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Advances in Mathematical Physics
Volume 2015, Article ID 905978, 7 pages
http://dx.doi.org/10.1155/2015/905978
Research Article

Geometry of the Solutions of Localized Induction Equation in the Pseudo-Galilean Space

1Department of Mathematics, Faculty of Science, Firat University, 23200 Elazig, Turkey
2Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania
3Department of Mathematics, Faculty of Science and Letters, Namik Kemal University, 59030 Tekirdağ, Turkey

Received 26 January 2015; Accepted 21 May 2015

Academic Editor: Soheil Salahshour

Copyright © 2015 Muhittin Evren Aydin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the surfaces corresponding to solutions of the localized induction equation in the pseudo-Galilean space . We classify such surfaces with null curvature and characterize some special curves on these surfaces in .

1. Introduction

The localized induction equation (LIE), also called the vortex filament equation or the Betchov-Da Rios equation, is an idealized model of the evolution of the centerline of a thin vortex tube in a three-dimensional inviscid incompressible fluid. The connection of LIE with the theory of solitons was discovered by Hasimoto proving that the solutions of LIE are related to solutions of the cubic nonlinear Schrödinger equation, which is well known to be an equation with soliton solutions. For details, see [15].

The soliton surface associated with the nonlinear Schrödinger equation is called a Hasimoto surface. The geometric properties of such surfaces are investigated in [4, 6, 7]. By motivating the fact that the study of the Hasimoto surfaces can be interesting in the pseudo-Galilean space which is one of the Cayley-Klein spaces, we are mainly interested in the geometric properties of these surfaces in the pseudo-Galilean space.

Let be a 3-dimensional Riemannian manifold and the Levi-Civita connection with respect to . Note that the cross product of two vector fields on may be defined aswhere is the volume form of the manifold and . Denote by the vortex filament; then LIE or the Betchov-Da Rios filament equation is

On the other hand, the Galilean space which can be defined in three-dimensional projective space is the space of Galilean Relativity. The geometries of Galilean space and pseudo-Galilean space have similarities but, of course, are different. For study of surfaces in the pseudo-Galilean space, we refer to Šipuš and Divjak’s series of works [813].

In classical geometry of surfaces, it makes sense to classify the surfaces having null curvature. In particular, a surface is said to be developable if it has null Gaussian curvature. In this case the surface can be flattened onto a plane without distortion. We remark that cylinders and cones are examples of developable surfaces, but the spheres are not under any metric.

There exist significant applications of the results obtained on the surfaces of null curvature in different fields, for example, in microeconomics. When the graphs of production functions in microeconomics have null Gaussian curvature, one can realize a “good” analysis of isoquants by projections, without losing essential information about their geometry (see [14, 15]). By targeting that, we study the Hasimoto surfaces in the pseudo-Galilean space .

In this paper, we classify the Hasimoto surfaces having null curvature in . Then we analyze the parameter curves on such surfaces from the point of view to be the geodesics, asymptotic lines, and principal curves in .

2. Preliminaries

The pseudo-Galilean space is one of the Cayley-Klein spaces with absolute figure that consists of the ordered triple , where is the absolute plane in the three-dimensional real projective space , is the absolute line in , and is the fixed hyperbolic involution of points of [8, 9, 13].

Homogenous coordinates of can be given in the following way: The absolute plane is given by , the absolute line by , and the hyperbolic involution by , which is equivalent to the requirement for the conic to be the absolute conic. The metric connections in are introduced with respect to the absolute figure. According to the affine coordinates given by , the distance between the points and is defined by (see [16])The pseudo-Galilean scalar product of the vectors and is defined byIn this sense, the pseudo-Galilean norm of a vector is . A vector is called isotropic if ; otherwise it is called nonisotropic. All unit nonisotropic vectors are of the form . The isotropic vector is called spacelike, timelike, and lightlike if ,, and , respectively. The pseudo-Galilean cross product of and on is given bywhere , .

Let be an admissible curve given by in ; that is, it has no isotropic tangent vectors for . Note that an admissible curve has nonzero curvature. Then its curvature and torsion functions are, respectively, defined bywhere and so forth, . If is an admissible curve parameterized by the arc length , it is of the formThe associated Frenet frame field of is the trihedron   such that where . In this sense, the Frenet formulas for the curve are [17, 18]

Let be a surface in the pseudo-Galilean space parameterized byDenote and , and . Then such a surface is admissible if and only if for some . The coefficients of the first fundamental form of areor, in matrix form,where and .

Let us define the function asThus a side tangential vector is defined bywhich can be a spacelike isotropic or a timelike isotropic vector. The unit normal vector field of is given byThe second fundamental form of and its coefficients are defined bywhere . Two types of admissible surfaces can be distinguished: spacelike surfaces having timelike unit normals and timelike ones having spacelike unit normals . The third fundamental form of iswhereThe Gaussian curvature and the mean curvature of are, respectively, defined byA surface in is said to be minimal if its mean curvature vanishes.

3. Hasimoto Surfaces with Null Curvature in

Let and be the open intervals of and open domain of .

A Hasimoto surface is the surface traced out by a curve in as it evolves over time according to this evolution equation:where is smooth and regular mapping such that .

In order to parameterize by arc length the curve for , the parameterization of the surface in may be chosen as follows:where is a smooth function of one variable on . After using the pseudo-Galilean cross product given by (5) and (21), it can be easily seen that the function is a constant function. In this sense, we get the parameterization of a Hasimoto surface in asfor arbitrary constant . Note that the Hasimoto surface given by (23) is always admissible; that is, it has no pseudo-Euclidean tangent planes, if .

Now, let be a Hasimoto surface given by (23) in . Denote the associated Frenet frame field of the curve for by . Then, the derivative of with respect to has the expressionfor some smooth functions on . By (21) and (24), we getMoreover, applying the compatibility condition yields that In similar ways, we deduce The compatibility conditions and give

Summing up, we have the following result.

Lemma 1. Let be a Hasimoto surface given by (23) in . Then the following equations hold:(i) and ;(ii)where is the Frenet frame field of the curve for .

Example 2. Let us consider the spacelike admissible Hasimoto surface in parameterized byIt can be easily seen that We plot the surface as in Figure 1.

Figure 1

Example 3. Take the timelike admissible Hasimoto surface in as It follows that We plot the surface as in Figure 2.

Figure 2

Theorem 4. Let be a Hasimoto surface given by (23) in . Then(i) has null Gaussian curvature in if and only if ;(ii) is a minimal Hasimoto surface in if and only if , where and are, respectively, the curvature and the torsion of the curve for .

Proof. Let be a Hasimoto surface given by (23) in . Then the unit normal vector field of iswhere is the principal normal vector field of the curve for . Thus, we haveFrom (19) and (35), we deduce that which gives the first statement.
Next by substituting (35) into (20), we obtain that is a minimal Hasimoto surface if and only if Thus the proof is completed.

Gheorghe Tzitzeica (1873–1939) introduced a class of curves, nowadays called Tzitzeica curves, and a class of surfaces of the 3-dimensional Euclidean space, called Tzitzeica surface.

A Tzitzeica surface is a spatial surface for which the ratio of its Gaussian curvature and the distance from the origin to the tangent plane at any arbitrary point of the surface satisfy for a constant . This class of surface is of great interest, having important applications both in mathematics and in physics (see [15]). The relation between Tzitzeica curves and surfaces is the following: For a Tzitzeica surface with negative Gaussian curvature, the asymptotic lines are Tzitzeica curves [1921].

It is easy to prove that the tangent plane at an arbitrary point of any Hasimoto surface passes through the origin of . That is why all Hasimoto surfaces having null Gaussian curvature satisfy Tzitzeica condition in .

Therefore the following result can be given without proof.

Theorem 5. Let be a Hasimoto surface given by (23) in . Then it is a Tzitzeica surface if and only if , where and are, respectively, the curvature and the torsion of the curve for .

On the other hand, the third fundamental form of a surface in may be introduced in the analogous way as in Euclidean space. Let be a surface and its unit normal vector in . If is spacelike (timelike) in , then the end points of associated position vectors of lie on a unit spacelike sphere (unit timelike sphere ). The mapping obtained in such a way is called the Gauss mapping or the spherical mapping in . The set of all end points of is called the spherical image of in . In this sense, the third fundamental form is indeed the first fundamental form of the spherical image (cf. [13]). Thus the following result for the Hasimoto surfaces in can be given.

Corollary 6. Let be a Hasimoto surface given by (23) in . Then its third fundamental form is singular; that is, .

Proof. Let us consider the Hasimoto surface given by (23) in . Then from (35) we have as one of the coefficients of the first fundamental form of . This immediately implies from (18) that , which completes the proof.

4. Curves on Hasimoto Surfaces in

There exists a frame field, also called the Darboux frame field, for the curves lying on surfaces apart from the Frenet frame field. For details, see [22, 23]. Let be a curve lying on the surface with unit normal vector field . By taking one can get a new frame field which is the Darboux frame field of with respect to .

On the other hand, the second derivative of the curve on has a component perpendicular to and a component tangent to ; that is, where the dot “” denotes the derivative with respect to the parameter of the curve. The norms and are called the geodesic curvature and the normal curvature of on , respectively. The curve is called geodesic (resp., asymptotic line) if and only if its geodesic curvature (resp., normal curvature ) vanishes.

In our framework, the following results provide some characterizations for the parameter curves of the Hasimoto surfaces to be geodesics and asymptotic lines in .

Theorem 7. Let be a Hasimoto surface given by (23) in . Then(i)the -parameter curves on are geodesics of ,(ii)the -parameter curves on are geodesics of if and only if vanishes, where and are, respectively, the curvature and the torsion of the curve for .

Proof. Let us assume that is a Hasimoto surface given by (23) in . Then the geodesic curvature of the -parameter curves on is the tangential component of ; that is,From (21), we derive that the geodesic curvature of the -parameter curves vanishes. This gives first statement.
Similarly, the geodesic curvature of the -parameter curves on is given byBy considering Lemma 1 into (40), we deduce that is identically zero if and only if vanishes, which implies statement (ii).

Theorem 8. Let be a Hasimoto surface given by (23) in . Then(i)the -parameter curves on cannot be asymptotic lines of ,(ii)the -parameter curves on are asymptotic lines of if and only if vanishes, where and are, respectively, the curvature and the torsion of the curve for .

Proof. Suppose that is a Hasimoto surface given by (23) in . Then the normal curvature of the -parameter curves on is the normal component of ; that is,This means that the -parameter curves are asymptotic lines if and only if , which is not possible since the fact that is a regular mapping. This gives the first statement.
The normal curvature of the -parameter curves on is defined byEquation (42) yields that the -parameter curves are asymptotic lines if and only if , which completes the proof.

Comparing Theorems 4 and 8, we have the following.

Corollary 9. A Hasimoto surface given by (23) in is minimal if and only if its -parameter curves are asymptotic lines.

A curve on a regular surface is called a principal curve if and only if its velocity vector field always points in a principal direction. Moreover, a surface is called a principal surface if and only if its parameter curves are principal curves.

A principal curve on a surface in is determined by the following formula:where is the unit normal vector field of the surface. Thus, we have the following results.

Theorem 10. Let be a Hasimoto surface given by (23) in . Then(i)the -parameter curves on are principal curves of if and only if vanishes, where is the torsion of the curve for all ;(ii)the -parameter curves on are principal curves of .

Proof. Assume that is a Hasimoto surface given by (23) in . Then the -parameter curves are the principal curves if and only ifwhere is the torsion and is the Frenet frame field of the curve for all such that . By (43), it follows that the -parameter curves are principal if and only if their torsions vanish, which gives the first statement.
In a similar way, by using (24), it is easily seen that , which specify that -parameter curves are the principal curves.
Therefore the proof is completed.

As a consequence of Theorem 10, the following result can be given.

Corollary 11. A Hasimoto surface given by (23) in is a principal surface if and only if the torsions of s-parameter curves vanish.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The second author’s work was supported by the DAAD Fellowship for University Professors and Researchers A/14/02337 at Technical University Berlin, December 2014 to January 2015. She thanks Professor Udo Simon for being her host Professor and for his kind hospitality. The drawings in the present paper are made with Wolfram Mathematica 7.0.

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