#### Abstract

In this study, we developed a mathematical framework for studying spherical and null curves in dual Lorentzian 3-space . Considering the causal character of the dual curve, sufficient and necessary conditions for spacelike curves to be dual Lorentzian spherical were given. Also, new sufficient and necessary conditions for non-null curves to be hyperbolic dual spherical were presented. Then, an insight on curves with null-principal normal is reported.

#### 1. Introduction

Nowadays, with the evolution of the relativity theory, geometers have extended some subjects’ geometrical studies of Riemannian manifolds to that of Lorentzian manifold. For example, in the Euclidean 3-space , a curve for which the tangent vector has a constant angle with a fixed direction is named a helix (distinguished by is a nonconstant linear function of the arc length parameter), spherical curves (distinguished by constant, with , ), and rectifying curves (distinguished by is a nonconstant linear function of the arc length parameter), where and are the torsion and curvature of the curve, respectively (see [1–5]). Due to its connection with physical sciences in Minkowski space, the helices, spherical, normal, and rectifying curves have been studied widely in [6–13].

Dual numbers were originally introduced to the subject of geometrical studies and later exploited to deal with the spatial kinematics and applied mechanics [14–17]. E. Study used it as an instrument for his studies on the differential line geometry. He dedicated specific attention to the representation of directed lines by dual unit vectors and considered the mapping that is recognized by his name. The set of dual points on the dual unit sphere in dual 3-space is in one-to- one correspondence with the set of all directed lines in Euclidean 3-space . Specific details on the needful fundamental concepts concerning with the dual elements and the one-to-one correspondence between the ruled surface and the one-parameter dual spherical motion can be found in [14–17]. If we take the Minkowski 3-space instead of , the E. Study map can be stated as follows: The set of dual points on the hyperbolic (Lorentzian) dual unit sphere (resp., ) in the Lorentzian dual 3-space is in one-to-one correspondence with the set of all timelike (resp., spacelike) directed lines in Minkowski 3-space .

In this work, we give some new characterizations of the spherical and null curves in . Then, several known properties of the helices, spherical, normal, and rectifying curves at the Euclidean 3-space are extended to the dual Lorentzian 3-space . In addition, new sufficient and necessary conditions for non-null curves to be hyperbolic dual spherical were presented. Also, the case of null dual curves to be spherical in dual Lorentzian 3-space was examined. Therefore, this work gives a connection with the classical surface theory since a differentiable curve on the hyperbolic (resp., Lorentzian) dual unit sphere (resp., ) corresponds to a timelike (resp., spacelike or timelike) ruled surface in [18–21].

#### 2. Preliminaries

It is started with some basic concepts on the theory of dual numbers, dual Lorentzian vectors and E. Study map (see [18–21]). If and refer to real numbers, the combination is named a dual number. At this point, is the dual unit subject to the rules . The set of dual numbers, , presents the commutative ring with the numbers as divisors of zero, not a field. No number has inverse in algebra. However, the algebra of dual numbers has similar lows as the complex numbers. Therefore, this setin addition to the Lorentzian scalar productresulted in what is named dual Lorentzian 3-space . This yieldswhere , , and are the dual base at the origin point of the dual Lorentzian 3-space . Then, this point has dual coordinates . If , the norm of is defined byThen, the vector is named the spacelike (resp., timelike) dual unit vector in case (resp., ). It is clear that

The six components , of and are named the normed Plücker coordinates of the line.

Assume is the fixed dual point in . Therefore,defines the Lorentzian dual unit sphere with the center anddefines the hyperbolic dual unit sphere that has the center . Then, the hyperbolic and Lorentzian (de Sitter space) dual unit spheres that has the center areandrespectively. As a result of this, the following map (E. Study map) is given as follows: The dual unit spheres are shaped as a pair of conjugate hyperboloids. The combined asymptotic cone presents the set of null (lightlike) lines, the ring-shaped hyperboloid represents the set of spacelike lines, the oval-shaped hyperboloid forms the set of timelike lines, and opposite points of each hyperboloid present the pair of opposite vectors on the line (see Figure 1).

Assume and are two real smooth differentiable curves at Minkowski 3-space . Therefore, the differentiable curverepresents the curve in the dual Lorentzian 3-space and is named the dual space curve. The pseudo dual arc length of from to is defined aswhere defines the unit tangent vector of . From here, the pseudo dual arc length of will be used as the parameter rather than . Therefore, is named the dual arc-length parameter curve. Also, we shall often not write the dual parameter explicitly in our formulae.

Let {, , } be the moving dual Serret–Frenet frame along , such that , , and are the unit tangent, the principal normal, and the binormal vector, respectively. The dual arc-length derivative of the dual Serret–Frenet frame iswhere , , and are nowhere pure dual curvature, and is nowhere pure dual torsion. Here “prime” indicates the derivative respecting to the pseudo dual parameter. Furthermore, in case is the unit speed spacelike dual curve with the null principal normal , we have the following equation:whereHere, will be just one of the two values: in case is the spacelike oriented line or otherwise it will be . Furthermore, in case is the null dual curve, i.e., if is a null dual vector, therefore, the Serret–Frenet formulae is read asand

The curvature takes just one of the two values: in case is the null oriented line or otherwise it will be .

#### 3. Explicit Characterizations of Dual Spherical Curves

Let be the non-null unit speed dual curve with timelike or spacelike rectifying dual plane in , with and are nowhere pure duals for each . Then, we have the following equation:where , and are differentiable dual functions of . Differentiating equation (17) with respect to , with equation (12), we obtain the following equation:

From equation (18), we have the following equation:

If the unit speed non-null dual curve with a spacelike or a timelike rectifying dual plane lies at the hyperbolic or Lorentzian dual unit sphere, then , for or , respectively. So, considering equation (17), it gives us the following equation:

Differentiating equation (20), with using equation (19), we obtain the following equation:where and are nowhere pure duals. If we substitute equation (21) in equation (17), we have the following equation:

Thus, from the fact that , we obtain the following equation:

Therefore, the following corollaries will be given:

Corollary 1. *There is no unit speed timelike dual curves which lies on .*

*Proof. *Suppose is the unit speed timelike dual curve which lies at . Therefore, we have the following equation:and this is a contradiction.

Corollary 2. *Suppose is the unit speed timelike dual curve lies on , with and are nowhere pure duals for each , then*

Theorem 1. *A unit speed spacelike dual curve with a spacelike principal normal lies at the hyperbolic dual sphere with imaginary dual radius for iffwhere and are nowhere pure duals for all .*

*Proof. *Assume that is the unit speed spacelike dual curve with the spacelike principal normal lying on the hyperbolic dual sphere with imaginary dual radius for . Then, if and nowhere pure duals for all , we have the following equation:It follows thatThis inequality requires that ; i.e., it is not constant. Furthermore, the differentiation of relation (27) equals toand consequentlyConversely, assume that the unit speed spacelike dual curve with a spacelike principal normal satisfies the relationssuch that and are nowhere pure duals for all . From the inequality, we know that , and also, by the fact that and are nowhere pure duals for all , a simple calculation gives us the following equation:which is the differential of the equationso that we may take . This implies that lies at the hyperbolic sphere of with imaginary dual radius for .

In a similar form, the following corollaries are given.

Corollary 3. *Suppose is the unit speed spacelike dual curve with a timelike principal normal. If lies at the hyperbolic dual sphere with imaginary dual radius for , thensuch that and are nowhere pure duals for all . This relation is satisfied iff**The Lorentzian spherical spacelike curves with spacelike principal normal were studied by Pekmen and Pasali [13]. The following corollary is the dual version of the corollary given in [13].*

Corollary 4. *Let be the unit speed spacelike curve with the timelike principal normal. If lies on the dual Lorentzian sphere of radius for , thensuch that and are nowhere pure duals for all . This relation exists iff**The Lorentzian spherical spacelike curves with timelike principal normal were studied by Petrovic–Torgasev and Sucurovic. The following corollary is the dual version of the given in Theorems 2 and 3 of [14].*

Corollary 5. *Let be the unit speed spacelike dual curve with the spacelike principal normal. If lies at the Lorentzian dual sphere of radius for , then ; thensuch that and are nowhere pure duals for all . This relation exists iff*

Afterwards, some necessary and sufficient conditions of lie at (or ) for and are nowhere pure duals for all . Here, if we make a parameter change likewe obtain the following theorem:

Theorem 2. *A non-null unit speed dual curve with timelike or spacelike rectifying plane lies on a Lorentzian dual sphere with radius (or a hyperbolic dual sphere with imaginary dual radius ) iff there exists a C^{1}-dual function , such thatwhere is nowhere pure dual.*

*Proof. *Assume that the unit speed non-null dual curve with spacelike or timelike rectifying dual plane lies on a Lorentzian dual sphere with radius (or a hyperbolic dual sphere with imaginary dual radius ). So, for , by equations (22) and (23), we can introduce *C*^{1} dual functions = and defined byrespectively. From equations (22) and (42), we have or , which is the first equation in the theorem. Differentiation of equation (42) with respect to , and making use of equations (12), (22), and (40), we obtain , which is the second equation in the theorem.

On the other hand, suppose is the non-null unit speed dual curve in and satisfies the condition (41). Consider a dual curve defined byand a vector dual function is defined by , thenDifferentiating equations (43) and (44) with respect , and making use of equation (8), we obtain the following equation:These show that the dual vectors and are constant dual vectors; therefore, the non-null dual curve lies on a Lorentzian dual sphere with radius (or a hyperbolic dual sphere with imaginary dual radius ).

According to Theorem 2, we have the following theorems.

Theorem 3. *The unit speed timelike dual curve lies on a Lorentzian dual sphere of radius iff there are dual constants , ; that is,holds for every .*

*Proof. *First assume that lies on a Lorentzian dual sphere of radius ; that is, it satisfies the conditions in Theorem 3. Thus, there is a differential dual function such thatFurthermore, the dual functions and are defined byDifferentiation of the dual functions and with respect to *s* easily gives . Also, we get , which is the radius of Lorentzian dual sphere. Hence, ; that is, and are dual constants, so the above relation becomesMultiplying the first previous equations with , multiplying the second by , and then adding them together resulted in . Thus, considering equation (27) gives equation (46).

On the other hand, suppose and are the dual constants; that is, equation (46) holds for all . Then, is nowhere pure dual, such that . The differentiation with respect to of the above relation (46) givesFurthermore, the differentiable function is defined byThen, the above two relations give . By differentiation with respect to of relation (51) and using relation (46), we obtain . Therefore, by Theorem 2, we obtain that this curve lies on the Lorentzian dual sphere with radius .

Theorem 4. *The unit speed spacelike dual curve with timelike or spacelike rectifying dual plane lies at the Lorentzian dual sphere of radius iff there are duals , such thatholds for all .*

*Proof. *Assume that is the unit speed spacelike dual curve with timelike or spacelike rectifying dual plane lies at the Lorentzian dual sphere of radius . Also, by Theorem 2, for a Lorentzian dual sphere of radius , there exists a differential dual function ; that is,Let us define the dual functions and bySince is a spacelike dual curve, that is, , we know . So, the last functions becomeIt is easily seen that , and this means that .

In these regards, by our assumption we obtain Also, if we differentiate the dual functions and with respect to , we obtain Thus, and are dual constants, such that . Thus, the above relations becomeMultiplying the first previous equations with , and multiplying the second by -, and then adding them together resulted in . Consequently, by equation (40), we findConsidering completes the proof of the necessary condition.

Conversely, let and be the real constants that are satisfying relation (52) for all . Then, obviously is nowhere pure dual for all . The differentiation with respect to of the above relation givesAlso, the differentiable dual function is defined asThen, the above two relations give . By differentiation with respect to of relation (62) and using relation (52), we get . Therefore, by Theorem 2, this spacelike dual curve with a spacelike or the timelike rectifying dual plane lies on the Lorentzian dual sphere of radius .

The real parts of the special case of and in Theorem 4 correspond to Theorem 5 of [13] that can be seen in the following corollary.

Corollary 6. *A unit speed spacelike dual curve with spacelike principal normal lies on the Lorentzian dual sphere of radius iff there are dual constants , such thathold for all .*

The real parts of the special case of and in Theorem 4 correspond to Theorem 5 of [14] that can be seen in the following corollary.

Corollary 7. *A unit speed spacelike dual curve with timelike principal normal lies on the Lorentzian dual sphere of radius iff there are constants and such thathold for all .**By interchanging the roles of hyperbolic dual sphere with Lorentzian dual sphere, we state the following theorems.*

Theorem 5. *The unit speed spacelike dual curve with a spacelike or a timelike rectifying dual plane lies at the hyperbolic dual sphere of imaginary dual radius iff there are dual constants and such thathold for all .*

*Proof. *Assume that is the unit speed spacelike dual curve with spacelike principal normal lies a hyperbolic dual sphere of imaginary dual radius . By Theorem 2 for a hyperbolic dual sphere of imaginary dual radius , there is a differential dual function ; that is,Let us define the dual functions and byIt is easily seen that . By our assumption, we get . Also, if we differentiate and with respect to , we get . Thus, and are dual constants, such that . Thus, from the above relations, one can sayMultiplying the first previous equations with , and multiplying the second by , and then adding them together resulted in . Consequently, by equation (19), we find the following equation:Considering completes the proof of the necessary condition.

Conversely, let and be the dual constants that are satisfying the relation (60) for all . Then, obviously is nowhere pure dual . The differentiation with respect to of the above relation givesFurthermore, the differentiable dual function is defined byThen, the above two relations give . By differentiation with respect to of relation (68) and using relation (60), we get . Therefore, by Theorem 2, this spacelike dual curve with a spacelike or a timelike rectifying dual plane lies at the hyperbolic dual sphere of imaginary dual radius .

Corollary 8. *The unit speed spacelike dual curve with timelike principal normal lies at the hyperbolic dual sphere of imaginary dual radius iff there are dual constants and such thatholds for all *

Corollary 9. *A unit speed spacelike dual curve with timelike principal normal lies on the Lorentzian dual sphere of dual radius iff there are dual constants and such thathold for all .*

##### 3.1. Spacelike Dual Spherical Curves with Null Principal Normal

The geometric place of the curves with null principal normal can be given by the following theorem.

Theorem 6. *Let be the unit speed spacelike dual curve, with the null principal normal in . Therefore, lies on (or ) iff is the planar dual curve andis satisfied for all .*

*Proof. *Assume is the dual curve satisfying the mentioned conditions which lies on (or ). As stated in the above section, we have the following equation:where , and are differentiable dual functions of . Differentiating equation (72) with respect to , and by the use of corresponding Serret–Frenet formulae, it is found thatand since in this case we have , for all , it follows thatIn view of equation (72), and from the fact thatwe get the following equation:Differentiation of equation (76), with account of equation (74), we get the following equation:Consequently, is the planar dual curve, and by replacing equation (77) in equation (72), we obtainConversely, assume that is the unit speed spacelike planar dual curve, with the null principal normal , which satisfies equation (78) for all . By differentiation of equation (78) with respect to , it is found thatSince in this case we have and for all , the corresponding Serret–Frenet formulae read asCombining this equation with equation (79) yields ; i.e., the vector is constant dual vector. Thus, we find , so that (or ).

The Lorentzian spherical spacelike curves with null principal normal was studied by Petrovic–Torgasev and the following theorem was also proved by [14].

Theorem 7. *Suppose is the unit speed spacelike dual curve, with the null principal normal , for all . iff there are dual constants and such that*

*Proof. *Suppose is the dual curve satisfying the mentioned conditions, which lies on . Taking into consideration Theorem 6, we see that , for all . Also consider the dual function , the functions , and bySince , for all , we find thatTherefore, there are dual constants and ; that is, the equationholds for all .

Conversely, assume that here exist dual constants and , such that the dual torsion of the dual curve satisfies equation (84). By differentiation of equation (84) with respect to , it is found thatDifferentiating equation (85) with respect to , it is found thatApplying equation (84) to equation (86), it gives directly:Multiplying the last equation with , and using equation (74), we get , and consequently , for every . Moreover, consider the dual curve defined bySince , for all , it follows that , thereby is dual constant. Finally, we easily obtain , so that

By a similar procedure as in Theorem 7, we can give the following theorem:

Theorem 8. *Suppose ) is a unit speed spacelike dual curve, with the null principal normal , for all . lies on iff there are dual constants and such that*

##### 3.2. Spherical Null Dual Curves

Now, let us give the characterization for spherical null dual curves with the following theorem.

Theorem 9. *There is no null dual curves lying on (or ).*

*Proof. *Suppose is the null dual curve lying on (or ). Then,where , and are differentiable functions of . By a similar procedure as in Theorem 6, we have the following equation:If is an oriented line, i.e., , then we have , , , and thereforewhere , , and are dual constants of integrations. Then,More specific, using the fact that in equation (92) we obtain , and consequently is a constant, which is a contradiction. On the other hand, if is not an oriented line, i.e., , we find thatConsequently, we have and , where . Then,which together with gives . This contradiction completes the proof.

#### 4. Conclusions

In the present paper, we dealt with spherical and null curves in dual Lorentzian 3-space . Then, without the precondition on the dual torsion is nowhere pure dual, a necessary and sufficient condition for a non-null curve to be dual Lorentzian spherical is presented. Also, new necessary and sufficient conditions for null curves to be hyperbolic dual spherical were reported. Hopefully, these results will be useful to physicists and those studying the general relativity theory.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest.

#### Acknowledgments

The second author expresses her gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R27), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project (PNURSP2022R27), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.