On the Geometry of Curves in Minkowski 3-Space and Its Foldings ()
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
. Let
and
, 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
.