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

Abstract

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.

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El-Ahmady, A. and Al-Hesiny, E. (2013) On the Geometry of Curves in Minkowski 3-Space and Its Foldings. Applied Mathematics, 4, 746-752. doi: 10.4236/am.2013.45103.

1. 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 [1-3].

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, where (x1, x2, x3) is a rectangular coordinate system of. Since g is an indefinite metric, recall that a vector can have one of three Lorentzian causal characters, it can be space like if 0 or, time like if and light like if and. Similarly, an arbitrary curve in can locally be space like, time like or light like, if all of its velocity vectors are respectively, space like, time like or light like respectively. A curve in Lorentzian space Ln is a smooth map where I is the open interval in the real line. The interval has a coordinate system consisting of the identity map u of I. The velocity of at is.

A curve is said to be regular if does not vanish for all. is space like if its velocity vectors are space like for all, similarly for timelike and null. If is a null curve, we can reparametrize it such that and 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 [8-10] i.e. A map, where M and N are -Riemannian manifolds of dimension m, n respectively is said to be an isometric folding of M into N, iff for any piecewise geodesic path, the induced path is a piecewise geodesic and of the same length as. If does not preserve length, then is a topological folding El-Ahmady [10-12].

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 [7-12] and Gregory [13]. This can be restated as follows. If is the inclusion map, then is a map such that Miles [14] and Martin [15]. If, in addition, we call r a deformation retract and A a deformation retract of X Jeffrey [16] and John [17].

2. Main Result

Let be a curve in the space-time, parameterized by arc length function Lopez [18] and Formiga [19]. Then for the unit speed curve with non-null frame vectors the following Frenet equations are given in

(1)

We write following subcases.

1) If is space-like curve in, then T is a space-like vector. Thus, we distinguish according to N.

Case 1: If N is space-like vector, then B is time-like vector, then read,. And T, N and B are mutually orthogonal vectors satisfying equations, ,.

Case 2: is time-like vector, then read.

And T, N and B are mutually orthogonal vectors satisfying equations

.

2) If is time-like curve in, then T is timelike vector. Then read. And T, N and B are mutually orthogonal vectors satisfying equations,

.

3) If is light-like curve in then the following Frenet equations are given in

(2)

Also, if be a curve in the space-time, parameterized by arc length function . Then for the unit speed curve with non-null frame vectors the following Frenet equations are given in

(3)

Due to character of, we write following subcases.

1) If is space-like curve in, then T is a space-like vector. Thus, we distinguish according to N.

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

Case 1.1: is space-like vector, then read,. And and are mutually orthogonal vectors satisfying equations

Case 1.2: is time-like vector, then read. And and are mutually orthogonal vectors satisfying equations

Case 2: N is time-like vector, then read. And and are mutually orthogonal vectors satisfying equations

.

2) If is time-like curve in, then T is timelike vector. Then read,. And and are mutually orthogonal vectors satisfying equations

Hence, we can formulate the following theorems.

Theorem 1. Under the retraction, a spacelike curve and a timelike curve in the space-time parameterized by arc length function s, where and with non-vanishing curvature, lies in ahyperplane if and only if the torsion vanishes identically.

Proof. Suppose the curve lies in a hyperplane. Let us assume that we can bring to lie in the -hyperplane. Then the parametric equations of are of the form

. Let denote the vectors of the canonical coordinate basis. Thus, in these coordinates,

and

.

From (1) we have. Since then has no components in the -direction, i.e.

.

Thus, hence from the equation  we conclude that, where

However, cannot be zero. Otherwise the set of vectors would not be linearly independent. Then must vanish.

Suppose that. Since, then B is a constant vector. 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 lies in the hyperplane.

Corollary 1. Under the folding, , a spacelike curve and a timelike curve in the space-time parametrized by arc length function s, with non-vanishing curvature, lies in a hyperplane if and only if the torsion vanishes identically.

Theorem 2. Under the retraction, if the curve is a lightlike curve in the space-time parameterized by arc length function s, where then the curve is not lies in a hyperplane.

Proof. Suppose the curve lies in a hyperplane. Let us assume that we can bring to lie in the -hyperplane. Then the parametric equations of are of the form

Let denote the vectors of the canonical coordinate basis. Thus, in these coordinates,

and

.

From (2) we have. Then has no components in the -direction, i.e. Thus, , hence from the equation, we conclude, where

. But cannot be zero because the set of vectors would not be linearly independent, then the curve is not lies in a hyperplane.

Corollary 2. Under the folding, if the curve is a lightlike curve in the spacetime parameterized by arc length function s, then the curve is not lies in a hyperplane.

Theorem 3. Under the retraction, a spacelike curve and a timelike curve in the space-time parameterized by arc length function s, where and with 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 lies in a hyperplane. Let us assume that we can bring to lie in the

-hyperplane. Then the parametric equations of are of the form

. Let denote the vectors of the canonical coordinate basis. Thus, in these coordinates,

and

.

From (3) we have. Since then N has no components in the -direction, i.e.

Thus, , hence from the equation and we conclude that

, where. If, then also must vanish, for in this case is chosen to be constant. If, then, hence Also,

and the third Serret-Frenet equation we are led to conclude that, where

.

However, cannot be zero. Otherwise the set of vectors would not be linearly independent. Then must vanish.

Suppose that. Since, then

is a constant vector. Let us conveniently choose our coordinate system in such a way that. Now, since is orthogonal to we must have, which means that lies in the hyperplane.

Corollary 3. Under the contraction, a spacelike curve and a timelike curve in the space-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. Under the folding, a spacelike curve and a timelike curve in the space-time parameterized by arc length function s, with non-vanishing curvature, lies in a hyperplane if and only if the second torsion vanishes identically.

Corollary 5. Under the folding, , a spacelike curve and a timelike curve in the space-time parameterized by arclength function s, with non-vanishing curvature, is lies in a hyperplane if the first and second torsions vanishes identically.

Theorem 4. Given differentiable functions and such that s is the arc length, there exists a regular parameterized spacelike curve under the folding with the spacelike vector N, , in the space-time. Also, is the curvature and is the torsion of. Moreover, any other spacelike curve with the spacelike vector satisfying the same conditions and at then

and the Frenet trihedrons of and is identically.

Proof. Now, assume that two curves and satisfy the conditions and,. Let T0, N0, B0 and, , be the Frenet trihedrons of and at, respectively. Since then, and, where, , and, , are the Frenet trihedrons of and, respectively. We now observe, by using the Frenet equations at (1), that

for all. Thus, the above expression is constant, and, since it is zero for, it is identically zero. It follows that, , for all.

Since, we obtain.

Thus, where a is a constant vector. Since, we have; hence, for all.

Corollary 6. Given differentiable functions and, such that s is the arc length, there exists a regular parameterized spacelike curve under the contraction with the spacelike vector, , in the space-time. Also, is the curvature and is the torsion of. Moreover, any other spacelike curve with the spacelike vector satisfying the same conditions and at then and the Frenet trihedrons of and is identically.

Theorem 5. Given differentiable functions, and such that s is the arc length, there exists a regular parameterized spacelike curve under the folding with the spacelike vectors and, in the space-time. Also, is the curvature, is the first torsion, and is the second torsion of. Moreover, any other spacelike curve with the spacelike vectors and satisfying the same conditions and where then and the FrenetSerret formulas of and is identically.

Proof. Now assume that two curves and satisfy the conditions,  and,. Let, , , and, , , be the Frenet-Serret formulas of and at, respectively. Since then, , , and. And, , ,

and, , , are the Frenet-Serret formulas of and , respectively. We now observe, by using the Frenet equations at (3), that

for all. Thus, the above expression is constant and, since it is zero for, it is identically zero. It follows that

for all. Since. We obtain. Thus, where

is a constant vector. Since, we have; hence, for all.

Theorem 6. Given differentiable functions, and such that s is the arc length, there exists a regular parameterized spacelike curve under the deformation retract with the spacelike vectors and, , in the space-time. Also, is the curvature, is the first torsion, and is the second torsion of. Moreover, any other spacelike curve with the spacelike vectors and satisfying the same conditions and

where then and the Frenet-Serret formulas of and is identically.

Theorem 7. Let be a simple closed hyperplane curve under the folding in with length, and let A be the area of the region bounded by. Then

(4)

and equality holds if and only if  is a circle.

Proof. Let E and be two parallel lines which do not meet the closed curve and moves them together until they first meet We thus obtain two parallel tangent lines to, L and, so that the curve is entirely contained in the strip bounded by L and. Consider a circle which is tangent to both L and and does not meet. Let O be the center of and take a coordinate system with origin at O and the axis perpendicular to L and. Parameterize by arc length, since simple closed a hyperplane curve, then , so that it is positively oriented and the tangency points of L and are and, respectively. We can assume that the equation of is, where 2r is the distance between L and. Denoting by the area bounded by, we have

thus

(5)

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

(6)

Therefore,

Now, assume that equality holds in Equation (4). Then equality must hold everywhere in Equations (5) and (6). From the equality in Equation (6) it follows that. Thus, and does not depend on the choice of the direction of. Furthermore, equality in Equation (5) implies that, or; that is,

Since does not depend on the choice of the direction of, we can interchange and in the last relation and obtain. Thus,

and is a circle.

Theorem 8. Let be a simple closed a hyperplane curve under the deformation retract in with length, and let be the area of the region bounded by. Then , and equality holds if and only if is a circle.

Corollary 7. Let be a simple closed a hyperplane curve under the contraction in with length, and let be the area of the region bounded by. Then, and equality holds if and only if is a circle.

Any n vectors forming a basis for will be written, i.e. the basis will be written. Relative to a basis, any vector in is uniquely expressible in the form

The numbers, where, are called the components of relative to the basis. If are the components of another vector relative to the same basis. Let the vectors form another basis of. Since each vector is uniquely expressible as a linear combination of the vectors, we have

(7)

where is an matrix, non-singular because the vectors, are linearly independent. Similarly, the vector is uniquely expressible in the form

(8)

where is a non-singular matrix. Then

The linear independence of the basis vectors implies that, where called the kroneckel delta, takes the value 1 if and is otherwise zero.

Theorem 9. The components of a vector in where defined relative to the basis, and 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 to according the Equation (7). If the vector has components relative to the basis, it is convenient to write for it’s components relative to the new basis, related to the former by (7). Then

(9)

Equations (7), (9) give

From which, since the basis vectors are linearly independent, it follows that

(10)

Similarly, substitute in (9) for from (8) to get

From which, since the basis vectors are linearly independent

(11)

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 functions, and, there exists a retraction of regular parameterized timelike curve, , such that is the curvature, and are, respectively, the first and second torsion of. Any other curve satisfying the same conditions, different from By a Poincaré transformation.

Proof. Let us assume that two timelike curves and satisfy the conditions, and, with, where is an open interval of, and , and are, respectively, the curvature, first and second torsion of. Letand

, where be the Serret-Frenet tetrads at of and, respectively. Now, the two Serret-Frenet tetrads of and satisfy the equations

and

This can be written in a more compact form as

(12)

where, with and denoting the elements of the Serret-Frenet matrix. Clearly, the two tetrads, are related by an equation of the type

(13)

with the elements of the matrix satisfying the condition

Since we are assuming that

From (12) and (13) we obtain a system of first-order differential equations for the elements of given by

(14)

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

defined in an open interval containing. On the other hand, it is easily seen that is a solution of (14). Therefore, we conclude that

.

Conflicts of Interest

The authors declare no conflicts of interest.