On the Geometry of Curves in Minkowski 3-Space and Its Foldings

We will introduce a new connection between some transformations and some aspects of differential geometry of some curves in Minkowski space. The concept of folding, retractions and contraction on some curves in Minkowski space will be characterized by using some aspects of differential geometry. Types of the deformation retracts of some curves in Minkowski 3-space are obtained. The relations between the foldings and the deformation retracts of some curves are deduced. The connections between some transformations and time like, space like, light like of some curves in Minkowski 3-space are also presented.


Introduction and Definitions
. As is well known, the theory of deformation retract is always one of the interesting topics in Euclidian and Non-Euclidian space and it has been investigated from the various viewpoints by many branches of topology and differential geometry El-Ahmady [1][2][3].
Minkowski space is originally from the relativity in physics.In fact, a time like curve corresponds to the path of an observer moving at less than the speed of light, a light like curve corresponds to moving at the speed of light and a space like curve moving faster than light El-Ahmady [4,5].
The Minkowski 3-space is the Euclidean 3-space provided with the standard flat metric given by El-Ahmady [5].
Most folding problems are attractive from a pure mathematical standpoint, for the beauty of the problems themselves.The folding problems have close connections to important industrial applications.Linkage folding has applications in robotics and hydraulic tube bending.Paper folding has application in sheet-metal bending, packaging, and air-bag folding El-Ahmady [6].Following the great Soviet geometer El-Ahmady [5], also, used folding to solve difficult problems related to shell structures in civil engineering and aero space design, namely buckling instability El-Ahmady [7].Isometric folding between two Riemannian manifold may be characterized as maps that send piecewise geodesic segments to a piecewise geodesic segments of the same length [8,9].For a topological folding the maps do not preserves lengths El-Ahmady [8][9][10] i.e.A map : M N   , where M and N are C  -Riemannian manifolds of dimension m, n respectively is said to be an isometric folding of M into N, iff 2 g dx dx dx     , where (x 1 , x 2 , x 3 ) is a rectangular coordinate system of .Since g is an indefinite metric, recall that a vector 1 can have one of three Lorentzian causal characters, it can be space like if , time like if and light like if and .Similarly, an arbitrary curve  does not preserve length, then is a topological folding El-Ahmady [10][11][12].
A subset A of a topological space X is called a retract of X if there exists a continuous map such that , , where A is closed and X is open.El-Ahmady [7][8][9][10][11][12] and Gregory [13].This can be restated as follows.If is the inclusion map, then is a map such that A Miles [14] and Martin [15].If, in addition x ri  id , we call r a deformation retract and A a deformation retract of X Jeffrey [16] and John [17].

Main Result
Let   s  be a curve in the space-time , parameterized by arc length function 3 1 E s Lopez [18] and Formiga [19].Then for the unit speed curve   with non-null frame vectors the following Frenet equations are given in We write following subcases.

1) If
s is space-like curve in , then T is a space-like vector.Thus, we distinguish according to N.
. And T, N and B are mutually orthogonal vectors satisfying equations, , .

  g T T N
 , , is time-like vector, then  


. And T, N and B are mutually orthogonal vectors satisfying equations  , , And T, N and B are mutually orthogonal vectors satisfying equations, Due to character of   s  , we write following subcases.
, then T is a space-like vector.Thus, we distinguish according to N. is space-like vector, then . And 1 and are mutually orthogonal vectors satisfying equations , , . And 1 and are mutually orthogonal vectors satisfying equations , , And 1 and are mutually orthogonal vectors satisfying equations , , 2) If   s  is time-like curve in , then T is timelike vector.Then , , e e e denote the vec- tors of the canonical coordinate basis.Thus, in these coordinates, From (1) we have Thus, Otherwise the set of vectors   , , T N B would not be linearly independent.Then  must vanish.
Suppose that 0 Let us conveniently choose our coordinate system in such a way that .Now, since T is orthogonal to B we must have , which means that parametrized by arc length function s, with non-vanishing curvature, lies in a hyperplane if and only if the torsion vanishes identically.
would not be linearly independent, then the curve is not lies in a hyper-plane.
Corollary 2. Under the folding, parameterized by arc length function s, then the curve is not lies in a hyperplane.
, , , e e e e de- note the vectors of the canonical coordinate basis.Thus, in these coordinates, However,   , respectively.We now ob- serve, by using the Frenet equations at (1), that

T s N s N s B s B s s T s s T s T s N s N s N s N s B s B s B s B s s T s T s T s N s N s N s N s B s B s B s B s T s kN s kN s N s N s B s B B s s N s N s T s kN KT s KT s kN s N s N s s s kT
for all s I  .Thus, the above expression is constant, and, since it is zero for , where a is a constant vector.
for all s I  .

T s T s N s N s B s B s B s B s T s T s T s T s
for all s I  .Thus, the above expression is constant and, since it is zero for , B satisfying the same conditions and  be a simple closed hyperplane curve under the folding in with length , and let A be the area of the region bounded by and equality holds if and only if Let O be the center of and take a coordinate system with origin at O and the 1

S
x axis perpendicular to L and L .Parameterize , so that it is positively oriented and the tangency points of L and are L 0 s  and 1 s s  , respectively.We can assume that the equation of is x y s lr We now notice the fact that the geometric mean of two positive numbers is smaller than or equal to their arithmetic mean, and equality holds if and only if they are equal.It follows that Therefore,  4).Then equality must hold everywhere in Equations ( 5) and (6).From the equality in Equation ( 6) it follows that 2 π A r  .Thus, 2πr l  and does not depend on the choice of r the direction of L .Furthermore, equ implies that ality in Equation ( 5) . thus, x y r x ry x y

Since
does not depend on the choice of the direction of L , we can interchange x and in the last relation and obtain . Thus,  be a simple closed a hyperplane curve under the deformation retract in with length , and let The linear independence of the basis vectors implies that i i p e , related to the for- mer by (7).Then Equations ( 7), (9) give 1 , From which, since the basis vectors   i e are linearly independent, it follows that Similarly, substitute in (9) for from (8) to get From which, since the basis vectors  i e  are linearly independent The Equation (11) expresses the new components in terms of the old component, while Equation (10) expresses the old components in terms of the new component.
Theorem 10.Under the retraction, given differentiable This can be written in a more compact form as By assumption, are differentiable functions of the proper parameter s.From the theory of ordinary differential equations, we know that if we are given a set of initial conditions then the above system admits a unique solution  (14).Therefore, we conclude that u of I.The velocity of A curve  is said to be regular if tively, space like, time like or light like respectively.A curve in Lorentzian space L n is a smooth map , : n I L    where I is the open interval in the real line .The interval has a coordinate system consisting of the identity for any piecewise geodesic path : geodesic and of the same length as  .If


be a curve in the space-time , parameterized by arc length function 4 1 E s .Then for the unit speed curve   s  with non-null frame vectors the fol-lowing Frenet equations are given in

4 1 E 1 : 1 B
Case If N is space-like vector, then can have two causal characters.Case 1.1:

3 1 E 2 .
Theorem Under the retraction, if the curve is a lightlike curve length function s, where then the curve is not lies in a hyperplane.
canonical coordinate basis.Thus, in these coordinates,

3 1 E 3 .
Theorem Under the retraction, a spacelike curve and a timelike curve non-vanishing curvature, lies in ahyperplane if and only if the second torsion vanishes identically.Proof.Let us start with the necessary condition.Suppose the curve

2 BTheorem 4 .
is a constant vector.Let us conveniently choose our coordinate system in such a way that .Now, since Given differentiable functions such that s is the arc length, there exists a regular parameterized spacelike curve under the folding with the spacelike vector N, -time parameterized by arc length function s, where and with non-vanishing curvature, is lies in a hyperplane if the first and second torsions vanishes identically.

Corollary 4 . 1 E 5 .
Under the folding, a spacelike curve and a by arc length function s, with non-vanishing curvature, lies in a hyperplane if and only if the second torsion vanishes identically.4 Corollary Under the folding, parameterized by arclength function s, with non-vanishing curvature, is lies in a hyperplane if the first and second torsions vanishes identically.

Corollary 6 .Theorem 5 . 2 B
Given differentiable functions   0  such that s is the arc length, there exists a regular parameterized spacelike curve under the contraction with the spacelike vector , Given differentiable functions   k s 0  , such that s is the arc length, there exists a regular parameterized spacelike curve under the folding with the spacelike vectors and 1 s are the Frenet-Serret formulas of respectively.We now observe, by using the Frenet equations at(3), that

Theorem 6 .
where a is a constant vector.Since     Given differentiable functions   0 k s  , s is the arc length, there exists a regular parameterized spacelike curve under the deformation retract with the spacelike vectors and 1 , E and be two parallel lines which do not meet the closed curve E together until they first meet    .


, L and L , so that the curve is entirely contained in the strip bounded by L and L .Consider a circle which is tangent to both L and 1 S L and does not meet that equality holds in Equation (


called the kroneckel delta, takes the value 1 if and is otherwise zero.i  j Theorem 9.The components of a vector  in where defined relative to the basis 1 a change of basis will induce a change of components.Proof.The law of transformation for the components of the vector  will now be found when the basis is change from   i e to   i e  according the Equation (7).If the vector  has components i  relative to the basis   i e , it is convenient to write i   for it's components relative to the new basis   ) and (13) we obtain a system of first-order differential equations for the elements of  