# Integral curves of a linear vector field in semi-Euclidean spaces.

1. Introduction

A vector field is an assignment of a vector to each point in a subset of Euclidean space. As vector fields exist at all points of space, they can be specified along curves and surfaces as well. This is especially important because all laws of electricity and magnetism can be formulated through the behavior of vector fields along curves and surfaces. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point, .

Several authors studied the integral curves by using matrix of a linear vector field in Euclidean and 3-dimensional Lorentz-Minkowski spaces. Karger and Novak  classified the integral curves of a linear vector field in 3-dimensional Euclidean space. They showed that the integral curves of the linear vector field in E3 are helixes, circles or parallel straight lines. Acratalishian  has shown that these results of Karger and Novak are extended to (2n + 1)-dimensional Euclidean space. In , Yaylaci has given a classification of the integral curves of a linear vector field in 3-dimensional Lorentz-Minkowski space.

In this paper, we investigate integral curves of a linear vector field in (2n + 1)-dimensional semi-Euclidean space [E.sup.2n+1.sub.v]. The results can be easily transferred to the Euclidean and Lorentz-Minkowski spaces.

Section 2 is concerned with some basic geometric notations.

Section 3 deals with the classification of the integral curves of a linear vector field on [E.sup.2n+1.sub.v]. This classification depends on whether the number of timelike vectors (equivalently the index of the (2n + 1)- dimensional semi-Euclidean space [E.sup.2n+1.sub.v]) is odd or even as well as the rank of the matrix of the linear vector in [E.sup.2n+1.sub.v]. Also in this section, it is given some examples with respect to special cases of n and v.

2. Preliminaries

We review briefly the basic concepts of the semi-Euclidean space that will be required in this paper.

Definition 2.1. The semi-Euclidean space [E.sup.2n+1.sub.v] is the (2n + 1)-dimensional vector space [E.sup.2n+1] endowed with the pseudo scalar product

<v, w> [v.summation over (i=1)] [v.sub.i][w.sub.i] + [2n+1.summation over (j=v+1)] [v.sub.j][w.sub.j], [for all]v, w [member of] [E.sup.2n+1.sub.v].

We say that the vector v [member of] [E.sup.2n+1.sub.v] is spacelike, lightlike or timelike if (v,v) > 0 or v = 0, (v,v) = 0 and v [not equal to] 0, and (v,v) < 0, respectively, . We define the signature of a vector v as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The norm of a vector v [member of] [E.sup.2n+1.sub.v] is defined by [parallel]v[parallel] = [square root of [absolute value of <v, v>]].

Definition 2.2. A frame field {[u.sub.1], ..., [u.sub.n], ..., [u.sub.2n], [u.sub.2n+1]} in [E.sup.2n+1.sub.v] is called a pseudo orthonormal frame field, , if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 2.3. Let [alpha](s), s being the arclength parameter, be a non-null regular curve in semi-Euclidean space [E.sup.2n+1.sub.v]. The changing of a pseudo orthonormal frame field {[u.sub.1](s),...,[u.sub.n](s),...,[u.sub.2n](s), [u.sub.2n+1](s)} of [E.sup.2n+1.sub.v] along [alpha] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

These equations are called the Frenet-Serret type formulae for [alpha](s), where [[kappa].sub.i](s), 1 [less than or equal to] i [less than or equal to] 2n, is the curvature function of [alpha], [[kappa].sub.i](s) = [[epsilon].sub.i+1] < [u.sup.'.sub.i](s), [u.sub.i+1] (s) >, and [[epsilon].sub.i] is the signature of the vector [u.sub.i], 1 [less than or equal to] i [less than or equal to] 2n, .

Definition 2.4. The signature matrix S in the (2n + 1)-dimensional semi-Euclidean space [E.sup.2n+1.sub.v] is the diagonal matrix whose diagonal entries are [s.sub.1] = [s.sub.2] = ... = [s.sub.v] = -1 and [s.sub.v+1] = [s.sub.v+2] = ... = [s.sub.2n+1] = +1. We call that A is a skew-symmetric matrix in (2n + 1)-dimensional semi-Euclidean space if its transpose satisfies the equation [A.sup.t] = -SAS, .

Let X be a vector field in the [E.sup.2n+1.sub.v]. By an integral curve of the vector field X we understand a curve [alpha] : (a, b) [right arrow] [E.sup.2n+1.sub.v] such that its every tangent vector belongs to the vector field X. If d[alpha]/dt = X([alpha](t)), [for all]t [member of] I, is satisfied, then the curve [alpha] is called an integral curve of the vector field X. A vector field X in [E.sup.2n+1.sub.v] is called linear if [X.sub.v] = SAS(v) for all v [member of] [E.sup.2n+1.sub.v], where A is a linear mapping from [E.sup.2n+1.sub.v] into [E.sup.2n+1.sub.v] and S is the signature matrix.

Definition 2.5. A curve is called a general helix or cylindrical helix if its tangents makes a constant angle with a fixed line in space. A curve is a general helix if and only if the ratio of curvature to torsion is constant, .

For n-dimensional case, we know from Hayden in 1831 that if

[[kappa].sub.n-1]/[[kappa].sub.n-2] = cons., [[kappa].sub.n-3]/[[kappa].sub.n-4] = cons., ... [[kappa].sub.2]/ [[kappa].sub.1] = cons.,

the curve is called as generalized helix where [[kappa].sub.1], [[kappa].sub.2] ,..., [[kappa].sub.n-1] are curvatures of the curve, .

3. Classification of Integral Curves of A Linear Vector Field

In this section, we will classify the integral curves of a linear vector field X in [E.sup.2n+1.sub.v].

Let [E.sup.2n+1.sub.v] be a (2n + 1)-dimensional semi-Euclidean vector space over R and X be a linear vector field in [E.sup.2n+1.sub.v]. The classification will be done in either case in terms of whether the number of timelike vectors is odd and even.

* Case 1. Let the number of timelike vectors be odd: Let X be a linear mapping in [E.sup.2n+1.sub.v] given by a skew- symmetric matrix A with respect to a pseudo- orthonormal basis [empty set]. So, the normal form of this matrix can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[lambda].sub.i] [member of] R - {0}, 1 [less than or equal to] i [less than or equal to] n. So, we have the following theorem.

Theorem 3.1. Let X be a linear vector field in [E.sup.2n+1.sub.v] determined by the matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with respect to a pseudo-orthonormal frame {O; [u.sub.1], [u.sub.2],..., [u.sub.2n+1]}, whose A is the normal formed skew-symmetric matrix and C is a (2n + 1) x 1 column matrix such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then the integral curves of X have the following properties:

i) If the rank of the matrix [AC] is equal to 2n + 1, then the integral curves are the generalized helixes,

ii) If the rank of the matrix [AC] is equal to 2k, 1 [less than or equal to] k [less than or equal to] n, then the integral curves are Lorentzian circles in parallel planes whose centres lie on a same straight line perpendicular to those planes,

iii) If the rank of the matrix [AC] is equal to 2k + 1, 1 [less than or equal to] k [less than or equal to] n, then the integral curves are the generalized helixes,

iv) If the rank of the matrix [AC] is equal to 1, then the integral curves are the parallel straight lines.

Proof. i) Let X be a linear vector field in [E.sup.2n+1.sub.v]. Then the value of the linear vector field X for all points P = ([x.sub.1],[x.sub.2],...,[x.sub.2n+1]) [member of] [E.sup.2n+1.sub.v] can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, for the sake of simplicity, we choose [[lambda].sub.i] = 1, 1 [less than or equal to] i [less than or equal to] n. If [alpha]: I [subset] R [right arrow] [E.sup.2n+1.sub.v] is an integral curve of the linear vector field X, then by the definition of the integral curve we can write

d[alpha]/dt = X ([alpha](t)), [for all]t [member of] I.

So, the integral curve with the initial condition [alpha](t) = P = ([x.sub.1],[x.sub.2],...,[x.sub.2n+1]) is a solution of the differential equation

d[alpha](t)/dt = X (P).

Hence, we get the system of differential equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If we solve this system of differential equations, we get the integral curve of the linear vector field X as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 1 [less than or equal to] i [less than or equal to] v-1/2 and v+3/2 [less than or equal to] j [less than or equal to] n. Now, we can examine the character of the integral curve. If we take into consideration the derivations of [alpha](t), we get linearly independent vectors [alpha]', [alpha]", [alpha]'", [[alpha].sup.(4)] and [[alpha].sup.(5)]. The other higher order derivations are linear dependent. So, it can be constructed only Frenet quintette on the curve [alpha](t).

Therefore, there exists four curvature functions [k.sub.1], [k.sub.2], [k.sub.3] and [k.sub.4]. Now we show that [alpha](t) is a generalized helixes in [E.sup.2n+1.sub.v]. For this aim, firstly, let us calculate the velocity of [alpha](t). The velocity of the curve is obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Assume that [alpha](t) is a timelike curve, that is <[alpha]'(t), [alpha]'(t)> = -1. Thus, we have a pseudo-orthogonal system by the Gramm-Schmidt method:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and 1 [less than or equal to] i [less than or equal to] v-1/2 and v+3/2 [less than or equal to] j [less than or equal to] n. If we use

[[kappa].sub.i] (s) = [[epsilon].sub.i+1] <[u.sup.'.sub.i](s), [u.sub.i+1](s)>

for the curvature functions [k.sub.i] (s), 1 [less than or equal to] i [less than or equal to] 4, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So, we obtain

[k.sub.1]/[k.sub.2] = const and [k.sub.3]/[k.sub.4] = const.

This means that the curve [alpha](t) is the generalized helix.

ii) Let rank [AC] = 2k, 1 [less than or equal to] k [less than or equal to] n, then:

a) If rank [AC] = 2n, k = n, then the linear first order system of differential equations became

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, the solution of this system is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where 1 [less than or equal to] i [less than or equal to] v-1/2 and v+3/2 [less than or equal to] j [less than or equal to] n. It is easy to show that the curve [alpha](t) is a Lorentzian circle.

b) Let rank [AC] = r, r = 2, 4, ... , 2n - 2. Then, the linear first order system of differential equations became

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we solve this system, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where 1 [less than or equal to] i [less than or equal to] v-1/2 and v+3/2 [less than or equal to] j [less than or equal to] r/2. So, the curve [alpha](t) is a Lorentzian circle.

iii) Let rank [AC] = 2k + 1,1 [less than or equal to] k [less than or equal to] n. In this case,

a) If rank [AC] = 2n + 1, for k = n, then [alpha](t) is the same as the first part of the theorem.

b) If rank [AC] = 2k +1 = r + 1, r = 2, 4,..., 2n - 2, then the linear first order system of differential equations became

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This system of differential equations has the solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where 1 [less than or equal to] i [less than or equal to] v-1/2 and v+3/2 [less than or equal to] j [less than or equal to] r/2. It is easy to show that the curve [alpha](t) is the generalized helix.

iv) If rank [AC] = 1, then [[lambda].sub.i] = 0 which gives us a linear first order system of the differential equations. This system has the solution [alpha](t) which is the parallel straight lines in [E.sup.2n+1.sub.v].

* Case 2. Let the number of timelike vectors be even: In this case, a skew-symmetric matrix A with respect to a pseudo-orthonormal basis [empty set] can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[lambda].sub.i] [member of] R - {0}. So, we have the following theorem.

Theorem 3.2. Let X be a linear vector field in [E.sup.2n+1.sub.v]. Then the integral curves of X have the following properties:

i) If the rank of the matrix [AC] is equal to 2n + 1, then the integral curves are same parametrized circular helixes,

ii) If the rank of the matrix [AC] is equal to 2k, 1 [less than or equal to] k [less than or equal to] n, then the integral curves are Lorentzian circles in parallel planes,

iii) If the rank of the matrix [AC] is equal to 2k + 1, 1 [less than or equal to] k [less than or equal to] n, then the integral curves are circular helixes,

iv) If the rank of the matrix [AC] is equal to 1, then the integral curves are the parallel straight lines.

Proof. i) Let X be a linear vector field in [E.sup.2n+1.sub.v]. Then the value of the linear vector field X for all points P = ([x.sub.1],[x.sub.2],..., [x.sub.2n+1]) [member of] [E.sup.2n+1.sub.v] can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

X(P) = ([[lambda].sub.1][x.sub.2] + [a.sub.1], - [[lambda].sub.1][x.sub.1] + [a.sub.2],...,[[lambda].sub.n][x.sub.2n] + [a.sub.2n-1], - [[lambda].sub.n][x.sub.2n-1] + [a.sub.2n], [a.sub.2n+1]).

If we choose [[lambda].sub.i] = 1, 1 [less than or equal to] i [less than or equal to] n, and use the definition of the integral curve, we get the system of differential equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we solve the differential equation [d[alpha].sub.2n+1]/dt = c, we get [[alpha].sub.2n+1] = ct + d. The other 2n equations can be solved in pairs. So, the general solution of these differential equations are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, we can examine the character of the integral curve . If we take into consideration the derivations of [alpha](t), we get linearly independent vectors [alpha]', [alpha]" and [alpha]"'. The other higher order derivations are linear dependent. Therefore, there are two curvature functions [k.sub.1] and [k.sub.2]. Now we show that [alpha](t) is a circular helix in [E.sup.2n+1.sub.v]. Let us calculate the velocity of [alpha](t). The velocity of the curve is obtained as

<[alpha]'(t), [alpha]'(t)) = - [v/2.summation over (i=1)]([A.sup.2.sub.i] + [B.sup.2.sub.i]) + [n.summation over (j=v+2/2)] ([A.sup.2.sub.j] + [B.sup.2.sub.j]) + [c.sup.2].

Assume that [alpha](t) is a timelike curve, that is <[alpha]'(t), [alpha]'(t)) = -1. Thus, we have an orthogonal system by the Gramm-Schmidt method:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[gamma] = [v/2.summation over (i=1)] ([A.sup.2.sub.i] + [B.sup.2.sub.i]) - [n.summation over (j=v+2/2)] ([A.sup.2.sub.j] + [B.sup.2.sub.j]).

So, we have

[k.sub.1] = -[[member of].sub.2][gamma], [k.sub.2] = [[member of].sub.3][gamma]([gamma] - 1) and [k.sub.1]/[k.sub.2] = [[member of].sub.2]/[[member of].sub.3] 1/(1-[gamma]).

This means that the curve [alpha](t) is the helix. If the curve [alpha](t) is translated by T = (-[a.sub.1], [a.sub.2], -[a.sub.3], [a.sub.4],..., -[a.sub.2n-1], [a.sub.2n], 0) we get

[[alpha].sup.2.sub.1] + [[alpha].sup.2.sub.2] + ... + [[alpha].sup.2.sub.2n] = -[gamma] = const.

So, we can say that [alpha] (t) is a circular helix.

ii) Let rank [AC] = 2k, 1 [less than or equal to] k [less than or equal to] n, then:

a) If rank [AC] = 2n, k = n, then the differential equations system became

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This system has the solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is easy to show that the curve [alpha](t) is a Lorentzian circle.

b) Let rank [AC] = r, r = 2, 4, ... , 2n - 2. Then, the differential equations system became

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we solve this system, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The curve a(t) is a Lorentzian circle.

iii) Let rank [AC] = 2k + 1, 1 [less than or equal to] k [less than or equal to] n. In this case,

a) If rank [AC] = 2n +1, for k = n, then [alpha](t) is the same as the first part of the theorem.

b) If rank [AC] = 2k + 1 = r + 1, r = 2, 4,..., 2n - 2, then the differential equations system became

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This system has the solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is easy to show that the curve [alpha](t) is the circular helix.

iv) If rank [AC] = 1, then [[lambda].sub.i] = 0 which gives us a system of the differential equations. This system of differential equations has the solution [alpha](t) which are parallel straight lines in [E.sup.2n+1.sub.v].

Now, let us give two examples for some special cases.

Example 3.3. The case v = 0 and n =1. Consider the vector field X : [E.sup.3] [right arrow] [E.sup.3] defined by X(x, y, z) = (y, -x, 0). We can take skew-symmetric matrix A and 3 x 1 column matrix C as below

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then the value of the vector field X for all points P = (x,y,z) [member of] [E.sup.3] can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If [alpha] : I [subset] R [right arrow] [E.sup.3] is an integral curve of the vector field X, then by the definition of the integral curve, we get the system of differential equations as

dx/dt = y + a; dy/dt = -x + b, dz/dt = 0

So, the solution of this system is

[alpha](t) = (B sin t - D cos t + b, B cos t + D sin t - a, d),

where B and D are any constants. The curve [alpha](t) is a circle in [E.sup.3]. In the following figure, the vector field X and its integral curve is demonstrated.

[ILLUSTRATION OMITTED]

Example 3.4. The case v = 1 and n = 1. Consider the vector field X : [E.sup.3.sub.1] [right arrow] [E.sup.3.sub.1] defined by X(x,y,z) = (y,x,0). We can take skew- symmetric matrix A and 3 x 1 column matrix C as below

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then the value of the vector field X for all points P = (x,y,z) [member of] [E.sup.3.sub.1] can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If [alpha] : I [subset] R [right arrow] [E.sup.3.sub.1] is an integral curve of the vector field X, then by the definition of the integral curve, we get the system of differential equations as

dx/dt = y + a, dy/dt = x + b, dz/dt = 0.

So, the solution of this system is

[alpha](t) = (B sinh t + D cosh t - b, B cosh t + D sinh t + a, d),

where B and D are any constants. The curve a(t) is a Lorentzian circle in [E.sup.3.sub.1]. Figure of vector field X and its integral curve is as follows:

[ILLUSTRATION OMITTED]

TUNAHAN TURHAN AND NIHAT AYYILDIZ

Necmettin Erbakan University, Seydisehir Vocational School 42360 Konya, Turkey

Suleyman Demirel University, Department of Mathematics, 32260 Isparta, Turkey

REFERENCES

 A. Acratalishian, On the Linear Vector Field in [E.sup.2n+1.sub.v], Commun. Fac. Sci. Ank., 39, 21-35, 1989.

 A. Ferrandez, A. Gimenez, and P. Lucas, Null Helices in Lorentzian Space Forms, International Journal of Modern Physics, 16, 4845-4863, 2001.

 A. Ferrandez, A. Gimenez, and P. Lucas, Null Generalized Helices in Lorentz-Minkowski Spaces, J. Phys. A: Math. Gen., 35, 8243-8251, 2002.

 A. Galbis M. Maestre, Vector Analysis Versus Vector Calculus, Springer, London, 375 pp, 2012.

 A. Karger, J. Novak, Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, 422 pp, 1978.

 A. Yucesan, A. C. Coken, N. Ayyildiz, G.S. Manning, On the Relaxed Elastic Line on Pseudo- Hypersurfaces in Pseudo-Euclidean Spaces, Applied Mathematics and Computation (AMC), 155(2), 353-372, 2004.

 B. O'Neill, Elementary Differential Geometry Second Edition, Academic Press, Inc., New York, 482 pp, 1997.

 B. O'Neill, Semi-Riemann Geometry: with Applictions to Relativity. Academic Pres, New York, 469 pp, 1983.

 K. L. Duggal, A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers, The Nedherlands, 303 pp, 1996.

 T. Yaylaci, Linear Vector Fields and Applications, MSc thesis, Ankara University, The Institute of Science, Ankara, 49 pp, 2006.