On differential analysis of spacelike flows on normal congruence of surfaces

1.   Introduction

In general, differential geometry of surfaces examines the geometric structure of smooth surfaces. In this context, there are many studies on differential geometry of surfaces in different spaces such as Euclidean and non-Euclidean spaces. In many of these studies, the arc length of the curves on the surface is used to obtain the distance over the surfaces. For this reason, curves on the surface have an important role in the field of differential geometry. Investigations of curves on the surface, not only their geometric properties but also their physical structures are considered. From this point of view, it is the most preferred way of examining the local differential geometric structure of the curve. In many studies dealing with differential geometric properties of curves, some methods and tools of differential calculus are used. This review makes use of the well-known Frenet-Serret frame $\{\overrightarrow{t}, \overrightarrow{n}, \overrightarrow{b}\}$. Considering that $\sigma = \sigma(s, n, b)$ is a given curve in Euclidean space, where $s,$ $n$ and $b$ are the distance along $s-lines$, $n-lines$ and $b-lines$ respectively. The main object is to obtain the system by the directional derivatives of Frenet-Serret frame ^[1]. The quantities, the normal deformations of the vector-tube in the directions $\overrightarrow{n}$ and $\overrightarrow{b},$

are firstly introduced in ^[2], respectively.

Since it has many physical applications, Lorentzian geometry is the most studied geometry among non-Euclidean geometries ^{[3,4,5,6,7,8]}. Also, this geometry is a very common research area of physical problems on integrable systems, soliton theory, fluid dynamics, field theories, etc. ^[1,9,10,11]. Since Lorentz-Minkowski spacetime was extended to a curved spacetime by A. Einstein in order to model nonzero gravitational fields, this geometry has been the mathematical theory which is used by general relativity.

In this study, it is aimed to examine spacelike curve flow on Lorentzian space from a different perspective. The three dimensional real vector space equipped with Lorentzian metric

is named after Lorentzian space and denoted by $\mathbb{E}_{1}^{3}.$ In second section, the three dimensional vector field and the differential geometric aspects of curvature and torsion of vector lines are investigated by means of anholomonic coordinates. We describe Frenet-Serret frame $\{\overrightarrow {t}, \overrightarrow{n}, \overrightarrow{b}\}$ of a given spacelike space curve in $\mathbb{E}_{1}^{3}$ in terms of anholonomic coordinates which includes eight parameters, related by three partial differential equations. It is proved that the $\operatorname{curl}$ of tangent vector field has no component in the direction of principal normal vector field. Thus, there exists a surface which contains both $s-lines$ and $b-lines.$ For this reason, the expression of this normal congruence is also discussed in the last section. Then, intrinsic geometric properties of this normal congruence of surfaces are also given.

2.   Differential analysis of spacelike curve with spacelike normal vector

Suppose that $\sigma = \sigma(s, n, b)$ is a given spacelike curve with spacelike normal vector in three dimensional Lorentzian space. And the distance along $s-lines,$ $n-lines$ and $b-lines$ of the curve $\sigma$ are denoted by $s, n$ and $b$, respectively. The unit spacelike tangent vector of $s-lines$ and $n-lines$ of the curve $\sigma$ are given by

respectively. Then the unit timelike tangent vector of $b-lines$ is given by ^[12,13]

A three-dimensional vector field can be considered in terms of anholonomic coordinates which includes eight parameters, related by three partial differential equations ^[14].

Theorem 1. Suppose that $\sigma = \sigma(s, n, b)$ is a given spacelike curve Lorentzianspace. Directional derivatives of the unit vectors $\{\overrightarrow{t}, \overrightarrow{n}, \overrightarrow{b}\}$ are given as follows :

i)

ii)

iii)

The normal deformations of the vector-tube in the directions $\overrightarrow{n}$ and $\overrightarrow{b}$ are given as

and abnormality of $\overrightarrow{n}$ and $\overrightarrow{b}$ are statedas

respectively. Curvature and torsion function of the unit speed spacelike curve$\sigma = \sigma(s, n, b)$ are denoted by $\kappa = \kappa(s, n, b)$ and $\tau = \tau(s, n, b)$ respectively.

Proof. Proof of ⅰ) is clear by Frenet-Serret equation for unit speed spacelike curve. So, the proof of ⅱ) and ⅲ) will be given. It is known that for $i = 1, 2, 3$ there exist smooth functions; $\alpha_{i}$ and $\beta_{i}$ where

We need to find these functionals. Firstly, we get

by our assumptions. Then, we also obtain divergence of Serret-Frenet frame fields as follows

and

Therefore, we get

On the other hand, we also obtain

and

Therefore, we get

Thus, we obtain

Finally, if we substitute obtained values of the smooth functions; $\alpha _{i}$ and $\beta_{i}$ for $i = 1, 2, 3,$ then we get

Corollary 2. The following relation between abnormalities of $\overrightarrow{t}, \overrightarrow{n}$ and $\overrightarrow{b}$ is given by

Above equation proves that important results involving Dupin theorem. This means that all coordinate surfaces intersect along common curvature lines in a triply orthogonal coordinate system.

Remark 3 In vectorial analysis, the vector operator $\operatorname{curl}$ describes the infinitesimal circulation of a vector field in three-dimensional space. So, the $\operatorname{curl}$ of a vector field refers to the idea of how a fluid may rotate. It is seen that

by proof of above theorem. The results of Eq (4) will be discussed in the next sections. Since $\operatorname{curl}\overrightarrow{t}$ does not include any component in the direction of principal normal $\overrightarrow{n},$ then there exists a surface which contains both $s-lines$ and $b-lines.$

Considering the identity $\operatorname{curl}\operatorname{grad}f = 0$, we obtain

By using of above relations, we get

This gives the following relations

In general, the mixed derivatives of order two don't commute. This means that $s,$ $n$ and $b$ represent anholonomic coordinates.

Theorem 4. The intrinsic representations of $\operatorname{grad}\overrightarrow{t}$, $\operatorname{grad}\overrightarrow{n}$ and $\operatorname{grad}\overrightarrow{b}$ give following conditions on geometric parameters$\ \kappa,$ $\tau,$ $\mu_{s},$ $\mu_{b},$ $\operatorname{div}\overrightarrow{n},$ $\operatorname{div}\overrightarrow{b},$ $\xi_{ns},$ $\xi_{bs}$ by thecompatibility of the linear systems

Proof. By using Eq (5), we may write

By compatibility of the linear systems in Eqs (2) and (3), we obtain

Therefore, we get

On the other hand, we have

Therefore, we get

By equality of the coefficient of binormal vector fields, we obtain

By using Eq (5), we have

and

By using coefficient of binormal vector field, we get

Similarly, we have

by Eq (6). By compatibility of the linear systems in Eqs (1) and (3), we obtain

Then, we get

Moreover, we have

This gives the following equation

And we obtain

by coefficients of binormal vector fields. Similarly, we get

Then, we also have

We obtain

by Eq (6). Then, we get

The last three equations can be obtained by similar way to the others.

3.   Normal congruence of surfaces containing $s-lines$ and $b-lines$

There exists a normal congruence of surfaces including the $s-lines$ and $b-lines$ if and only if

Theorem 5. Gauss-Mainardi-Codazzi equations are obtained as follows:

Proof. The proof can be stated by using the compatibility conditions of Eqs (10), (11) and (16) in Theorem 4. In the case of $\mu_{n} = 0,$ these equations reduces to Gauss-Mainardi-Codazzi equations for this normal congruence of surfaces.

Corollary 6. In the case of $\mu_{n} = 0$, since the $s-lines$ and $b-lines$ lie on theconstituent surfaces $\Psi$, this means that $\overrightarrow{n}$ isperpendicular to surface. Thus $\overrightarrow{n}$ parallel to the normalvector field $\overrightarrow{N}$ of the surfaces $\Psi.$

Proof. By definitions of the vector fields $\overrightarrow{t}$ and $\overrightarrow {b},$ we obtain

Then, we have

Thus we obtain

Remark 7. The one-parameter family of surfaces $\Psi,$ which contain the $s-lines$ and $b-lines,$ are timelike surfaces, since $\overrightarrow{n}$ is a spacelike vector field.

Theorem 8. The geodesic curvature of $b-lines$ of the surface $\Psi$ is given as follows

And $s-lines$ are the geodesics of the surface $\Psi.$

Proof. It is known that

by Eq (3) in Theorem 1. Then we get the geodesic curvatures of $b-lines$ as follows

Similarly, we get

by Eq (1) in Theorem 1. So, we get the geodesic curvatures of $s-lines$ as follows

This implies that $s-lines$ are the geodesics of the surface $\Psi.$

Theorem 9. The normal curvatures of $b-lines$ and $s-lines$ of the surface $\Psi$ aregiven as follows

respectively.

Proof. Again by using the equation

we obtain the normal curvatures of $b-lines$ as follows

And so, the normal curvatures of $s-lines$ are obtained as follows

Theorem 10. The geodesic torsion of $b-lines$ and $s-lines$ of the surface $\Psi$ areobtained as follows

respectively.

Proof. We obtain the geodesic torsion of $b-lines$ as follows:

by Eq (3) in Theorem 1. Similarly, the geodesic torsion of $s-lines$ is given as follows

by Eq (1) in Theorem 1.

Theorem 11. Gaussian and mean curvatures of the surface $\Psi$ are given as follows

respectively.

Proof. The first fundamental form of the surface $\Psi$ is obtained as

We get $g_{11} = 1,$ $g_{12} = 0$ and $g_{22} = -1.$ Since normal vector field of the surface $\Psi$ is equal to $\overrightarrow{n},$ we find the second fundamental form as follows

We have $l_{11} = -\kappa,$ $l_{12} = 0$ and $l_{22} = \operatorname{div} \overrightarrow{n}+\kappa.$ Thus, Gaussian curvature $K$ of the surface $\Psi$ is given

And the mean curvature $H$ of the surface $\Psi$ is obtained as

Corollary 12. If the following equality is satisfied

then the surface $\Psi$ is developable.

Remark 13. We know that Gaussian curvature of the surface $\Psi$ is found as

By following equation

we obtain that

If $b-lines$ are geodesics and $s-lines$ are plane curves, then the surface $\Psi$ is developable.

Corollary 14. The surface $\Psi$ is minimal if and only if

Corollary 15. The surface $\Psi$ is a NLS surface if and only if $\kappa = 1.$

Proof. It is easily seen that the equality

is satisfied if and only if $\kappa = 1.$

Example 16. Let the surface $\Psi = \Psi(s, b)$ containing the $s-lines$ and $b-lines$ be given as follows

where

is a unit spacelike vector field. We obtain that

And we also see that

Thus, we get

Furthermore, we obtain that

Then, we also have

which implies that

Gaussian and mean curvature of the surface $\Psi = \Psi(s, b)$ are obtained as follows

respectively.

4.   Conclusions

This study investigates spacelike curves with spacelike normal vector field by means of anholomonic coordinates on Lorentzian space. Frenet-Serret formulas $\{\overrightarrow {t}, \overrightarrow{n}, \overrightarrow{b}\}$ of a given spacelike space curve are described which includes eight parameters related to three partial differential equations. It is proved that the $\operatorname{curl}$ of tangent vector field has no component in the direction of principal normal vector field. This means that there exists a surface which contains both $s-lines$ and $b-lines$. For this reason, the expression of this normal congruence is also discussed with intrinsic geometric properties. Finally, an example is stated to explain the obtained results.

Conflict of interest

All authors declare no conflicts of interest in this paper.