Melek Erdoğdu

• Professor
• Professor at Necmettin Erbakan University

The study focuses on surface motion in three-dimensional Minkowski space, using anholonomic coordinates to analyze unit-speed spacelike curves with timelike normality. It highlights the importance of anholonomic coordinates in investigating important ideas and conclusions based on differential geometry. The study also aims to osculate motion in spa...

This study progresses to a detailed analysis of three-dimensional vector fields, focusing on the differential geometric properties of vector lines, including curvature and torsion, using anholonomic coordinates. Further, it investigates the curl of the binormal vector field in specific configurations, leading to the identification of surfaces with...

Bu çalışmada, Minkowski uzayında birim hızlı spacelike eğri ile ilişkilendirilmiş dinamik bir sistem olarak ele alarak duman halkası denklemi olarak bilinen girdap filaman denkleminin araştırılması verilmiştir. Darboux çatısı kullanılarak, doğrusal olmayan Schrödinger (NLS) denklemine karşılık gelen soliton yüzeyinin diferensiyel geometrik özellikl...

In the present paper, we investigate differential geometric properties the soliton surface M associated with the Betchow-Da Rios Equation. Then, we give derivative formulas of Frenet frame of unit speed curve Φ=Φ(s,t) for all t. Also, we discuss the linear map of Weingarten type in the tangent space of the surface that generates two invariants: k a...

This paper is investigated geometry of vector fields along spacelike curve with timelike normal vector by using anholonomic coordinates. Derivative formulas of Frenet Serret frame of the curve are stated which includes eight parameters. Surfaces with vanishing abnormality of normal direction in Minkowski space are examined. Intrinsic geometric prop...

The aim of this paper is to investigate the nonnull soliton surfaces associated with Betchov–Da Rios equation in Minkowski space-time. The differential geometric properties of these kind of nonnull soliton surfaces are examined with respect to the Lorentzian casual characterizations. Moreover, the linear maps of Weingarten type are obtained which a...

Considering the importance of Minkowski space in physics, it is an incomplete approach to deal with EM waves only in Euclidean space. For this reason, this paper deals with EM waves along pseudo null curves in Minkowski space. The main purpose of this study is to examine electromagnetic waves by defining an adapted orthogonal frame along the EM wav...

In this paper, we take advantage of envelope theory and singularity theory to study the evolutoids and pedaloids in Minkowski plane. We illustrate an internal correlation from algebraic and geometric viewpoints, and give the geometric explanation of evolutoids and pedaloids. Then, we generalize the notions of evolutoids and pedaloids to the categor...

This paper is devoted to the geometry of vector fields and timelike flows in terms of anholonomic coordinates in three dimensional Lorentzian space. We discuss eight parameters which are related by three partial differential equations. Then, it is seen that the curl of tangent vector field does not include any component in the direction of principa...

The present paper examines the differential analysis of flows on normal congruence of spacelike curves with spacelike normal vector in terms of anholonomic coordinates in three dimensional Lorentzian space. Eight parameters, which are related by three partial differential equations, are discussed. Then, it is seen that the curl of tangent vector fi...

In this paper, we investigate some important properties of split quaternion matrices by using real matrices. Firstly, we introduce real matrix representation of a split quaternion matrix. Then we prove the existence of left eigenvalue of a split quaternion matrix. Moreover, we give the relation between complex left eigenvalues of a split quaternion...

This paper is devoted to the geometry of pseudo null curve in terms of anholonomic coordinates in Minkowski space. Firstly, extended Frenet formulas for pseudo null curves are deeply discussed. Then binormal congruence of degenerate surfaces containing the s−lines and n−lines are investigated with the condition μb=0. This condition represents the n...

The main purpose of this study is to examine curves lying on a given non-lightlike surface with the help of its position vectors. For this purpose, the darboux frame is used and the position vector of the curve is expressed as a linear combination of the darboux frame with differentiable functions. Then, nonhomogeneous systems of differential equat...

This paper is devoted to the geometry of vector fields and timelike flows in terms of anholo-nomic coordinates in three dimensional Lorentzian space. We discuss eight parameters which are related by three partial di¤erential equations. Then, it is seen that the curl of tangent vector field does not include any component in the direction of principa...

The present paper examines the di¤erential analysis of fows on normal congruence of space-like surfaces with spacelike normal vector in terms of anholonomic coordinates in three dimensional Lorentzian space. Eight parameters which are related by three partial di¤erential equations are discussed. Then, it is seen that the curl of tangent vector …eld...

This paper is devoted to the geometry of null Cartan flows and Landau-Lifshitz equations in three dimensional Minkowski space. We discuss the parameters which are related by three partial di¤erential equations. Then, it is seen that the curl of binormal vector field does not include any component in the direction of principal normal vector field. T...

This paper is investigated to the geometry of vector fields and spacelike fows with time-like normal vector by using anholonomic coordinates in three dimensional Lorentzian space. We describe Frenet Serret frame of given a spacelike curve with timelike normal in terms of anholonomic coordinates which includes eight parameters, related by three part...

The main scope of this paper is to examine null Cartan curves especially the ones with constant torsion. In accordance with this scope, the position vector of a null Cartan curve is stated by a linear combination of the vector fields of its pseudo-orthogonal frame with differentiable functions. However, the most important difference that distinguis...

In this paper, some special types of surfaces with null Cartan base curve are investigated. The generating lines of the surfaces are chosen as a linear combination of Cartan frame fields with non-constant differentiable functions. Firstly, the surfaces whose generating lines have the same direction of Cartan frame fields B; N and T are examined res...

The purpose of this study is to obtain a characterization of spacelike Bertrand curve mate with constant curvature and torsion in Minkowski space. According to this purpose, the position vector of a spacelike Bertrand curve mate is obtained by a linear combination of its Serret Frenet Frame with differentiable functions. Then we investigate the spa...

The main scope of this presentation is to explain the smoke ring (or vortex filament) equation which can be viewed as a dynamical system on the space curve in E3. The differential geometric properties the soliton surface associated with Nonlinear Schrödinger (NLS) equation, which is called NLS surface or Hasimoto surface, are investigated by using...

The purpose of this study is to obtain a characterization of unit speed spacelike curve with constant curvature and torsion in Minkowski 3 - space. According to this purpose, the position vector of a spacelike curve is expressed by a linear combination of its Serret Frenet Frame with differentiable functions. Since a spacelike curve has different k...

The aim of this presentation is to characterize the position vectors of the timelike Bertrand mate in Minkowski space by means of differentiable functions. Therefore, the position vector of a timelike Bertrand curve is obtained by a linear combination of its Frenet frame with differentiable functions. Depending on the curvature and torsion value, d...

The purpose of this study is to obtain a characterization of unit speed spacelike curve with constant curvature and torsion in the Minkowski 3-space. According to this purpose, the position vector of a spacelike curve is expressed by a linear combination of its Serret Frenet Frame with differentiable functions. Since a spacelike curve has different...

The main scope of this paper is to examine the smoke ring (or vortex filament) equation which can be viewed as a dynamical system on the space curve in E³. The differential geometric properties the soliton surface associated with Nonlinear Schrödinger (NLS) equation, which is called NLS surface or Hasimoto surface, are investigated by using Darboux...

In this study, the characterization of position vectors belonging to non-lightlike Bertrand W curve mate with constant curvature are obtained depending on differentiable functions. The position vector of Bertrand W curve is stated by a linear combination of its Frenet frame with differentiable functions. There exist also different cases for the cur...

In this paper, we introduce 8×8 real matrix representations of complex split quaternions. Then, the relations between real matrix representations of split and complex split quaternions are stated. Moreover, we investigate some linear split and complex split quaternionic equations with split Fibonacci and complex split Fibonacci quaternion coefficie...

The main topic of this study is to investigate rotation matrices in four dimensional Euclidean space in two different ways. The first of these ways is Rodrigues formula and the second is Cayley formula.The most important common point of both formulas is the use of skew symmetric matrices. However, depending on the skew symmetric matrix used, it is...

Bu çalışmada, Minkowski uzayında sabit eğrilikli İnvolüt-Evolüt eğri çiftlerine ait pozisyon vektörlerinin karakterizasyonları diferansiyellenebilir fonksiyonlara bağlı olarak elde edilmiştir. Eğri çiftleri Frenet vektör alanlarının Minkowski uzayında sahip olduğu karakterlere göre ayrı ayrı ele alınmıştır. Aynı zamanda bu eğri çiftleri ile ilgili...

Bu çalışmada, Minkowski uzayında sabit eğrilikli İnvolüt-Evolüt eğri çiftlerine ait pozisyon vektörlerinin karakterizasyonlarını diferansiyellenebilir fonksiyonlara bağlı olarak elde edilmiştir. Eğri çifti Frenet vektör alanlarının karakterlerine göre ayrı ayrı ele alınmıştır. Elde edilen bu ifadeler örnekler verilerek açıklanmıştır. Aynı zamanda b...

In this study, the reflections in and are investigated by unit quaternions. Firstly, a linear transformation is defined to describe reflections in with respect to the plane passing through the origin and orthogonal to the quaternion. Then some examples are given to discuss obtained results. Similarly, two linear transformations are stated which cor...

In this study, the characterization of position vectors belonging to non-lightlike Bertrand curve couples with constant curvature are obtained depending on differentiable functions. The position vector of Bertrand curve is stated by a linear combination of its Frenet frame with differentiable functions. There exist also different cases for the curv...

The main scope of this paper is to examine null Cartan curves with constant curvature and torsion. In accordance with this scope, the position vector of a null Cartan curve is stated by a linear combination of its pseudo orthogonal frame with di¤erentiable functions. However, the most important di¤erence that distinguishes this study from the afore...

In this paper, we give characterization of timelike unit speed curve with constant curvature and torsion in the Minkowski 3-space. In accordance with this scope, the position vector of a curve is stated by a linear combination of its Serret Frenet Frame with differentiable functions. There exist also two different types depending on the values of c...

The purpose of this study is to obtain a characterization of spacelike unit speed curve with constant curvature and torsion in the Minkowski 3-space. According to this purpose, the position vector of a spacelike curve is expressed by a linear combination of its Serret Frenet Frame with di¤erentiable functions. Since a spacelike curve has different...

In this study, the reflections in í µí´¼ 3 and í µí´¼ 4 are investigated by unit quaternions. Firstly, a linear transformation is defined to describe reflections in í µí´¼ 3 with respect to the plane passing through the origin and orthogonal to the quaternion. Then some examples are given to discuss obtained results. Similarly, two linear transform...

In this study, the reflections in í µí´¼ 3 and í µí´¼ 4 are investigated by unit quaternions. Firstly, a linear transformation is defined to describe reflections in í µí´¼ 3 with respect to the plane passing through the origin and orthogonal to the quaternion. Then some examples are given to discuss obtained results. Similarly, two linear transform...

In this study, the rotations in E 3 and E 4 are investigated by unit quaternions. Firstly, the eigenvalues of rotations in Euclidean 3-space are investigated by means of the scalar part of corresponding unit quaternions. Then, the eigenvalues and eigenvectors of the simple, double and isoclinic rotations are stated in tems of the corresponding quat...

Bu çalışmada, uzayında sabit oranlı Bertrand eğri çiftleri ele alınmıştır. Sabit oranlı eğrileri tanıtıp ve bunların bazı karakterizasyonları ifade edilmiştir. Bununla birlikte burulmuş (twisted) eğrisi, eğrisi, - sabit ve - sabit eğrisi üzerine çalışılmıştır. Ayrıca bir eğrisini, eğrinin eğrilik ve burulma fonksiyonlarına bağlı diferansiy...

Minkowski 3-uzayında null olmayan eğriler için Tzitzeica eğrisi olma şartı yeniden formülize edildi. Buna bağlı olarak null ve pseudo-null eğriler için de Tzitzeica eğrisi olma koşulu ifade edildi. Ayrıca; hiç bir null rektifiyan Tzitzeica eğrisi olmadığı, sabit burulmaya sahip hiç bir pseudo-null Tzitzeica eğrisi olmadığı ispatlanmıştır.

Bu çalışmanın amacı, Minkowski uzay-zamanda timelike eğriler arasında Bäcklund dönüşümünü tanımlamaktır. Bu amaç doğrultusunda, timelike Bäcklund eğrilerin Frenet çatıları arasında ilişkiyi ortaya koyan dönme matrisinin seçimine bağlı olarak dönüşümü inceledik. İkisi spacelike hiperdüzlemde küresel dönme ve biri ise timelike hiperdüzlemde hiperboli...

In this paper, we present some important properties of matrices over hyperbolic split quaternions. We examine hyperbolic split quaternion matrices by their split quaternion matrix representation.

In this paper, we investigate the reflections in Minkowski 3-space by three different appoach. Firstly, we define Lorentzian reflections with Lorentzian inner product. Then, we examine Lorentzian reflections in view of Lorentzian House-holder matrices. Finally, we use pure split quaternions to derive Lorentzian reflections. For each case, we find t...

In this paper, we investigate the reflections in Minkowski 3-space by three different appoach. Firstly, we define Lorentzian reflections with Lorentzian inner product. Then, we examine Lorentzian reflections in view of Lorentzian House-holder matrices. Finally, we use pure split quaternions to derive Lorentzian reflections. For each case, we find t...

In this paper, we investigate the reflections in Minkowski 3-space by three di¤er-ent appoach. Firstly, we define Lorentzian reflections with Lorentzian inner product. Then, we examine Lorentzian reflections in view of Lorentzian Householder matrices. Finally, we use pure split quaternions to derive Lorentzian reflections. For each case, we find th...

In this paper, real matrix representations of split quaternions are examined in terms of the casual character of quaternion. Then, we give De-Moivre' s formula for real matrices of timelike and spacelike split quaternions, separately. Finally, we state the Euler Theorem for real matrices of pure split quaternions

The main purpose of this paper is to investigate parallel frames of nonlightlike curves in Minkowski space-time. For this purpose, the relations between Frenet-Serret frame and parallel frame of a given nonligthlike curve are discussed in four different cases. Then, the curvature, first and second torsion functions κ(s), τ(s), σ(s) and principal cu...

The main topic of this study is to investigate rotation matrices in four dimensional Euclidean space in two different ways. The first of these ways is Rodrigues formula and the second is Cayley formula.The most important common point of both formulas is the use of skew symmetric matrices. However, depending on the skew symmetric matrix used, it is...

In this paper, we investigate the surfaces generated by binormal motion of Bertrand curves, which is called Razzaboni surface, in Minkowski 3-space. We discussed the geometric properties of these surfaces in M^3 according to the character of Bertrand geodesics. Then, we define the Razzaboni transformation for a given Razzaboni surface. In other wor...

In this paper, real matrix representations of split quaternions are examined in terms of the casual character of quaternion. Then, we give De-Moivre' s formula for real matrices of timelike and spacelike split quaternions, separately. Finally, we state the Euler theorem for real matrices of pure split quaternions.

In this paper, Cayley formula is derived for 4 x 4 semi-skew-symmetric real matrices in E-1(4). For this purpose, we use the decomposition of a semi-skew-symmetric matrix A = theta(1)A(1) + theta(2)A(2) by two unique semi-skew-symmetric matrices A(1) and A(2) satisfying the properties A(1)A(2) = 0, A(1)(3) = A(1) and A(2)(3) = -A(2). Then, we find...

In this paper, we present some important properties of matrices over dual split quaternions. We examine dual split quaternion matrices by their split quaternion matrix representation. Then, we give some interesting results for 2×2 split quaternion matrix representation of a dual split quaternion. Finally , we prove that the group H_{D} and SC(2,H)...

In this paper, we investigate linear split quaternionic equations with the terms of the form axb. We give a new method of solving general linear split quaternionic equations with one, two and n unknowns. Moreover, we present some examples to show how this procedure works.

In this paper, we investigate the reflections in Minkowski three-space by three different approaches. Firstly, we define Lorentzian reflections with Lorentzian inner product. Then, we examine Lorentzian reflections in view of Lorentzian Householder matrices. Finally, we use pure split quaternions to derive Lorentzian reflections. For each case, we...

In this paper, we investigate the Hasimoto surfaces in Minkowski 3-space. We discussed the geometric properties of Hasimoto surfaces in \(\mathbb {M}^{3}\) for three cases. The Gaussian and mean curvature of Hasimoto surface are found for each case. Then, we give the characterization of parameter curves of Hasimoto surfaces in \(\mathbb {M}^{3}.\)

In this paper, a Rodrigues-like formula is derived for 4 × 4 semi skew-symmetric real matrices in E41. For this purpose, we use the decomposition of a semi skew-symmetric matrix A = θ1A1 + θ2A2 by two unique semi skew-symmetric matrices A1 and A2 satisfying the properties A1A2=0, A31=A1 and A32=-A2. Then, we find Lorentzian rotation matrices with s...

In this paper, we examine eigenvalue problem of a rotation matrix in Minkowski 3 space by using split quaternions. We express the eigenvalues and the eigenvectors of a rotation matrix in term of the coefficients of the corresponding unit timelike split quaternion. We give the characterizations of eigenvalues (complex or real) of a rotation matrix i...

The main purpose of this paper is to construct a Bäcklund transformation between two spacelike curves with the same constant curvature in Minkowski space-time by considering some assumptions. Moreover, we give the relations between curvatures of these two spacelike Bäcklund curves.

The main purpose of this paper is to set a method of finding eigenvalues of split quaternion matrices. In particular, we will give an extension of Gershgorin theorem, which is one of the fundamental theorems of complex matrix theory, for split quaternion matrices.

In this paper, we present some important properties of complex split quaternions and their matrices. We also prove that any complex split quaternion has a 4 × 4 complex matrix representation. On the other hand, we give answers to the following two basic questions “If AB = I, is it true that BA = I for complex split quaternion matrices?” and “How ca...

The main purpose of this paper is to examine the linear split quaternionic systems with n unknowns. We investigate these kind of systems with left and right coe¢ cients, seperately. Then, we give the classi…cations of these systems of equations by using complex adjoint matrix of split quaternion matrix. Besides, a method of …nding solution of these...

The main purpose of this paper is to examine the linear split quaternionic systems with n unknowns. We investigate these kind of systems with left and right coe¢ cients, seperately. Then, we give the classifications of these systems of equations by using complex adjoint matrix of split quaternion matrix. Besides, a method of finding solution of the...

In this paper, we investigate the Hasimoto surfaces in Minkowski 3-space by using parallel frames. The smoke ring equation is given by parallel frames. Then, the differential geometry of Hasimoto surfaces is examined with parallel frames. Finally, we give the relation between Hasi-moto surfaces and split quaternionic NLS equation in M 3 .