^{2}

_{0}

^{1}

^{1}

^{*}

In this paper, we define dual curvature motion on the dual hyperbolic unit sphere H<sup>2</sup><sub style="margin-left:-8px;">0</sub> in the dual Lorentzian space D<sup>3</sup><sub style="margin-left:-8px;">1</sub> with dual signature (+,+-) . We carry the obtained results to the Lorentzian line space R<sup>3</sup><sub style="margin-left:-8px;">1</sub> by means of Study mapping. Then we make an analysis of the orbits during the dual hyperbolic spherical curvature motion. Finally, we find some line congruences, the family of ruled surfaces and ruled surfaces in R<sup>3</sup><sub style="margin-left:-8px;">1</sub>.

Dual numbers had been introduced by W.K. Clifford (1845-1849) as a tool for his geometrical investigations. After him, E. Study (1860-1930) used dual numbers and dual vectors in his research on the geometry of lines and kinematics [

E. Study’s mapping plays a fundamental role in the real and dual Lorentzian spaces [

Real spherical curvature motion had been introduced by A. Karger and J. Novak [

In this section, we give a brief summary of the theory of dual numbers, dual Lorentzian vectors and Study’s mapping.

Let

where

A vector

The norm of a vector

tors in

If

The set of all dual numbers forms a commutative ring over the real numbers field and is denoted by 𝔻. Then the set

is a module over the ring 𝔻 which is called a 𝔻-module or dual space. The elements of

where

If

Let

where

The norm

Then we can write

The Lorentzian inner product of two dual vectors

where

A dual vector

where

The set of all dual Lorentzian vectors is called dual Lorentzian space and it is denoted by

The Lorentzian cross product of dual vectors

Lemma 2.1. Let

where

Lemma 2.1. Let

1)

2)

3)

4)

Let

, (resp.,),

The set of all dual timelike unit vectors (resp., all dual spacelike unit vectors) is called the dual hyperbolic unit sphere (resp., dual Lorentzian unit sphere) and is denoted by

Theorem 2.2. (E. Study Map) [

Definition 2.1. A directed timelike line in

is called the moment of the vector

This means that the direction vector

Let

and

[

A ruled surface is a surface generated by the motion of a straight line in

Definition 2.2. A ruled surface is said to be timelike if the normal of surface at every point is spacelike, and spacelike if the normal of surface at every point is timelike [

Let

Let us consider a fixed dual orthonormal frame

endpoints of segment

where the vector

(timelike).

where

i.e.

Thus, we have the orthonormal dual frame

Denote the dual hyperbolic angles of

where

where _{1} and Ө_{2} are constant (i.e.

Let

Since

and

If

From Equation (9) we have

which represents a line congruence. Thus, we have the following theorem.

Theorem 3.1. During the dual hyperbolic spherical curvature motion

If we take

Thus, we have the following theorem.

Theorem 3.2. During the dual hyperbolic spherical curvature motion

In addition, if we take

which represents a right helicoid.

If

If we put

From Equation (14) we have

which represents a cone whose axis is the vector

Theorem 3.3. During the dual hyperbolic spherical curvature motion, the orbit drawn on

If we put

From Equation (16) we have

which represents a cone whose axis is the vector

Theorem 3.4. During the dual hyperbolic spherical curvature motion, the orbit drawn on

Seperating real and dual parts of

Equations (18) and (19) have only two parameters

Since

In the case of

From Equation (22) we obtain

which represents an one-parameter family of cone in

If we put

which represents an elliptic cone, whose axis is the vector

Theorem 4.1. During the dual hyperbolic spherical curvature motion, the orbit drawn on

In addition, putting various values of parameters in the Equations (21) or (22) we have different line congruences or ruled surfaces in

This paper presents the curvature motion on the dual hyperbolic unit sphere