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On non-unit speed curves in Minkowski 3-space.

[section]1. Introduction

Suffice it to say that the many important results in the theory of the curves in [E.sup.3] were initiated by G. Monge and the moving frames idea was due to G. Darboux. In the classical differential geometry of curves in [E.sup.3], the unit speed curve which is obtained by using the norm of the curve, is a well known concept. In the light of this idea, there are many studies on the unit speed curve in Minkowski space. Frenet apparatus for non-unit speed curve [E.sup.3] has been studied by Sabuncuoglu in [3]. Also he has studied Darboux frame and its derivative formulas for non-unit speed curves and surface pair in [E.sup.3] [3].

In this work, by means of the method used to obtain Frenet apparatus and Darboux frame for non-unit speed curves in [E.sup.3] in [3], we present Frenet apparatus for non-unit speed curves and Darboux frame and its derivative formulas for non-unit speed curves and surface pairs in Minkowski 3-space [E.sup.3.sub.1].

[section]2. Preliminaries

Let [E.sup.3.sub.1] be the three-dimensional Minkowski space, that is, the three-dimensional real vector space [E.sup.3] with the metric

<dx,dx>=[dx.sup.2.sub.1] + [dx.sup.2.sub.2] - [dx.sup.2.sub.3],

where ([x.sub.1], [x.sub.2], [x.sub.3]) denotes the canonical coordinates in [E.sup.3]. An arbitrary vector x of [E.sup.3.sub.1] is said to be spacelike if < x, x >>0 or x = 0 , timelike if < x, x ><0 and lightlike or null if < x, x >=0 and x= 0. A timelike or light-like vector in [E.sup.3.sub.1] is said to be causal. For x [member of] [E.sup.3.sub.1], the norm is defined by [parallel]x[parallel] = [square root of [absolute value of< x, x >]], then the vector x is called a spacelike unit vector if < x, x >=1 and a timelike unit vector if < x, x >= -1. Similarly, a regular curve in [E.sup.3.sub.1] can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors are spacelike, timelike or null (lightlike), respectively [2]. For any two vectors x = ([x.sub.1],[x.sub.2],[x.sub.3]) and y = ([y.sub.1],[y.sub.2],[y.sub.3]) of [E.sup.3.sub.1], the inner product is the real number < x, y >= [x.sub.1][y.sub.1] + [x.sub.2][y.sub.2] - [x.sub.3][y.sub.3] and the vector product is defined by x x 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.1][y.sub.2] - [x.sub.2][y.sub.1])).

The unit Lorentzian and hyperbolic spheres in [E.sup.3.sub.1] are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Let {T, n, b} be the moving Frenet frame along the curve a with arc-length parameter s. For a spacelike curve [alpha], the Frenet Serret equations are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where (T, T) = 1, (N, N) = [epsilon] = [+ or -]1, (B, B) = -[epsilon], (T, N) = (T, B) = (N, B) = 0. Furthermore, for a timelike non-unit speed curve [alpha] in [E.sup.3.sub.1], the following Frenet formulae are given in as follows,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where (T,T) = -1, (N, N) = (B,B) = 1, (T,N) = (T,B) = (N,B) = 0 [1].

Let M be an oriented surface in three-dimensional Minkowski space [E.sup.3.sub.1] and let consider a non-null curve [alpha](t) lying on M fully. Since the curve [alpha](t) is also in space, there exists Frenet frame {T, N, B} at each points of the curve where T is unit tangent vector, N is principal normal vector and B is binormal vector, respectively. Since the curve [alpha](t) lies on the surface M there exists another frame of the curve [alpha](t) which is called Darboux frame and denoted by {T, b, n}. In this frame T is the unit tangent of the curve, n is the unit normal of the surface M and b is a unit vector given by b = n x T. Since the unit tangent T is common in both Frenet frame and Darboux frame, the vectors N, B, b and n lie on the same plane. Then, if the surface M is an oriented timelike surface, the relations between these frames can be given as follows.

(i) If the surface M is a timelike surface, then the curve [alpha](t) lying on M can be a spacelike or a timelike curve. Thus, the derivative formulae of the Darboux frame of [alpha](t) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where (T,T) = [member of] = [+ or -]1, (b,b) = -[member of], (n,n) = 1.

(ii) If the surface M is a spacelike surface, then the curve [alpha](t) lying on M is a spacelike curve. Thus, the derivative formulae of the Darboux frame of [alpha](t) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where (T,T) = 1, (b,b) = 1, (n,n) = -1 [1].

In these formulae at (4) and (5) , the functions [[kappa].sub.n], [[kappa].sub.g] and [[tau].sub.g] are called the geodesic curvature, the normal curvature and the geodesic torsion, respectively. Here and in the following, we use "dot" to denote the derivative with respect to the arc length parameter of a curve. Here, we have the following properties of [alpha] characterized by the conditions of [[kappa].sub.g], [[kappa].sub.n] and [[tau].sub.g]:

[alpha] is a geodesic curve if and only if [[kappa].sub.g] = 0.

[alpha] is a asymptotic curve if and only if [[kappa].sub.n] = 0.

[alpha] is line of curvature if and only if [[tau].sub.g] = 0.

(see [2]).

[section]3. Some properties of non-unit speed curves in [E.sup.3.sub.1]

Let [alpha](t) be an arbitrary curve with speed

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theoretically we may reparametrize a to get a unit speed curve [bar.a](s(t)), so, we define curvature and torsion of [alpha] in terms of its arclength reparametrization [bar.a](s(t)). Moreover, the tangent vector [alpha]'(t) is in the direction of the unit tangent T(s) of the reparametrization, so T(s(t)) = [alpha]'(t)[absolute value of [alpha]'(t)]. This says that we should define a non-unit speed curve's invariants in terms of its unit speed reparametrization's invariants.

Definition 3.1. Let [alpha] be any non unit speed curve in Minkowski 3-space [E.sup.3.sub.1], so

(1) The unit tangent of [alpha](t) is defined to be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(2) The curvature of [alpha](t) is defined to be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(3) If [kappa] > 0, then the principal normal of [alpha](t) is defined to be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(4) If [kappa] > 0, then the binormal of [alpha](t) is defined to be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(5) If [kappa] > 0, then the torsion of [alpha](t) is defined to be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 3.1. If the curve [alpha]: I [right arrow] [E.sup.3.sub.1] is a regular non-unit speed curve in [E.sup.3.sub.1], then there is a parameter map h : I [right arrow] J so that [parallel]([alpha] o h)'[parallel] = 1 for all s [member of] J.

Proof. Let the arc-length function of the curve [alpha] be f. It is known that f' = [parallel]a'[parallel]. The function f' has values differed from zero since [alpha] is a regular curve. Hence f is a one to one function because it is an increasing function. Let's say f (I) = J. Consequently, f becomes regular and bijective function. So has get the inverse, let h be the inverse of f and [alpha] * h = [beta]. So if we show that [beta] : J [right arrow] [E.sup.3.sub.1] is a unit speed curve, the proof is completed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 3.2. (The Frenet formulas for non-unit speed curves in [E.sup.3.sub.1]) For a regular curve [alpha] with speed

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and curvature [kappa] > 0,

(i) if [alpha] is a spacelike non-unit speed curve, then the derivative formula of Frenet frame is as follows:

T' = v[kappa]N,

N' = v(- [epsilon][kappa]T + [tau]B),

B' = v[tau]N.

(ii) if [alpha] is a timelike non-unit speed curve, then the derivative formula of Frenet frame is as follows:

T' = v[kappa]N,

N' = v([kappa]T + [tau]B),

B' = v[tau]N.

Proof. (i) For non-unit speed spacelike curve, the unit tangent T(t) is [bar.T](s) by the definition 3.1. Now T'(t) denotes differentiation with respect of t, so we must use the chain rule on the righthand side to determine k and [tau]. From the definition 3.1,

T (t)= [bar.T](s(t)). (6)

If we differentiate (6), the expression is obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

From (7), we get

T' = [kappa] (t)vN(t).

So the first formula of (i) is proved. For the second and third,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by the unit speed Frenet formulas, we have

N'(t) = -K(t)vT (t) + t (t)vB(t).

And we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii) For non-unit speed timelike curve, if the computations are made as follows: by the definition 3.1,

T (t)= [bar.T](s(t)). (8)

If we differentiate (8), the expression is found as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

From (9), we obtain

T' = k(t)vN(t).

So the first formula of (ii) for non-unit speed timelike curve is proved. For the second and third, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and finally

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 3.1. For a non-unit speed curve a in [E.sup.3.sub.1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Since [alpha](t) = [bar.[alpha]](s(t)), the first calculation is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

while the second is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[section]4. Non-unit speed curve and surface pair in [E.sup.3.sub.1]

In this part, we will define the curvatures of curve-surface pair ([alpha], M) while a is a non-unit speed curve in [E.sup.3.sub.1]. Let's consider the non-unit speed curve [alpha] : I [right arrow] M. Previously, we proved that there is a parameter map h : J [right arrow] I so that [alpha] o h : J [right arrow] M is a unit speed curve. The function h is the inverse of f(t) = [[integral].sup.t.sub.t0] [parallel][alpha]'(r)[parallel]dr. Let's say [alpha] o h = [beta]. [beta] is a unit speed curve. Let's the unit tangent vector field of [beta] be denoted by T. The set {T,b, n * [} is the frame of curve-surface pair ([, M), where b = (n * [) x [bar.T]. Let s [member of] J, h(s) = t. s = f (t) because of h(s) = [f.sup.-1](s). So [beta](s) = [alpha](h(s)) = [alpha](t).

Definition 4.1. Let [beta] be the unit speed curve obtained from the non-unit speed curve [alpha] :I [right arrow] M in [E.sup.3.sub.1]. While the frame of curve-surface pair ([, M) is {T, b, n * [}, the set {T, b, no[alpha]} defined by

T (t)= [bar.T] (f(t)), b(t) = [bar.b](f(t)), n([alpha](t)) = n([beta](f(t)))

is called Darboux frame of curve-surface pair ([alpha], M) in [E.sup.3.sub.1]. Also, while the curvatures of curvesurface pair ([beta], M) are [[bar.K].sub.n], [[bar.K].sub.g], [[bar.[tau]].sub.g], the curvatures of curve-surface pair ([alpha], M) are as follows:

[K.sub.n](t) = [[bar.K].sub.n](f(t)), [K.sub.g](t) = [[bar.K].sub.g](f(t)), [[tau].sub.g](t) = [[bar.[tau]].sub.g] (f(t)).

Theorem 4.1. Given the non-unit speed curve [alpha] : I [right arrow] M, the curvatures [[kappa].sub.g], [[kappa].sub.n] and [[tau].sub.g] of curve-surface pair ([alpha], M) in [E.sup.3.sub.1] are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Let f (t) = s, by definition 4.1, we get

[K.sub.n](t) = [[bar.K].sub.n](s) = ([beta]"(s), (n * [beta])(s)). (10)

Let s calculate the components of (10),

[beta]' (s)= [bar.T] (s)= T (t) = 1/v(t)[alpha]' (t), (11)

we write (11) as follows:

[alpha]'(t) = v(t)[beta]'(s) = v(t)[beta]'(f(t)) = v(t)([beta]' * f)(t). (12)

If we differentiate (12), we get

[alpha]"(t) = v'(t)([beta]' * f)(t) + v(t)f'(t)[beta]"(f(t)) = v'(t)[beta]'(s) + [(v(t)).sup.2][beta]"(s). (13)

By differentiating (11) and using (13), we get

[beta]"(s) = 1/[v.sup.2](t)(a"(t) - v'(t)[beta]'(s)). (14)

By using (14) in (10),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let's see the equation of [[kappa].sub.g](t). So

[K.sub.g](t)= [[bar.K].sub.g](s) = ([beta]"(s), [bar.b](s)). (15)

By using (14) in (15), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Finally, let's see the equation of [[tau].sub.g](t). So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Also we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

(n * [beta])'(s) = 1/v(t)(n * [alpha])'(t). (17)

By using (17) in (16), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 4.2. Given non-unit speed curve [alpha] : I [right arrow] M,

(i) If ([alpha], M) is timelike or spacelike curves on timelike surface pair, the derivative formula of Darboux frame {T, b, n * [alpha]} is as follows:

T'(t) = v(t)[[K.sub.g](t)b(t) - [member of][[kappa].sub.n](t)(n * [alpha])(t)], b'(t) = v(t)[[K.sub.g](t)T(t)+ [member of][[tau].sub.g](t)(n * [alpha])(t)], (n * [alpha])'(t) = v(t)[[K.sub.n](t)T (t) + [[tau].sub.g](t)b(t)],

for curve-surface pair ([alpha], M) in [E.sup.3.sub.1].

(ii) If ([alpha], M) is spacelike curve on spacelike surface pair, the derivative formula of Darboux frame {T, b, n * [alpha]} is as follows:

T'(t) = v(t)[[K.sub.g](t)b(t) + [K.sub.n](t)(n * [alpha])(t)], b'(t) = v(t)[--[K.sub.g](t)T(t) + [[tau].sub.g](t)(n * [alpha])(t)], (n * [alpha])'(t) = v(t)[[K.sub.n](t)T(t) + [[tau].sub.g](t)b(t)],

for curve-surface pair ([alpha], M) in [E.sup.3.sub.1].

Proof. (i) For timelike surface:

T(t)= [bar.T](f (t)) = ([bar.T] * f)(t). (18)

By differentiating (18), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

b(t) = [bar.b](f(t)) = ([bar.b] * f)(t). (19)

By differentiating (19), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the third formula of (i),

(n * [alpha])(t) = (n * [beta] * f)(t). (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii) For spacelike surface:

T(t) = [bar.T](f (t)) = ([bar.T] * f)(t). (11)

By differentiating (11), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

b(t) = [bar.b](f (t)) = ([bar.b] * f)(t). (12)

By differentiating (12), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the third formula of (ii),

(n * [alpha])(t) = (n * [beta] * f)(t). (13)

By differentiating (13), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Corollary 4.1. Given non-unit speed curve [alpha] : I right arrow M, the curvatures of curve-surface pair ([alpha], M) [in E.sup.3.sub.1] as follows:

(i) If ([alpha], M) is timelike or spacelike curves on timelike surface pair, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii) If ([alpha], M) is spacelike curve on spacelike surface pair, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. The direct result of Theorem 4.3.

References

[1] R. Lopez, Differential geometry of curves and surfaces in Lorentz-Minkowski space, Mini-Course taught at the Instituto de Matematica [member of] Estatistica (IME-USP), University of Sao Paulo, Brasil, 2008.

[2] B. O'Neill, Semi Riemannian geometry with applications to relativity, Academic Press, Inc. New York, 1983.

[3] A. Sabuncuoglu, Diferensiyel Geometri, Nobel Yayin Dagitim, Ankara, 2001, (in Turkish).

Yasin Unluturk ([dagger])] and Umit Ziya Savci ([double dagger])

([dagger]) Department of Mathematics Kirklareli University, 39100, Kirklareli-Turkey

([double dagger]) Department of Mathematics Karabuk University, 78050, Karabuk-Turkey E-mail: yasinunluturk@kirklareli.edu.tr zsavci@hotmail.com
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Author:Unluturk, Yasin; Savci, Umit Ziya
Publication:Scientia Magna
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Date:Dec 1, 2012
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