Kadri Arslan

• Prof. Dr.
• Lecturer at Bursa Uludağ Üni̇versi̇tesi̇

The rotational embedded submanifold of $\mathbb{E}^{n+d}$ first studied by N. Kuiper. The special examples of this type are generalized Beltrami submanifolds and toroidals submanifold. The second named authour and at. all recently have considered $3-$dimensional rotational embedded submanifolds in $\mathbb{E}^{5}$. They gave some basic curvature pr...

The general rotational surfaces of [Formula: see text] were first studied by Moore. The Vranceanu surfaces are special examples of this kind of surfaces. These constant-ratio surfaces are surfaces for which the ratio of the norms of the tangent and normal components of the position vector fields is constant. However, spherical surfaces and conical...

Matematik ve Astronomi

In the present study we consider rotational surfaces in Euclidean 4-space whose canonical vector field xT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^{T}$$\end{doc...

In this study, geometrical and mechanical identification of the talus and cochlea tibiae in horse and ox is presented. The shape of the expressed bones of these animals can be considered as rotational surfaces of planar curves. The model is established based on the bolt-nut mechanism while interpreting the relationship between talus and tibiae, and...

In this study, the representations of the ribbons in the 3-dimensional Euclidean space as the developable ruled surface are given. By calculating the average curvature of the ribbon surface, the results regarding the mean curvature according to the character of the centerline are obtained. In addition, examples supporting these results are given.

Self-similar flows arise as special solution of the mean curvature flow that preserves the shape of the evolving submanifold. In addition, \lambda -hypersurfaces are the generalization of self-similar hypersurfaces. In the present article we consider \lambda -hypersurfaces in Euclidean spaces which are the generalization of self-shrinkers. We obtai...

In this work, we deal with a curve whose position vector can be expressed with the help of Frenet Frame in n-dimensional Euclidean space. We classify this type of curve with regards to curvature functions and get certain consequences for T-constant, N-constant and constant ratio curves in E^n .

In this study, we attend to the curves whose position vectors are written as alinear combination of their Serret-Frenet vectors in Minkowski 4-Space E_1^4. We characterize such curves with regard to their curvatures. Further, we get certain consequences of T-constant and N-constant types of curves in E_1^4.

Bu çalışmada, E^4 4-boyutlu Öklid uzayında, merkez eğrisinin paralel öteleme çatısı vektörleri yardımıyla tanımlanan kanal yüzeyini örneği ile verdik. Bu yüzeyin eğrilik özelliklerini paralel öteleme çatısına göre eğrilik fonksiyonları cinsinden araştırdık. Daha sonra, Weingarten tipindeki kanal ve tüp yüzeyleri hakkında bazı sonuçlar verdik. Son o...

In the present paper we consider weak biharmonic rotational surfaces in Euclidean 4-space E⁴. We have proved that the general rotational surface of parallel mean curvature vector field is weak biharmonic then either it is minimal or a constant mean curvature. Further, we show that if Vranceanu surface of constant mean curvature is weak-biharmonic t...

In the present study we give some corrections for our paper which published in the first volume of this journal.

In this study, we give a geometric description of the tracheal elements of the chard (Beta vulgaris var. cicla L.), which is a widespread cultivated plant in Turkey. It is used as an edible plant and its leaves are used as antidiabetic in traditional medicine plant. We have shown that the tracheal elements, which are taxonomic value of the plant, c...

The rotational embedded submanifold was first studied by Kuiper as a submanifold in n+d. The generalized Beltrami submanifolds and toroidal submanifold are the special examples of these kind of submanifolds. In this paper, we consider 3-dimensional rotational embedded submanifolds in Euclidean 5-space 5. We give some basic curvature properties of t...

In the present study we consider surfaces in Euclidean (n+2)-space Eⁿ⁺². Firstly, we introduce some basic concepts of second fundamental form and curvatures of the surfaces in Eⁿ⁺². Further, we obtained some basic properties of surfaces in Eⁿ⁺² and some results related with their total shear curvatures. Finally, we give an example of generalized sp...

In this study firstly, we study with conchoid curves in Euclidean plane E2. We calculate the curvature of the conchoid curve and give some results. Furthermore, we consider the surface of revolution given with the conchoid curve in Euclidean 3-space E3. The Gaussian and mean curvature is calculated of these surfaces. Also we give some examples and...

In this study, we consider Tzitzeica curves (Tz-curves) in Euclidean 3-space E^3. We characterize such curves according to their curvatures. We show that there is no Tz-curve with constant curvatures (W-curves). We consider Salkowski (TC-curve) and Anti-Salkowski curves.

In the present paper we study with the convolution surface $C=M\star N$ of a paraboloid $M\subset \mathbb{E}^{3}$ and a parametric surface $N\subset \mathbb{E}^{3}$. We take some spacial surfaces for $N$ such as, surface of revolution, Monge patch and ruled surface and calculate the Gaussian curvature of the convolution surface $C$. Further, we giv...

The orthogonal trajectories of the first tangents of the curve are called the involutes of x. The hyperspheres which have higher order contact with a curve x are known osculating hyperspheres of x. The centers of osculating hyperspheres form a curve which is called generalized evolute of the given curve x in n-dimensional Euclidean space E(double-s...

We consider hypersurfaces of a semi-Euclidean spaces satisfying some curvature condition of pseudosymmetry type related to solutions of the P.J. Ryan problem of the equivalence of semisymmetry and Ricci-semisymmetry on hypersurfaces.

In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized tractrices in Euclidean (n + 1)-space 𝔼ⁿ⁺¹. Further, we introduce some kind of generalized rotational surfaces in Euclidean spaces 𝔼³ and 𝔼⁴, respectively. We have also obtained some basic properties of generalized rot...

In the present study, we characterize a regular curve whose position vector can be written as a linear combination of its Serret-Frenet vectors in Euclidean 4-space E⁴. We investigate such curves in terms of their curvature functions. Further, we obtain some results of T-constant, N-constant and constant ratio curves in E⁴.

In the present study, we consider canal surfaces imbedded in an Euclidean space of four dimensions. The curvature properties of these surface are investigated with respect to the variation of the normal vectors and curvature ellipse. We also give some special examples of canal surfaces in E^4. Further, we give necessary and sufficient condition for...

We consider translation surfaces in Euclidean spaces. Firstly, we give some results of translation surfaces in the 3-dimensional Euclidean space E³. Further, we consider translation surfaces in the 4-dimensional Euclidean space E⁴. We prove that a translation surface is flat in E⁴ if and only if it is either a hyperplane or a hypercylinder. Finally...

In the present study we consider the generalized rotational surfaces in Euclidean m-space \(\mathbb {E}^{m}\). Firstly, we introduce some basic concepts of second fundamental form and curvatures of the surfaces in \( \mathbb {E}^{m}\). Further, we obtained some basic properties of generalized rotational surfaces in \(\mathbb {E}^{m}\) and some resu...

In the present study we consider knotted spheres in Euclidean $4$-space $ \mathbb{E}^{4}$. Firstly, we give some basic curvature properties of knotted spheres in $ \mathbb{E}^{4}$. Further, we obtained some results related with the conjugate nets and Laplace transforms of these kind of surfaces.

In the present study we define a new kind of product surfaces namely mixed products which are product of two space curves in 4-dimensional Euclidean space . We investigate the Gaussian curvature, Gaussian torsion and mean curvature of these kind of surfaces. We obtain some original results of mixed product surfaces in . Further, we give some exampl...

In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized spherical curves in Euclidean $(n+1)-$space $\mathbb{E}^{n+1}$. Further, we introduce some kind of generalized spherical surfaces in Euclidean spaces $\mathbb{E}^{3}$ and $% \mathbb{E}^{4}$ respectively. We have shown that the...

In the present study we define a new kind of product surfaces namely mixed product which are product of plane curve and space curve in 3-dimensional Euclidean space. We give the original results of mixed product surface patches of flat or minimal type in $\mathbb{E}^{3}$. Further, we give some examples of these kind of surfaces and plot their graph...

The orthogonal trajectories of the first tangents of the curve are called the involutes of $x$. The hyperspheres which have higher order contact with a curve $x$ are known osculating hyperspheres of $x$. The centers of osculating hyperspheres form a curve which is called generalized evolute of the given curve $x$ in $n$-dimensional Euclidean space...

In the present paper we consider a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. They are called meridian surfaces of elliptic or hyperbolic type, respectively. We study these surfaces with respect to their Gauss map. We find...

In this study, by using Laplace and normal Laplace operators, we give some characterizations for the Darboux instantaneous rotation vector field of the curves in the Euclidean 3-space E³. Further, we give necessary and sufficient conditions for unit speed space curves to have 1-type Darboux vectors. Moreover, we obtain some characterizations of hel...

In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We classify semi-parallel meridian surfaces in 4-dimensional Euclidean space E4.

We study the Riemann curvature tensor of \((\kappa ,\mu ,\nu )\) -contact metric manifolds, which we prove to be completely determined in dimension 3. We also observe how this curvature tensor is affected by \(D_a\)-homothetic deformations, which will prompt the definition and study of generalized \((\kappa ,\mu ,\nu )\) -space forms and of the nec...

In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We classified semi-parallel meridian surface in 4-dimensional Euclidean space $E^4$.

In the present paper we study the problem of constructing a family of surfaces (surface pencils) from a given curve in 4-dimensional Euclidean space E 4. We have shown that generalized rotation surfaces in E 4 are the special type of surface pencils. Further, the curvature properties of these surfaces are investigated. Finally, we give some example...

In the present paper we study pseudo-Riemannian submanifolds which have 3-planar geodesic normal sections.We consider W-curves (helices) on pseudo-Riemannian submanifolds. Finally, we give neccessary and sufficient condition for a normal section to be a W-curve on pseudo-Riemannian submanifolds.

In the present paper we study the problem of constructing a family of surfaces (surface pencils) from a given curve in 4-dimensional Euclidean space $\mathbb{E}^{4}$. We have shown that generalized rotation surfaces in $\mathbb{E}^{4}$ are the special type of surface pencils. Further, the curvature properties of these surfaces are investigated. Fin...

In this paper we consider the focal curves of the curves in Euclidean n-space Rn. First we give some basic results on Darboux vector of these curves. Later, we realized some results of the order of contact of these curves. Further, we give necessary and sufficient conditions for focal curve to become 2-planar. We also show that if the ratios of the...

In the present paper we study normal transport surfaces in four-dimensional Euclidean space E 4 which are the generalization of surface offsets in E 3 . We find some results of normal transport surfaces in E 4 of evolute and parallel type. Further, we give some examples of these type of surfaces.

In this study we consider canal surfaces according to parallel transport frame in Euclidean space $\mathbb{E}^{4}$. The curvature properties of these surface are investigated with respect to $k_{1}$, $k_{2}$ and $k_{3}$ which are principal curvature functions according to parallel transport frame. Finally, we point out that if spine curve $\gamma $...

In this study we consider AW(k)-type curves according to parallel transport frame in Euclidean space E^4. We give the relations between the parallel transport curvatures of these kinds of curves.

A twisted curve in Euclidean 3-space E 3 can be considered as a curve whose position vector can be written as linear combination of its Frenet vectors. In the present study we study the twisted curves of constant ratio in E 3 and characterize such curves in terms of their curvature functions. Further, we obtain some results of T-constant and N-cons...

In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We show that a meridian surface has a harmonic Gauss map if and only if it is part of a plane. Further, we give necessary and su...

In the present paper we consider a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. They are called meridian surfaces of elliptic or hyperbolic type, respectively. We study these surfaces with respect to their Gauss map. We find...

A twisted curve in Euclidean 3-space E^3 can be considered as a curve whose position vector can be written as linear combination of its Frenet vectors. In the present study we study the twisted curves of constant ratio in E^3 and characterize such curves in terms of their curvature functions. Further, we obtain some results of T-constant and N-cons...

B.Y. Chen initiated the study of the tensor product immersion of two immersions of a given Riemannian manifold. Inspired by Chen's definition , F. Decruyenaere, F. Dillen, L. Verstraelen and L. Vrancken studied the tensor product of two immersions, in general, different manifolds; under certain conditions, this realizes an immersion of the product...

In the present study we consider weak biharmonic and harmonic 1-type curves in semi-Euclidean space E 4 1 . We give the classifications of these type curves.

In the present study we consider self-similar surfaces imbedded in Euclidean space. We give necessary and su¢ cient conditions for the surface of revolution and surfaces with Monge patch in E3 to become self-similar. Fur- ther, we investigate self-similar surfaces in Euclidean 4-space E4:Additionally we give necessary and su¢ cient condition of sph...

In this study, we give some characterizations on the Dar-boux instantaneous rotation vector field of the curves in Euclidean (2n + 1)-space E 2n+1 by using Laplacian operator. Further, we give necessary and sufficient conditions for unit speed space curves to have 1-type Darboux vector.

A depth surface of E^3 is a range image observed from a single view can be represented by a digital graph (Monge patch) surface . That is, a depth or range value at a point (u,v) is given by a single valued function z=f(u,v). In the present study we consider the surfaces in Euclidean 4-space E^4 given with a Monge patch z=f(u,v),w=g(u,v). We invest...

We give necessary and sufficient conditions for warped product manifolds (M, g), of dimension >= 4, with 1-dimensional base, and in particular, for generalized Robertson-Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R.C-C.R, formed from the curvature tensor R and the Weyl conformal curvature...

Submanifolds of coordinate finite-type were introduced in [11]. A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of �. In the present study we consider coordinate finite-type surfaces in E4. We give necessary and sufficient conditions for generalized rotation surfaces i...

In this paper, we study meridian surfaces of Weingarten type in Euclidean 4-space E^4. We give the neccessary and sufficient conditions for a meridian surface in E^4 to become Weingarten type.

Submanifolds of coordinate finite-type were introduced in HV1. A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of {\Delta}. In the present study we consider coordinate finite-type surfaces in E^4. We give necessary and sufficient conditions for generalized rotation sur...

Wintgen ideal surfaces in E-4 form an important family of surfaces, namely surfaces with circular ellipse of curvature. Obviously, Wintgen ideal surfaces satisfy the pointwise equality K + vertical bar K-N vertical bar = parallel to H parallel to(2). In the present study we consider the Wintgen ideal surfaces in n-dimensional Euclidean space E-n. W...

A focal representation of a generic regular curve {\gamma} in E^{m+1} consists of the centers of the osculating hyperplanes. A k-slant helix {\gamma} in E^{m+1} is a (generic) regular curve whose unit normal vector V_{k} makes a constant angle with a fixed direction U in E^{m+1}. In the present paper we proved that if {\gamma} is a k-slant helix in...

In this study, we consider curves of generalized AW(k)-type of Euclidean n-space. We give curvature conditions of these kind of curves.

In this study, by using Laplacian and normal Laplacian operators, some characterizations on the Darboux instantaneous rotation vector …eld of timelike and spacelike curves are given in Minkowski 3-space E31 .

In this study, by using Laplacian and normal Laplacian operators, some characterizations on the Darboux instantaneous rotation vector �eld of timelike and spacelike curves are given in Minkowski 3-space.

In the present study we consider generalized rotation surfaces imbedded in an Euclidean space of four dimensions. We also give some special examples of these surfaces in $${\mathbb E^4}$$. Further, the curvature properties of these surfaces are investigated. We give necessary and sufficient conditions for generalized rotation surfaces to become pse...

In the present study we calculate the coefficients of the second fundamental form and curvature ellipse of spherical product surfaces in E4. Otsuki rotational surfaces and Ganchev-Milousheva rotational surfaces are the special type of spherical product surfaces in E4. Further, we give necessary and sufficient condition for the origin of NpM to lie...

In the present paper we define a special class of surfaces de-termined by a given surface in the four-dimensional Euclidean space E 4 , which we call Benz surfaces following the idea of W. Benz and G. Stanilov in the three-dimensional case. We consider the class of Benz surfaces in-duced by surfaces of revolution in E 3 , and by standard rotational...

In the present study we consider generalized rotation surfaces imbedded in an Euclidean space of four dimensions. We also give some special examples of these surfaces in E 4 . Further, the curvature proper-ties of these surfaces are investigated. We give necessary and sufficient conditions for generalized rotation surfaces to become pseudo-umbilica...

In the present study we calculate the coefficients of the second fundamental form and curvature ellipse of spherical product surfaces in E 4. Otsuki rotational surfaces and Ganchev-Milousheva rotational surfaces are the special type of spherical product surfaces in E 4. Further, we give necessary and sufficient condition for the origin of NpM to li...

In the present study we calculate the coefficients of the second fundamental form and curvature ellipse of spherical product surfaces in E 4. Otsuki rotational surfaces and Ganchev-Milousheva rotational surfaces are the special type of spherical product surfaces in E 4. Further, we give necessary and sufficient condition for the origin of N pM to l...

We study the Riemann curvature tensor of (\kappa,\mu,\nu)-contact metric manifolds, which we prove to be completely determined in dimension 3, and we observe how it is affected by D_a-homothetic deformations. This prompts the definition and study of generalized (\kappa,\mu,\nu)-space forms and of the necessary and sufficient conditions for them to...

Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle c 1 centered at origin with an Euclidean planar curve c 2 has harmonic Gauss map if and only if M is a part of a pl...

In this article we investigate Vranceanu rotation surfaces with pointwise 1- type Gauss map in Euclidean 4-space $ \mathbb{E}^4 $ \mathbb{E}^4 . We show that a Vranceanu rotation surface M has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficent conditions for Vranceanu rotation surface to have pointw...

In this article we investigate Vranceanu rotation surfaces with pointwise 1-type Gauss map in Euclidean 4-space E-4. We show that a Vranceanu rotation surface M has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficent conditions for Vranceanu rotation surface to have pointwise 1-type Gauss map.

The object of the paper is to study some smooth surfaces M whose mean curvature vector H satisfies the H-recurrent condition DX H = λ(X)H in m-dimensional Euclidean space Em, where X is a tangent vector field of M and λ is a 1-form. First of all,we prove that the surfaces which satisfy the Hrecurrent condition in Em are R⊥-parallel (i.e., R⊥H = 0)....

For the Monge-Ampère Z xx Z yy -Z xy 2 =b 20 x 2 +b 11 xy+b 02 y 2 +b 00 we study the question of existence of a solution Z(x,y) in the class of polynomials, where Z(x,y) is the graph of a convex surface. If Z is a polynomial of odd order, a solution does not exist. If Z is a polynomial of fourth order and 4b 20 b 02 -b 11 2 >0, a solution also doe...

For the Monge-Ampere equation ZxxZyy-Z2xy = b20x2+b11xy + b02y2 + 600 we consider the question on the existence of a solution Z(x, y) in the class of polynomials such that Z = Z(x, y) is a graph of a convex surface. If Z is a polynomial of odd degree, then the solution does not exist. If Z is a polynomial of 4-th degree and 4b20b02 - b211 > 0, then...

An indefinite Riemannian manifold is called neutral it its index is equal to one half of its dimension and an indefinite Kaehler manifold is called neutral Kaehler if its complex index is equal to the half of its complex dimension. In the first part of this article, we extend the notion of slant sur-faces in Lorentzian Kaehler surfaces to slant sub...

In this study we consider the focal curve Cγ of a space curve γ and its focal curvatures. We characterize some special types of ruled surface, choosing one of the base curves or director curves as the focal curve of the space curve γ. Finally we construct new types of ruled surface and calculate their distinguished parameters. We give necessary and...

In the present article we study the rotational embedded surfaces in 4 . The rotational embedded surface was first studied by G. Ganchev and V. Milousheva as a surface in 4 . The Otsuki (non-round) sphere in 4 is one of the special examples of this surface. Finally, we give necessary and sufficient conditions for the flat Ganchev-Milousheva rotation...

In the present study we consider the submanifold M of Em sat- isfying the condition h¢H; eii = 0; where H is the mean curvature of M and ei 2 TM. We call such submanifolds tangentially cubic. We proved that every null 2- type submanifold M of Em is tangentially cubic. Further, we prove that the pointed helical geodesic surfaces of E5 with constant...

In the present paper we classify N(k)-contact metric manifolds which satisfy P(�,X) ·R = 0, R(�,X) · P = 0, P(�,X) · S = 0, P(�,X) · P = 0 and P(�,X) · Z = 0 where P is the Weyl projective curvature tensor and Z is the concircular curvature tensor.

In this study we consider the focal curve C γ of a space curve γ and its focal curvatures. We characterize some special types of ruled surface, choosing one of the base curves or director curves as the focal curve of the space curve γ. Finally we construct new types of ruled surface and calculate their distinguished parameters. We give necessary an...

In this paper we deal with the geometric properties of canal surfaces in E³. Further, the first and second fundamental form of canal surfaces are presented. By the use of the second fundamental form, the Gaussian and mean curvature of canal surfaces are obtained. Finally, the visualization of canal surfaces which their spine curves are unit circle...

In the present study we consider ruled surfaces imbedded in a Euclidean space of four dimensions. We also give some special examples of ruled surfaces in E 4 . Further, the curvature properties of these surface are investigated with respect to variation of normal vectors and curvature ellipse. Finally, we give a necessary and sufficient condition f...

The object of the present paper is to study certain curva-ture restriction on an LP-Sasakian manifold with a coefficient α. Among others it is shown that if an LP-Sasakian manifold with a coefficient α is a manifold of constant curvature, then the manifold is the product manifold. Also it is proved that a 3-dimensional Ricci semisymmetric LP-Sasaki...

The object of the present paper is to study a type of Riemannian manifolds called generalized recurrent manifolds. We have constructed two concrete examples of such a manifold whose scalar curvature is non-zero non-constant. Some other properties have been considered. Among others it is shown that on a generalized recurrent manifold with constant s...

We study on semiparallel and Weyl semiparallel Sasakian hypersurfaces of Kaehler manifolds. We prove that a (2n + 1)-dimensional Sasakian hypersurface M of a (2n+2)-dimensional Kaehler manifold fM2n+2 is semiparallel if and only if it is totally umbilical with unit mean curvature, if dimM = 3 and fM4 is a Calabi-Yau manifold, then fM is at at each...

In the present study we consider spherical product surfaces X = alpha otimes beta of two 2D curves in E3. We prove that if a spherical product surface patch X = alpha otimes beta has vanishing Gaussian curvature K (i.e. a flat surface) then either a or b is a straight line. Further, we prove that if alpha(u) is a straight line and b(v) is a 2D curv...

We study with biminimal curves, that is curves which are critical points of the bienergy for normal variations. We give a description of the Euler-Lagrange equation associated to biminimal curves on a Riemannian manifold. We describe curves of AW(k) type of Euclidean n-space E n . We show that every biminimal curves of E n are of AW(1)-type. We als...

In this study, a geometric and experimental analysis of the peritechia of Pseudonectria rousseliana (Mont.) Wollenw was presented. Experimentally, longitudinal length (AB) and the body (CD) of peritechia were measured. Geometrically, it was shown that the peritechia is comparable with a surface of revolution of a profile curve. On the same region...