [4,5]. In the same space the authors of [9] defined a vector product and gave a method to determine the Frenet-Serret invariants for an arbitrary curve by the following definition and the theorem.

Definition 2.2. Let a = ([a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4]), b = ([b.sub.1], [b.sub.2], [b.sub.3], [b.sub.4]), and c = ([c.sub.1], [c.sub.2], [c.sub.3], [c.sub.4]) be vectors in [E.sup.4.sub.1]. The vector product in the Minkowski space-time [E.sup.4.sub.1] is defined by the determinant

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

where [e.sub.1], [e.sub.2], [e.sub.3], and [e.sub.4] are mutually orthogonal vectors (coordinate direction vectors) satisfying the equations

[e.sub.1]^[e.sub.2]^[e.sub.3] = [e.sub.4], [e.sub.2]^[e.sub.3]^[e.sub.4] = [e.sub.1], [e.sub.3]^[e.sub.4]^[e.sub.1] = [e.sub.2], [e.sub.4]^[e.sub.1]^[e.sub.2] = -[e.sub.3].

Theorem 2.3. Let [alpha] = [alpha](t) be an arbitrary space-like curve in the Minkowski space-time [E.sup.4.sub.1]. The Frenet-Serret apparatus of a can be written as follows:

T = [alpha]'/[parallel][alpha]'[parallel], (2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.4)

[B.sub.1] = [mu]N^T^[B.sub.2], (2.5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.8)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.9)

where [mu] is taken -1 or +1 to make +1 the determinant of the [T,N,[B.sub.1],[B.sub.2]] matrix.

Here, we shall use Frenet-Serret invariants of a time-like curve. Therefore, our calculations do not contain null vectors.

3. TANGENT SPHERICAL IMAGE OF A TIME-LIKE CURVE LYING ON [H.sup.3.sub.0]

Following the paper [8], we first adapt the tangent spherical image definition to time-like curves of Minkowski space-time. Moreover, we give the definition of the trinormal spherical image for time-like curves at the beginning of Section 4.

Definition 3.1. Let [beta] = [beta](s) be a unit speed time-like curve in Minkowski space-time. If we translate the tangent vector to the centre O of the pseudohyperbolic space [H.sup.3.sub.0], we obtain a curve [delta] = [delta]([s.sub.[delta]]). This curve is called the tangent spherical image or tangent indicatrix of the curve [beta] in [E.sup.4.sub.1].

Theorem 3.2. Let [beta] = [beta](s) be a unit speed time-like curve and [delta] = [delta]([s.sub.[delta]]) be its tangent spherical image. Then

(i) [delta] = [delta]([s.sub.[delta]]) is a space-like curve;

(ii) the Frenet-Serret apparatus of [delta], {[T.sub.[delta]], [N.sub.[delta]], [B.sub.1[delta]], [B.sub.2[delta]], [K.sub.[delta]], [[tau].sub.[delta]], [[sigma].sub.[delta]] can be formed by the apparatus of [beta], {T,N,[B.sub.1],[B.sub.2],k,[tau],[sigma]}.

Proof. Let [beta] = [beta](s) be a unit speed time-like curve and [delta] = [delta]([s.sub.[delta]]) be its tangent spherical image. One can write

[delta] = T(s). (3.1)

By differentiation with respect to s we get

[delta]' = [delta] d[s.sub.[delta]]/ds = kN. (3.2)

Here we shall denote differentiation according to s by a dash, and differentiation according to [s.sub.delta]] by a dot. So, we immediately arrive at

[T.sub.[delta]] = N (3.3)

and

[parallel][delta]'[parallel] = d[s.sub.[delta]]/ds = k (3.4)

By the inner product g([delta]',[delta]') = [k.sup.2] > 0, thus the tangent spherical image [delta]([s.sub.[delta]) is a space-like curve. Considering the previous method, we form the following differentiations with respect to s:

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

By equation (2.4) we form

[[parallel][delta]'[parallel].sup.2] [delta]" - g([delta]', [delta]")[delta]' = [K.sup.3](KT + [tau][B.sub.1]). (3.6)

Then one can easily have the principal normal vector

[N.sub.[delta]] = KT + [tau][B.sub.1]/[sqaure root of ([absolute value of [[tau].sup.2] - [K.sup.2]])] (3.7)

and the first curvature

K.sub.[delta]] = [sqaure root of ([absolute value of [[tau].sup.2] - [K.sup.2]])]/K, (3.8)

where k [not equal to] [tau]. Now let us form the vector [T.sub.[delta]]^[N.sub.[delta]]^[delta]'''

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.9)

where [l.sub.i] are the components of the differentiable vector function of [delta]''' and X = [square root of ([absolute value of [[tau].sup.2] - [K.sup.2]])]. This product yields

[T.sub.delta]]^[N.sub.delta]]^[delta]''' = K[tau]/X{[tau][sigma]T + K[sigma][B.sub.1] + [tau](K/[tau])' [B.sub.2]}. (3.10)

Thus, we express the trinormal (second binormal) vector field of the curve [delta]([s.sub.[delta]]) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.11)

Taking the norm of both sides of (3.10), we have the second curvature

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.12)

To determine the binormal vector field, we express

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.13)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. So, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.14)

Finally, using (2.9) and the obtained equations, we have the third curvature

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.15)

Corollary 3.3. {[T.sub.[delta]], [N.sub.[delta]], [B.sub.1[delta]], [B.sub.2[delta]]} is an orthonormal frame ofMinkowski space-time.

Considering the above theorem, we also give:

Theorem 3.4. Let [beta] = [beta](s) be a time-like unit speed curve and [delta]([s.sub.[delta]]) be its tangent spherical image. If [beta] is a ccr-curve or a helix (i.e. W-curve), then [delta] is also a helix.

Proof. Let [beta] = [beta](s) be a time-like unit speed ccr-curve. Then we know K/[tau] = [c.sub.1] = constant and [tau]/[sigma] = [c.sub.2] = constant. Since, in terms of the above theorem, we have, respectively,

[K.sub.[delta]] = [square root of ([absolute value of 1 - 1/[c.sup.2.sub.1])]] = constant (3.16)

and

[[tau].sub.[delta]] = [c.sub.2]/[c.sub.1][square root of [c.sup.2.sub.1] - 1] = constant, (3.17)

[delta]([s.sub.[delta]]) is a spherical curve; so we may substitute [K.sub.[delta]] and [[tau].sub.[delta]] to the first part of formula (2.1). In this way we easily have

[[sigma].sub.[delta]] = [c.sub.2]/[c.sup.2.sub.1] = constant. (3.18)

Therefore, [delta]([s.sub.[delta]]) is also a helix. The case of [beta] being a helix can be immediately seen from the above equations.

By this theorem we present a characterization of the tangent spherical image with respect to constant Frenet-Serret curvature ratios (or helices). We observe that the mentioned indicatrix can be a helix, so, one can ask whether this tangent spherical image is a general helix or not. Therefore, we investigate it by the following statements.

Let [beta] = [beta](s) be a unit speed time-like curve and [delta] = [delta]([s.sub.[delta]) be its space-like tangent spherical image. If [delta] = [delta]([s.sub.[delta]]) is a general helix, then, for a constant space-like vector U, we may express

g([T.sub.[delta]],U)= cos [theta], (3.19)

where [theta] is a constant angle. Equation (3.19) is also congruent to

g(N,U) = cos [theta]. (3.20)

One can form a constant vectorU according to {T,N,[B.sub.1],[B.sub.2]} as follows:

U = [[epsilon].sub.1]T + [[epsilon].sub.2]N + [[epsilon].sub.3]B1 + [[epsilon].sub.4][B.sub.2]. (3.21)

Differentiating (3.21) with respect to s, we have the following system of ordinary differential equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.22)

We know that [[epsilon].sub.2] = c [not equal to] 0 is a constant. Using this system, we have two differential equations according to [[epsilon].sub.2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.23)

The solution of these two differential equations with variable coefficients is not easy. Besides, a general solution of it has not yet been found. Due to this, we shall prove the following theorem which contains a special solution of system (3.23).

Theorem 3.5. Let [beta] = [beta](s) be a unit speed time-like ccr-curve and [delta] = [delta]([s.sub.[delta]) be its space-like tangent spherical image. If [delta] is a general helix, then there exists a relation among Frenet-Serret curvatures of [beta]

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

where c, [c.sub.1] = K/[tau], [[alpha].sub.1], and [[alpha].sub.2] are constants.

Proof. Let us suppose [beta] = [beta](s) is a unit speed time-like ccr-curve and [delta] = [delta]([s.sub.[delta]]) is its space-like tangent spherical image; then relations (3.23) hold. We know that a ccr-curve has constant curvature ratios such that K/[tau] = [c.sub.1] and [tau]/[sigma] = [c.sub.2] are constants. Thus the differential equations (3.23) transform to

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

By the first equation we have

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

Using an exchange variable t = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] [sigma]ds in (3.25) (2), we obtain

[d.sup.2][[epsilon].sub.3]/[dt.sup.2] + [[epsilon].sub.3] = 0. (3.27)

The solution of it and (3.26) give us

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

for the real numbers c, c1, a1, and a2, as desired.

Corollary 3.6. The fixed direction (constant vector U) can be composed by the components

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

4. TRINORMAL SPHERICAL IMAGE OF A TIME-LIKE CURVE LYING ON [S.sup.3.sub.1]

Definition 4.1. Let [beta] = [beta](s) be a unit speed time-like curve in Minkowski space-time. If we translate the space-like trinormal vector fields to the centre O of the pseudohyperbolic space [H.sup.3.sub.0], we obtain a curve ([phi] = [phi]([s.sub.[phi]]). This curve is called the trinormal spherical image or trinormal indicatrix of the curve [beta] in [E.sup.4.sub.1].

We follow the same procedure to prove the following theorems. Owing to this, we express them without proofs.

Theorem 4.2. Let [beta] = [beta](s) be a unit speed time-like curve and [phi] = [phi]([s.sub.[phi]) be its trinormal spherical image. Then

(i) [phi] = [phi]([s.sub.[phi]) is a space-like curve;

(ii) the Frenet-Serret apparatus of [phi] can be determined by the apparatus of [beta] = [beta](s) by the following formulas:

[T.sub.[phi]] = [B.sub.1], (4.1)

[N.sub.[phi]] = [tau]N + [sigma][B.sub.2]/[square root of ([[tau].sup.2] + [[sigma].sup.2])], (4.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.7)

where [eta] = [square root of ([absolute value of [(k[tau][sigma]).sup.2] + (k[[tau].sup.2]).sup.2] - [[([tau][sigma])'].sup.2]], [phi] = [square root of ([[tau].sup.2] + [[sigma].sup.2]], and [[omega].sub.i] are the components of the differentiable function [[phi].sup.(IV)] with respect to s;

(iii) if [beta] is a helix, then [phi] is also a helix.

Theorem 4.3. Let [beta] = [beta](s) be a unit speed time-like ccr-curve and [phi] = [phi]([s.sub.[phi]]) be its space- like trinormal spherical image. If [phi] is a general helix, then

(i) there exists the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.8)

where c, [c.sub.2] = [tau]/[sigma], [[psi].sub.1] and [[psi].sub.2] are constants;

(ii) the fixed direction of the axis of the general helix is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.9)

5. CONCLUSION AND FURTHER REMARKS

In this work we extended the spherical image concept to the time-like curves of Minkowski space-time. We investigated tangent and trinormal spherical images of a time-like curve and observed that such spherical curves are space-like curves. Thereafter, we determined relations between Frenet-Serret invariants of spherical images and the base curve. In the light of the obtained results, we also express some open problems for further studies.

The involute of a given curve [alpha] = [alpha](s), which is a well-known concept in classical differential geometry, is defined by

[gamma] = [alpha] + [lambda][T.sub.[alpha]], (5.1)

where [lambda] is (c - [s.sub.[alpha]]) for constant c [6]. Recently, involutes of a space-like curve in Minkowski space-time were studied by [6]. Similary, if we express the involute of the tangent spherical image, we get

[gamma] = [delta] + [lambda][T.sub.[delta]] (5.2)

or, in other words,

[gamma] = T + [lambda]N. (5.3)

This equation belongs to the spherical image's involute with respect to the base curve. Besides, in an analogous way we may express the same case for the trinormal spherical image by

[zeta] = [phi] + [lambda][T.sub.[phi]] (5.4)

or

[zeta] = [B.sub.2] + [lambda][B.sub.1]. (5.5)

So, one can determine the relations between Frenet-Serret invariants of the involute of the tangent spherical image and the Frenet-Serret apparatus of the base curve.

doi: 10.3176/proc.2010.3.04

ACKNOWLEDGEMENT

The fourth author would like to thank Tubitak-Bideb for their financial support during his Ph.D. studies.

Received 9 September 2009, accepted 14 April 2010

REFERENCES

[1.] Camci, C., Ilarslan, K., and Sucurovic, E. On pseudohyperbolical curves in Minkowski space-time. Turk. J. Math., 2003, 27, 315-328.

[2.] Ekmekci, N., Hacisalihoglu, H. H., and Ilarslan, K. Harmonic curvatures in Lorentzian space. Bull. Malaysian Math. Soc. (Second Series), 2000, 23, 173-179.

[3.] Milman, R. S. and Parker, G. D. Elements of Differential Geometry. Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1977.

[4.] Monterde, J. Curves with constant curvature ratios. Bol. Soc. Mat. Mexicana, 2007, 3, 177-186.

[5.] Ozturk, G., Arslan, K., and Hacisalihoglu, H. H. A characterization of ccr-curves in Rm. Proc. Estonian Acad. Sci., 2008, 57, 217-224.

[6.] 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, 2008, 3, 793-801.

[7.] Walrave, J. Curves and Surfaces in Minkowski Space. Dissertation, K. U. Leuven, Fac. of Science, Leuven, 1995.

[8.] Yilmaz, S. Spherical Indicators of Curves and Characterizations of Some Special Curves in Four Dimensional Lorentzian Space [L.sup.4]. Dissertation, Dokuz Eylul University, 2001.

[9.] Yilmaz, S. and Turgut, M. On the differential geometry of the curves in Minkowski space-time I. Int. J. Contemp. Math. Sci., 2008, 3, 1343-1349.

[10.] Yilmaz, S., Ozyilmaz, E., and Turgut, M. On the differential geometry of the curves in Minkowski space-time II. Int. J. Comput. Math. Sci., 2009, 3, 53-55.

* Corresponding author, Melih.Turgut@gmail.com

Suha Yilmaz (a), Emin Ozyilmaz (b), Yusuf Yayli (c), and Melih Turgut (a) *

(a) Department of Mathematics, Buca Educational Faculty, Dokuz Eyliil University, 35160 Buca, Izmir, Turkey; suha.yilmaz@yahoo.com

(b) Department of Mathematics, Faculty of Science, Ege University, 35100 Bornova, Izmir, Turkey; eminozyilmaz@hotmail.com

(c) Ankara University, Facultyof Science, Department of Mathematics, 06100 Tandogan, Ankara, Turkey; yayli@science.ankara.edu.tr
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Title Annotation:MATHEMATICS
Author:Yilmaz, Suha; Ozyilmaz, Emin; Yayli, Yusuf; Turgut, Melih
Publication:Proceedings of the Estonian Academy of Sciences
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Date:Sep 1, 2010
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