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In this paper we consider the homothetic motion of Lorentzian circle by studying the scalar curvature for the corresponding cyclic surface locally. We prove that if the scalar curvature is constant, then . We describe the equations that govern such surfaces.

Homothetic motion is general form of Euclidean motion. It is crucial that homothetic motions are regular motions. These motions have been studied in kinematic and differential geometry in recent years. An equiform transformation in the n-dimensional Euclidean space

The number s is called the scaling factor. A homothetic motion is defined if the parameters of (1), including s, are given as functions of a time parameter t. Then a smooth one-parameter equiform motion moves a point x via

In this work we consider the homothetic motion of the hyperbolas(Lorentzian circles)

Let

The proof of our results involves explicit computations of the scalar curvature

This paper is organized as follows: In Section 2, we obtain the expression of the scalar curvature

In two copies

Under a one-parameter homothetic motion of

where

system. For varying t and fixed

where

As homothetic motion has an invariant point, we can assume without loss of generality that the moving frame

Thus

where

or in the equivalent form

For any fixed t in the above expression (3), we generally get an ellipse centered at the point

where

A straightforward computation leads to the coefficients of the first fundamental form defined by

is the sign matrix. Then we get

Under the conditions (4) a computation yields

and

The Christoffel symbols of the second kind are defined by

where

Although the explicit computation of the scalar curvature

The assumption of the constancy of the scalar curvature

Equation (8) means that if we write it as a linear combination of the functions

describe all cyclic surfaces with constant scalar curvature obtained by the homothetic motion of the Lorentzian circle

We distinguish the cases

In this section we assume that

We distinguish different cases that fill all possible cases (Note that we have all solutions by using the symbolic program Mathematica under the condition

At

Theorem 3.1. Let _{0} and given by (3) under condition (4). Assume

1)

2)

In particular, if

We have two possibilities:

1) If

coefficients

conditions

2) If

the coefficients

Theorem 3.2. Let _{0} and given by (3) under condition (4) hold:

1) Assume

2) Assume

If

Theorem 3.3. Let _{0} and given by (3) under condition (4). Assume

1)

2)

In this section we assume that the scalar curvature

Following the same scheme as in the case

1) CASE

If

1.

2.

3.

From (1), (2) and (3) we have

4.

If

2) CASE

1. If

2. If

3) CASE

Theorem 4.1. Let

and given by (3) under condition (4). Assume that

on the surface if and only if the following conditions hold:

In this section, we construct two examples of a cyclic surfaces

Example 1. Case

We assume

Theorem 3.3 says that

and

and both

Example 2. Case

Let

Theorem 4.1 says that

axonometric viewpoint

and

and both

M. M.Wageeda,E. M.Solouma, (2015) Local Study of Scalar Curvature of Cyclic Surfaces Obtained by Homothetic Motion of Lorentzian Circle. Applied Mathematics,06,1344-1352. doi: 10.4236/am.2015.68127