Local Study of Scalar Curvature of Cyclic Surfaces Obtained by Homothetic Motion of Lorentzian Circle

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 0  = . We describe the equations that govern such surfaces.


Introduction
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 n  is an affine transformation whose linear part is composed from an orthogonal transformation and a homothetical transformation add see [1]- [3].Such an equiform transformation maps points 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 ( ) ( ) ( ) ( ) ( ) . The kinematic corresponding to this transformation group is called similarity kinematic.See [4].Recently, the similarity kinematic geometry has been used in computer vision and reverse engineering of geometric models such as the problem of reconstruction of a computer model from an existing ob-ject which is known (a large number of) data points on the surface of the technical object [5] [6].Abdel-All and Hamdoon studied a cyclic surface in 5   .In this sense, they proved that such surface in 5   is in general contained in a canal hypersurface [7].Solouma ([8]- [10]) studied locally some geometric problems on surfaces obtained by the equiform motion up to the first order.In Minkowski (semi-Euclidean) space, hyperbolas (Lorentzian circles) play role in Euclidean space [11].
In this work we consider the homothetic motion of the hyperbolas(Lorentzian circles) 0 c .Let 0 Σ and Σ be two copies of Euclidean space n  .Under a one-parameter homothetic motion of moving space 0 Σ with respect to fixed space Σ , we consider 0 0 c ⊂ Σ which is moved according homothetic motion.The point paths of the Lorentzian circle generate a cyclic surface X, containing the position of the starting Lorentzian circle.At any moment, the infinitesimal transformations of the motion will map the points of the Lorentzian circle 0 c into the velocity vectors whose end points will form an affine image of 0 c that will be, in general, a Lorentzian circle in the moving space Σ .Both curves are planar and therefore, they span a subspace W of n  , with ( ) dim 5 W ≤ .This is the reason because we restrict our considerations to dimension 5 n = .
Let ( ) x φ be a parametrization of 0 c and ( ) , X t φ the resultant surface by the homothetic motion.We consider a certain position of the moving space, given by 0 t = , and we would like to obtain information about the motion at least during a certain period around 0 t = if we know its characteristics for one instant.Then we restrict our study to the properties of the motion for the limit case 0 t → .A first choice is then approximate ( ) , X t φ by the first derivative of the trajectories.The purpose of this paper is to describe the cyclic surfaces obtained by the homothetic motion of the Lorentzian circle and whose scalar curvature  is constant.
The proof of our results involves explicit computations of the scalar curvature  of the surface ( ) As we shall see, equation constant =  reduces to an expression that can be written as a linear combination of the hyperbolic functions cosh nφ and sinh nφ , n ∈  , namely, ( ) and n E and n F are functions on the variable t.In particular, the coefficients must vanish.The work then is to compute explicitly these coefficients n E and n F by successive manipulations.The authors were able to obtain the re- sults using the symbolic program Mathematica to check their work.The computer was used in each calculation several times, giving understandable expressions of the coefficients n E and n F .This paper is organized as follows: In Section 2, we obtain the expression of the scalar curvature  for the cyclic surfaces obtained by homothetic motion of Lorentzian circle.In successive Sections 3 and 4, we distinguish the cases 0 =  and 0 ≠  , respectively.Finally, in Section 5 explicit examples of surfaces with 0 =  and 0 ≠  are given.
Under a one-parameter homothetic motion of 0 c in the moving space 0 Σ with respect to fixed space Σ .
The position of a point ( ) 0 x φ ∈ Σ at "time" t may be represented in the fixed system as where ( ) describes the position of the origin of 0 Σ at the time t, ≤ is a semi orthogonal matrix and ( ) s t provides the scaling factor of the moving system.For varying t and fixed ( ) x φ , ( ) , X t φ gives a parametric representation of the path (or trajectory) of ( ) x φ .Moreover we assume that all involved functions are of class 1 C .Using the Taylor's expansion up to the first order, the representation of the cyclic surface is where ( ) ⋅ denotes the differentiation with respect to t.As homothetic motion has an invariant point, we can assume without loss of generality that the moving frame 0 Σ and the fixed frame Σ coincide at the zero position 0 t = .Then we have ( ) ( ) ( ) 0 , 0 1 and 0 0.
where ( ) ( ) is a semi skew-symmetric matrix.In this paper all values of , i s b and their derivatives are computed at 0 t = and for simplicity, we write s′ and i b′ instead of ( ) ( ) In these frames, the representation of ( ) , X t φ is given by ( ) 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 + ∈  .We now compute the scalar curvature of the cyclic surface ( ) , X t φ .The tangent vectors to the parametric curves of ( ) A straightforward computation leads to the coefficients of the first fundamental form defined by The scalar product in the above equation in Lorentzian metric.According to the inner product this equation tends to 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 i j k are indices that take the value 1 or 2 and ( ) lm g is the inverse matrix of ( ) ij g .From here, the scalar curvature of ( ) , , .
Although the explicit computation of the scalar curvature  can be obtained, for example, by using the Mathematica programme, its expression is some cumbersome.However, the key in our proofs lies that one can write  as The assumption of the constancy of the scalar curvature  implies that (7) converts into Equation ( 8) means that if we write it as a linear combination of the functions { } cosh ,sinh n n φ φ namely, ( ) , the corresponding coefficients must vanish.From here, we will be able to describe all cyclic surfaces with constant scalar curvature obtained by the homothetic motion of the Lorentzian circle 0 c .As we will see, it is not necessary to give the (long) expression of  but only the coefficients of higher order for the hyperbolic functions.
We distinguish the cases 0 =  and 0 ≠  .

Cyclic Surfaces with  = 0
In this section we assume that 0 =  on the surface ( ) , X t φ .From (7), we have We distinguish different cases that fill all possible cases (Note that we have all solutions by using the symbolic program Mathematica under the condition 0 s′ ≠ ).

Case ′ ′
2) Assume 1 0 b′ = and 2 0 b′ ≠ , then 0 =  on the surface if and only if the following conditions s means the equation ( 8) hold (i.e., ( ) From expression (6), we have the two conditions , X t φ be a cyclic surfaces obtained by the homothetic motion of Lorentzian circle c 0 and given by (3) under condition (4).Assume

Cyclic Surfaces with  ≠ 0
In this section we assume that the scalar curvature  of the cyclic surface ( ) , X t φ obtained by the homo- thetic motion of Lorentzian circle 0 c and given by (3) under condition ( 4) is a non-zero constant.The identity (8) writes then as Following the same scheme as in the case 0 =  studied in Section 3, we begin to compute the coefficients , we conclude that   From (1), ( 2) and (3) we have under the following conditions ) )   , implies that 2 0 b′ = which gives a contra- diction also.
3) CASE 1 2 0 b b ′ ′ ≠ .The computations of ( ) As conclusion of the above reasoning, we conclude the following theorem.

1 2 3 4 5 ,
, , , t b b b b b ′ ′ ′ ′ ′ .The latter ellipse reduce to a Lorentzian circle subject to the following conditions be a cyclic surfaces obtained by the homothetic motion of Lorentzian circle c 0 and given by (3) under condition (4then circles generating the cyclic surfaces are coaxial.
be a cyclic surfaces obtained by the homothetic motion of Lorentzian circle c 0 and given by (3) under condition(4 1 2 0 b b ′ ′ ≠ , then 0 = on the surface if and only if the following conditions hold: 1 the same result as in the above case. the same result as in cases from (1) and (2). conditions be a cyclic surfaces obtained by the homothetic motion of Lorentzian circle 0 c and given by (3) under condition (4).if and only if the following conditions hold:

.
Examples of a Cyclic Surfaces with  = 0 and  ≠ 0In this section, we construct two examples of a cyclic surfaces

Figure 1 ,
we display a piece of ( ) , X t φ of Example 1 in axonometric view- point ( ) , Y t φ .For this, the unit vectors

Figure 1 .
Figure 1.In (a), we have a piece of a cyclic surface foliated by a Lorentzian circle in axonometric view ( ) , Y t φ with zero Then

Figure 2 .
Figure 2. In (a), we have a piece of a cyclic surface foliated by a Lorentzian circle in axonometric view ( ) , Y t φ with non- . Let now the semi orthogonal matrix