On Special Curves According to Darboux Frame in the Three Dimensional Lorentz Space

H. S. Abdel-Azizand M. Khalifa Saad,

1 Introduction

The curves and their frames play an important role in differential geometry and in many branches of science such as mechanics and physics, so we are interested here in studying one of these curves which have many applications in Computer Aided Design (CAD),Computer Aided Geometric Design (CAGD) and mathematical modeling. Also, these curves can be used in the discrete model and equivalent model which are usually adopted for the design and mechanical analysis of grid structures [Dincel and Akbarov (2017)].Smarandache Geometry is a geometry which has at least one Smarandachely denied axiom.It was developed by Smarandache [Smarandache (1969)]. We say that an axiom is Smarandachely denied if the axiom behaves in at least two different ways within the same space (i.e. validated and invalided, or only invalidated but in multiple distinct ways).

As a particular case, Euclidean, Lobachevsky-Bolyai-Gauss, and Riemannian geometries may be united altogether, in the same space, by some Smarandache geometries.Florentin Smarandache proposed a number of ways in which we could explore “new math concepts and theories, especially if they run counter to the classical ones”.

In a manner consistent with his unique point of view, he defined several types of geometry that are purpose fully not Euclidean and that focus on structures that the rest of us can use to enhance our understanding of geometry in general.

To most of us, Euclidean geometry seems self-evident and natural. This feeling is so strong that it took thousands of years for anyone to even consider an alternative to Euclid’s teachings. These non-Euclidean ideas started, for the most part, with Gauss, Bolyai, and Lobachevski, and continued with Riemann, when they found counter examples to the notion that geometry is precisely Euclidean geometry. This opened a whole universe of possibilities for what geometry could be, and many years later, Smarandache’s imagination has wandered off into this universe [Howard (2002)]. Curves are usually studied as subsets of an ambient space with a notion of equivalence. For example, one may study curves in the plane, the usual three dimensional space, the Minkowski space, curves on a sphere, etc.In three-dimensional curve theory, for a differentiable curve, at each point a triad of mutually orthogonal unit vectors (Frenet frame vectors) called tangent, normal and binormal can be constructed. In the light of the existing studies about the curves and their properties, authors introduced new curves. One of the important of among curves called Smarandache curve which using the Frenet frame vectors of a given curve. Among all space curves, Smarandache curves have special emplacement regarding their properties,this is the reason that they deserve special attention in Euclidean geometry as well as in other geometries. It is known that Smarandache geometry is a geometry which has at least one Smarandache denied axiom [Ashbacher (1997)]. An axiom is said to be Smarandache denied, if it behaves in at least two different ways within the same space.

Smarandache geometries are connected with the theory of relativity and the parallel universes and they are the objects of Smarandache geometry.

By definition, if the position vector of a curve δis composed by Frenet frame’s vectors of another curveβ, then the curve δis called a Smarandache curve [Turgut and Yilmaz(2008)]. The study of such curves is very important and many interesting results on these curves have been obtained by some geometers [Abdel-Aziz and Khalifa Saad (2015, 2017);Ali (2010); Bektas and Yunce (2013); ?etin and Kocayi?it (2013); ?etin, Tun?er and Karacan (2014)); Khalifa Saad (2016)]. Turgut et al. [Turgut and Yilmaz (2008)]introduced a particular circumstance of such curves. They entitled it SmarandacheTB2curves in the space. Special Smarandache curves in such a manner that Smarandache curvesTN1,TN2,N1N2and TN1N2with respect to Bishop frame in Euclidean 3-space have been seeked for by ?etin et al. [?etin, Tun?er and Karacan (2014)]. Furthermore, they worked differential geometric properties of these special curves and they checked out first and second curvatures of these curves. Also, they get the centers of the curvature spheres and osculating spheres of Smarandache curves.

Recently, Abdel-Aziz et al. [Abdel-Aziz and Khalifa Saad (2015, 2016)] have studied special Smarandache curves of an arbitrary curve such asTN,TB and TNBwith respect to Frenet frame in the three-dimensional Galilean and pseudo-Galilean spaces. Also in Abdel-Aziz et al. [Abdel-Aziz and Khalifa Saad (2017)], authors have studied Smarandache curves of a timelike curve lying fully on a timelike surface according to Darboux frame in Minkowski 3-space.

In this work, for a given timelike surface and a spacelike curve lying fully on it, we study some special Smarandache curves with reference to Darboux frame in the threedimensional Minkowski space. We are looking forward to see that our results will be helpful to researchers who are specialized on mathematical modeling.

2 Basic concepts

Definition 2.1A surface Ψin the Minkowski 3-spaceis said to be spacelike, timelike surface if, respectively the induced metric on the surface is a positive definite Riemannian metric, Lorentz metric. In other words, the normal vector on the spacelike (timelike)surface is a timelike (spacelike) vector [O’'Neil (1983)].

3 Smarandache curves of a spacelike curve

Let Ψbe an oriented timelike surface in Minkowski 3-spaceand r=r(s)be a spacelike curve with timelike normal vector lying fully on it. Then, the Frenet equations of r(s)are given by

where a prime denotes differentiation with respect to s. For this frame the following are satisfying

Let {T,P,U}be the Darboux frame of r(s), then the relation between Frenet and Darboux frames takes the form [Do Carmo (1976); O’Neil (1983)]:

where T is the tangent vector of r and U is the unit normal to the surface Ψand P=U×T. Therefore, the derivative formula of the Darboux frame of r(s)is in the following form:

The vectors T,Pand U satisfy the following conditions:

In the differential geometry of surfaces, for a curve r=r(s)lying on a surface M, the following are well-known [Do Carmo (1976)]

1)r(s)is a geodesic curve if and only if κg=0,

2)r(s)is an asymptotic line if and only if κn=0,

3)r(s)is a principal line if and only if τg=0.

Definition 3.1A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache curve[Turgut and Yilmaz (2008)].

In the following, we investigate Smarandache curvesTP,TU,PUand TPU, and study some of their properties for a curve lies on a surface as follows:

3.1 TP-Smarandache curves

Definition 3.2Let Ψbe an oriented timelike surface inand the unit speed spacelike curver=r(s)lying fully on Ψwith Darboux frame {T,P,U}. Then the TPSmarandache curves ofr are defined by

Theorem 3.1Let r=r(s)be a spacelike curve lying fully on a timelike surface Ψinwith Darboux frame {T,P,U}, and non-zero curvatures;Then the curvature functions of the TP-Smarandache curves ofr satisfy the following equations:

Proof.Let α=α()be a TP-Smarandache curve reference to a spacelike curve r.From Eq.(4), we get

So, we have

this leads to

Differentiating Eq.(8)with respect to s and using Eq.(7), we obtain

where

Then, the curvature is given by

as denoted by Eq. (5).

And the principal normal vector field of the curve αis

On the other hand, we express

So, the binormal vector of αis given by

where

Now, in order to calculate the torsion ofα , we consider the derivatives α′,α′with respect tos as follows

where

In the light of the above calculations, the torsion of αis calculated as Eq. (6).

Lemma 3.1Let α()be a spacelike curve lies on a timelike surface Ψin Minkowski 3-space, then

1) If αis a geodesic curve, the following hold

2) If αis an asymptotic line, the following hold

3) If αis a principal line, the following hold

3.2 TU-Smarandache curves

Definition 3.3Let Ψbe an oriented timelike surface inand the unit speed spacelike curver=r(s)lying fully on Ψwith Darboux frame {T,P,U}. Then the TUSmarandache curves ofr are defined by

Theorem 3.2Let r=r(s)be a spacelike curve lying fully on a timelike surface Ψinwith Darboux frame {T,P,U}, and non-zero curvatures;Then the curvature functions of the TU- Smarandache curves ofr satisfy the following equations:

Proof.Let β=β()be a TU- Smarandache curve reference to a spacelike curve r.From Eq.（9）, we get

Differentiating Eq.(12)with respect to s , we get

where

as denoted by Eq. (10).

And principal normal vector fieldof βis

Besides, the binormal vector of βis

where

Differentiating βwith respect to s , we get

similarly,

where

It follows that, the torsion of βis expressed as in Eq. (11).

Lemma 3.2Letβ()be a spacelike curve lies on Ψin Minkowski 3-space, then

1) If βis a geodesic curve, the following are satisfied

2) If βis an asymptotic line, then

3) If βis a principal line, the following are satisfied

3.3 PU-Smarandache curves

Definition 3.4Let Ψbe an oriented timelike surface inand the unit speed spacelike curver=r(s)lying fully on Ψwith Darboux frame {T,P,U}. Then the PUSmarandache curves ofr are defined by

Theorem 3.3Let r=r(s)be a spacelike curve lying fully on a timelike surface Ψinwith Darboux frame {T,P,U}, and non-zero curvatures;Then the curvature functions of the PU-Smarandache curves ofr satisfy the following equations:

Proof.Let γ=γ()be a PU-Smarandache curve reference to a spacelike curve r. From Eq.(13), we obtain

Differentiating Eq.(16)with respect to s , we have

where

and then, the curvature of γis given by

which is denoted by Eq. (14).

Based on the above calculations, we can express the principal normal vector of as follows

Also,

where

The derivatives γ′and γ′as follows

where

According to the above calculations, we obtain the torsion of γas in Eq. (15).

Lemma 3.3Letγ()be a spacelike curve lies on Ψin Minkowski 3-space, then

1) If γis a geodesic curve, the curvature and torsion of γare, respectively

2) If γis an asymptotic line, we get

3)If γis a principal line, the following hold

3.4 TPU-Smarandache curves

Definition 3.5Let Ψbe an oriented timelike surface inand the unit speed spacelike curver=r(s)lying fully on Ψwith Darboux frame {T,P,U}. Then the TPUSmarandache curves ofr are defined by

Theorem 3.4Let r=r(s)be a spacelike curve lying fully on a timelike surface Ψinwith Darboux frame {T,P,U}, and non-zero curvatures;Then the curvature functions of the TPU-Smarandache curves ofr satisfy the following equations:

Proof.Let δ=δ()be a TPU-Smarandache curve reference to a spacelike curve r .

Differentiating (21)with respect to s , we get

where

Then, the curvature is given by

as denoted in Eq. (19).

And the principal normal vector field of δis

So, the binormal vector of δis

where

For computing the torsion of δ, we are going to differentiate δ′with respect to sas follows

and similarly

where

In the light of the above derivatives, the torsion of δis computed as in Eq. (20), where

Thus the proof is completed.

Lemma 3.4Let δ()be a spacelike curve lies on Ψin Minkowski 3-space, then

1) If δis a geodesic curve, the curvature and torsion can be expressed as follows

2) If δis an asymptotic line, we have

3) If δis a principal line, we obtain

4 Computational example

In this section, we consider an example for a spacelike curve lying fully on an oriented timelike ruled surface in(see Fig. 1(b)), and compute its Smarandache curves.

Suppose we are given a timelike ruled surface represented as

where the spacelike base curve is given by (see Fig. 1(a))

Figure 1: The spacelike curve r(s) on the timelike ruled surface Ψ

So, we can compute the Darboux frame of Ψas follows

where

and

where

According to Eq. (3), the geodesic curvatureκg, the normal curvature κnand the geodesic torsionτgof the curve r are computed as follows

In the case of (s =0and v =0), we have

TP-Smarandache curve

For this curveα=α(), (see Fig. 2(a)), we have

where

If we choose (s=0)and (v=0), the curvature and torsion of αare

As the above, we can calculate the other Smarandache curves as follows:

TU-Smarandache curve

For this curve (see Fig. 2(b)), we have

where

and therefore (s =0and v =0)

PU-Smarandache curve

For this curve (see Fig. 3(a)), we have

TPU-Smarandache curve

For this curve (see Fig. 3(b)), we have

it follows that (s =0and v =0)

Figure 2: The TP and TU-Smarandache curves α and β of the spacelike curve r

Figure 3: The PU and TPU-Smarandache curves γ and δ of the spacelike curve r

5 Conclusion

In this study, Smarandache curves of a given spacelike curve with timelike normal lying on a timelike surface in the three-dimensional Minkowski space are investigated.According to the Lorentzian Darboux frame the curvatures and some characterizations for these curves are obtained. Finally, for confirming our main results, an example is given and plotted.

Acknowledgment:The authors are very grateful to referees for the useful suggestions and remarks for the revised version.

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