# Murat Kemal KaracanUsak Üniversitesi · Department of Mathematics

Murat Kemal Karacan

Prof.Dr.

## About

84

Publications

8,422

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831

Citations

Citations since 2016

Introduction

Additional affiliations

November 2018 - present

**Uşak University**

Position

- Professor

## Publications

Publications (84)

In this paper, we classify conformal surfaces of revolution in hyperbolic3-space $\mathbb{H}^{3}(-c^{2})$ satisfying an equation in terms of the position vector field and the Laplace operators with respect to the first,the second and the third fundamental forms of the surface.

In this paper, we investigate Mannheim pairs, Frenet-Mannheim curves, and weakened Mannheim curves with respect to the modified orthogonal frame in Euclidean 3-space (E ³ ). We derive some characterization results of these curves.

In this study, we define the dual surfaces by z = f(u) + g(v) and also classify these surfaces in I(double-struck) 3¹ satisfying some algebraic equations in terms of the coordinate functions and the Laplace operators according to fundamental forms of the surface.

In this study, we investigated the natural mates of equiaffine curves with constant equiaffine curvatures, associated to equiaffine frame in affine 3-space and we gave the position vectors under some conditions.

In this paper, we study helices and the Bertrand curves. We obtain some of the classification results of these curves with respect to the modified orthogonal frame in Euclidean 3-spaces.

In [M. K. Karacan, D. W. Yoon and B. Bukcu, Translation surfaces in the three-dimensional simply isotropic space 𝕀31, Int. J. Geom. Methods Mod. Phys.13(7) (2016) 1650088], there is a mistake in Theorem 5 that appeared in the paper. We here provide a correct theorem.

In this paper, we study spherical images of the modified orthogonal vector fields and Darboux vector of a regular curve which lies on the unit sphere in Euclidean 3-space.

In this paper, we study helices and the Bertrand curves. We obtain some of the classification results of these curves with respect to the modified orthogonal frame in Euclidean 3-spaces.

In this paper, we investigate Mannheim pairs, Frenet-Mannheim curves and Weakened Mannheim curves with respect to the modified orthogonal frame in Euclidean 3-space(E 3 ). We obtain some characterizations of these curves.

In this paper, we study the dual translation surfaces in three dimensional simply isotropic space. We give classification of dual translation surface with constant dual isotropic mean curvature or constant dual isotropic Guassian curvature.

The characterizations of adjoint curves of a space curve in Euclidean 3-space have been done in the present study. A curve and its adjoint curve create new pairs like Bertrand and involute-evolute curves are obtained. Ruled surfaces such as normal and binormal surface with a base curve and adjoint of base curve are defined here. Tubular surface who...

In this paper, we classify translation surfaces of Type 2 in the three dimensional simply isotropic space ð¹3 satisfying some algebraic equations in terms of the coordinate functions and the Laplacian opera-tors with respect to the first, the second and the third fundamental form of the surface. We also give explicit forms of these surfaces.

In this paper, we classify helicoidal surfaces in the three dimensional simply isotropic space 〉3¹ satisfying some algebraic equations in terms of the coordinate functions and the Laplacian operators with respect to the first, the second and the third fundamental formof the surface. We also give explicit forms of these surfaces.

In this paper, we classify two types ruled surfaces in the three dimensional simply isotropic space I

In this paper, we study ruled surface in 3-dimensional almost contact metric manifolds by using surface theory defined by Gök [Surfaces theory in contact geometry, PhD thesis (2010)]. We also studied the theory of curves using cross product defined by Camcl. In this study, we obtain the distribution parameters of the ruled surface and then some res...

In this paper, we classify translation surfaces in the three dimensional Galilean space G3 satisfying some algebraic equations in terms of the coordinate functions and the Laplacian operators with respect to the second fundamental form of the surface. We also give explicit forms of these surfaces.

In this paper, we classify translation surfaces in the three-dimensional simply isotropic space 𝕀31 under the condition Δix i = λixi where Δ is the Laplace operator with respect to the first and second fundamental forms and λ is a real number. We also give explicit forms of these surfaces.

In this study, we gave an alternative kinematic model for two smooth submanifolds M and N both on another and inside of another, along given any two curves which are tangent to each other on M and N at every moment , which the motion accepted that these curves are trajectories of the instantaneous rotation centers at the contact points of these sub...

In this paper, we investigate the properties of curves of constant breadth in a 3-dimensional Lie group. Also, we find the condition of general helix as constant breadth curves and construct constant breadth curves which the tangent component of the curve vanishes.

In this study we have defined Bäcklund transformations of curves according to Bishop frame preserving the natural curvatures under certain assumptions in Minkowski 3-space.

In this study, non-null Frenet-Mannheim curves and non-null Weakened Mannheim curves are investigated in Minkowski 3-space. Some characterizations for this curves are obtained.

In this paper, we define translation surfaces in the 3-dimensional Heisenberg group obtained as a product of two planar curves lying in planes, which are not orthogonal, and study minimal translation surfaces in .

We study ruled surfaces in R3 which are obtained from dual spher- ical
indicatrix curves of dual Frenet vector fields. We find the Gaussian and mean
curvatures of the ruled surfaces and give some results of being Wein- garten
surface.

We study quadric surfaces in Euclidean 3-space with non-degenerate second fundamental form, and classify them in terms of the Gaussian curvature, the mean curvature, the second Gaussian curvature and the second mean curvature.

In this paper, we have defined canal surfaces in Galilean and Pseudo-Galilean 3-spaces.Then, we have studied Tubular surface in Galilean and Pseudo-Galilean 3-space satisfying some equations in terms of the Gaussian curvature and the mean curvature.We have discussed Weingarten, linear Weingarten conditions and HK−quadric type for this surface with...

In this paper, we define a generalized Mannheim quaternionic curve in the four-dimensional Euclidean space R4 and investigate the properties of it.

In this study, non-null Frenet-Mannheim curves and non-null Weakened Mannheim
curves are investigated in Minkowski 3-space. Some characterizations for this
curves are obtained.

In this work, we studied the properties of the spherical indicatrices of
involute curve of a space curve and presented some characteristic properties in
the cases that involute curve and evolute curve are slant helices and helices,
spherical indicatrices are slant helices and helices and we introduced new
representations of spherical indicatrices.

In this study, Frenet-Mannheim curves and Weakened Mannheim
curves are investigated in Galilean 3-space. Some characterizations for this
curves are obtained.

In the discussion of Bertrand and Mannheim curves, it is always assumed (explicitly or implicitly) that the curvature is nowhere zero. In this paper, we adapt this requirement on the Mannheim curves and investigate into the properties of two types of similar curves (the Frenet-Mannheim curves and the weakened Mannheim curves) under weakened conditi...

We study tubular surfaces in Euclidean 3-space satisfying some equations in terms of the Gaussian curvature, the mean curvature, the second Gaussian curvature, and the second mean curvature. This paper is a completion of Weingarten and linear Weingarten tubular surfaces in Euclidean 3-space.
1. Introduction
Let 𝑓 and 𝑔 be smooth functions on a sur...

In this study, we investigated the (K,H), (K,K_{II}), (H,K_{II})-Weingarten
and (K,H),(K,K_{II}),(H,K_{II}) and (K,H,K_{II})-linear Weingarten canal
surfaces in IR^3.

In this study, we analyze the general canal surfaces in terms of the features
flat, II-flat minimality and II-minimality, namely we study under which
conditions the first and second Gauss and mean curvature vanishes, i.e. K=0,
H=0, K_{II}=0 and H_{II} =0. We give a non-existence result for general canal
surfaces in E^3 with vanishing the curvatures...

In this paper, we investigate special Smarandache curves according to Bishop
frame in Euclidean 3-space and we give some differential geometric properties
of Smarandache curves. Also we find the centers of the osculating spheres and
curvature spheres of Smarandache curves.

In this paper, we study a tubular surfaces with Bishop frame in Euclidean 3-space satisfying some equations in terms of the Gaussian curvature, the mean curvature,the second Gaussian curvature and second mean curvature.

We study the relations between Frenet vector fields and curvatures and torsions of Bertrand curves at the corresponding points in Minkowski 3-space.

In this paper, the Darboux rotation axis for a curve in Galilean and
Pseudo-Galilean spaces are decomposed into two simultaneous rotation. The axes of
these simultaneous rotations are joined by a simple mechanism. One of these axes is a
parallel of the principal normal of the curve, the direction of the other is the direction of the
Darboux vectors...

A special motion by the form Y = AX + C with one parameter has been given by [5] in Euclidean n space.
In this paper, we find a geometrical meaning for the determinant of the derivative matrices A*,A** and A*** according to Bishop frame in Minkowski 3space. Then we search, in this case, the geometry of the 1s t and 2 nd order acceleration pole poin...

In this study, we have generalized for a spacelike curve with a spacelike binormal which is studied by Bishop [1] to Minkowski 3-Space. In addition, the Darboux vector (matrix) for spacelike curve with a spacelike binormal was found. Furthermore, using the derivative of the tangent vector T of the spacelike curve with a spacelike binormal, the rela...

In this study, we have defined slant helix according to Bishop frame in Euclidean 3-Space. Furthermore, we have given some necassary and sufficient conditons for the slant helix.

T.Ikawa obtained the following differential equation D T D T D T T − K D T T, K = κ 2 − τ 2 for the cırcular helix which corresponds the case that the curvature κ and torsion τ of timelike curve α on the Lorentzian manifold M 1 are constant [5]. In this paper, we have defined a slant helix according to Bishop frame of the timelike curve. Furthermor...

Singular points of a tubular surface which is found by using a two-parameter spatial motion along a curve in Euclidean 3-space and its parallel surface are investigated. When the rotation axis is chosen as T which is the Frenet vector of the curve, the characterization of the tubular surface in motion is obtained by a different method. Then, some t...

We generalize the notion of involute and evolute of the space-like curve α with a space-like principal normal in Minkowski 3-space. Firstly, we show that, the length between the space-like curves α and β is constant. Furthermore, the Frenet frame of the involute curve β is found as depending on the curvatures of the curve α. We have determined cond...

A. A special Frenet motion with a one parameter has been given by Bottema [5] in E 3. In this study, we have given a generalization of [5] to Bishop motion in Minkowski 3-space. Firstly, Bishop motion is defined for timelike curve α and then Bishop darboux vectors of this motion is calculated for fixed and moving spaces in E 3 1. The Bishop darboux...

The Bishop darboux rotation for spacelike curves with a spacelike principal normal in Minkowski 3-space E 1 3 is decomposed into two simultaneous rotations. The axes of these simultaneous rotations are joined by a simple mechanism. One of these axes is a parallel of the tangent vector of the timelike curve, the direction of the other is the directi...

In this paper, the Dual Bishop darboux rotation axis for dual timelike space curve in the semi-dual space D13 is decomposed in two simultaneous rotation. The axes of these simultaneous rotations are joined by a simple mechanism.

T.Ikawa obtained the following differential equation$$D_{T}D_{T}D_{T}T-KD_{T}T,K=\kappa ^{2}-\tau ^{2}$$for the c\i rcular helix which corresponds the case that the curvature $ \kappa $ and torsion $ \tau $ of timelike curve $ \alpha $ on the Lorentzian manifold $ M_{1} $ are constant [5]. In this paper, we have defined a slant helix according to B...

We describe a robust method for constructing a tubular surface surrounding a spacelike curve with a spacelike principal normal
in Minkowski 3-Space. Our method is designed to eliminate undesirable twists and wrinkles in the tubular surface’s skin at
points where the curve experiences high torsion. In our construction the tubular surface’s twist is...

In this study, we study the result which are obtained by BISHOP (1975) for a timelike curve in Minkowski 3-Space.In addition , the Darboux vector(matrix) for the timelike curve is found. Furthermore, using the derivative of the tangent vector T of the timelike curve, the relations between the curvature funtions τ κ , and 2 1 , k k are found.

In this study, we research geodesics of tubular surfaces which is founded by using two-parameter spatial motion along a curve in Minkowski 3-space. To do this, we solve dierential equation DTT = 0 of parametric curves on the tubular surface where D is the connection of tubular surface and ! T is the unit vector field of two parametric curves on the...

A special Frenet motion with a one parameter has been given by Bottema [5] in E 3. In this study, we have given a generalization of [5] to Bishop motion in Euclidean 3-space. Firstly, Bishop motion is defined for space curve α and then Darboux vector of this motion is calculated for fixed and moving spaces in E 3.The Bishop darboux rotation for spa...

In this study, we generalize for a spacelike curve with a spacelike principal normal which was studied by Bishop [1] to Minkowski 3-Space. In addition, the Bishop Darboux vector(matrix) for spacelike curve is found. Furthermore , using the derivative of the tangent vector T of the spacelike curve, the relations between the curvature funtions κ, τ a...

In this paper, the Dual Bishop Darboux rotation axis for dual space curve in the dual space D3 is decomposed in two simultaneous rotation. The axes of these simultaneous rotations are joined by a simple mechanism.

A. In this study, we have generalized the involute and evolute curves of the timelike curve in Minkowski 3-Space. Firstly, we have shown that, the length between the timelike curve α and the spacelike curve β is constant. Furthermore, the Frenet-Serret frame of the involute curve β has been found as depend on curvatures of the curve α. We have dete...

In this study, we have generalized the involute and evolute curves of the spacelike curve α with a spacelike binormal in Minkowski 3- Space. Firstly, we have shown that, the length between the spacelike curve α and the timelike curve β is constant. Furthermore, the Frenet frame of the involute curve β has been found as depend on curvatures of the c...

In this study, two parameter motions by using the rank of rotation matrix were analysed and some theorems were given. The locus of the instantaneous screw axes of two parameter motions for special case n=3 were investigated. Furthermore, the locus of the instantaneous screw axes is a ruled surface in the position (λ,µ) = (0,0) were shown.

We describe a robust method for constructing a tubular surface surrounding a timelike space curve in Minkowski 3-Space. Our method is designed to eliminate undesirable twists and wrinkles in the tubular surface's skin at points where the curve experiences high torsion. In our construction the tubular surface's twist is bounded by the timelike curve...

In this paper the formula of the exponential matrix A e when A is a semi skew-symmetric real matrix of order 4 is derived. The formula is a generalization of the Rodrigues formula for skew-symmetric matrices of order 3 in Minkowski 3-space.

In this study all one parameter motions obtained from two parameters motion on the plane, are investigated. It is shown that the pole points which on fixed and moving plane at any position of (λ,μ) are on a line. It is also shown that the velocity vector lengths of these axis are the same. Moreover, the locus of any Hodograph of any point, and acce...

We investigate all one-parameter motions obtained from two-parameter motion on the plane. It is shown that the pole points on fixed and moving plane at any position of parameters (λ,μ) are on a line. Moreover, we examine the locus of hodograph of any point and the acceleration poles of the motion.

Quaternion is a division ring. It is shown that planes passing through the origin can be made a field with the quaternion
product in R3. The Hamiltonian operators help us define the homothetic motions on these planes. New characterizations for these motions
are investigated.

In this study all one parameter motions obtained from two parameters motion on the Lorentzian plane, are investigated. It is shown that the pole points which on flxed and moving Lorentzian plane at any position of (‚;") are on a line. It is also shown the velocity vector lengths of these axis are the same. Moreover, the locus of any Hodograph of an...

In our day and new age, industrial and developing nations both find themselves in a world where such changes in the global political and economic environment as the result of the end of the Cold War and globalization of economies constitute significant realities that must be accommodated. Especially notable in developing nations is the projection o...

In this paper, we examine singular points of tubular sur-faces and its parallel surfaces, which is based on two-parameter spatial motion along a curve in Minkowski 3-space. Related results are pre-sented also.

We define Bäcklund transformations of curves according to the Bishop frame which preserve the natural curvatures under certain assumptions in Euclidean 3-space.