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Yusuf Yayli

Publications and source records attributed to Yusuf Yayli.

At least 19 recordsLinked to original sources
arXiv2024

Ruled surfaces and hyper-dual tangent sphere bundle

In this study, we define the unit hyper-dual sphere $S_{\mathbb{D} _{2}}$ in hyper-dual vectors $\mathbb{D}_{2}$ and we give E-Study map version in $\mathbb{D}_{2}$ which prove that $S_{\mathbb{D} _{2}}^{2} $ is isomorphism to the tangent bundle $TS_{\mathbb{D} }^{2}.$ Next, we define ruled surfaces in $\mathbb{D}$, we give its developability condition and a geometric interpretation in $\mathbb{R}^{3}$ of any curves in $\mathbb{D}_{2}$. Finally, we present a relationship between a ruled surfaces set in $\mathbb{R}^{3}$ and curves in hyper dual vectors $\mathbb{D}_{2}$. We close each study with examples.

math.DG↗
arXiv2018

Isophote curves on spacelike surfaces in Lorentz-Minkowski space E31

Isophote curve consists of a locus of surface points whose normal vectors make a constant angle with a fixed vector (the axis). In this paper, we define an isophote curve on a spacelike surface in Lorentz-Minkowski space and then find its axis as timelike and spacelike vectors via the Darboux frame. Besides, we give some characterizations concerning isophote curve and its axis.

math.DG↗
arXiv2017

Generalized Quaternion and Rotation in 3-space E (3-alfa,beta)

The paper explains how a unit generalized quaternion is used to represent a rotation of a vector in 3-dimensional space. We review of some algebraic properties of generalized quaternions and operations between them and then show their relation with the rotation matrix.

math-ph↗
arXiv2017

Spherical Cyclic Motions in Euclidean Space E3

By considering a spatial curve in a Euclidean space, we use its components, together with attaining a cyclic matrix, to show that this matrix is homothetic too and is in correspondence with a homothetic motion. Furthermore, if the curve lies on a unit sphere, then the motion is a spherical cyclic motion.

math-ph↗
arXiv2016

N-Legendre and N-Slant Curves in the Unit Tangent Bundle of Minkowski Surfaces

Let $(\mathbb{M}_{1}^{2},g)$ be a Minkowski surface and $(T_1\mathbb{M}_1^2, g_1)$ its unit tangent bundle endowed with the pseudo-Riemannian induced Sasaki metric. We extend in this paper the study of the N-Legendre and N-slant curves which the inner product of normal vector and Reeb vector is zero and nonzero constant respectively in $\left( T_1 \mathbb{M}_1^2, g_1 \right)$, given in \cite{hmy}, to the Minkowski context and several important characterizations of these curves are given.\newline

math.DG↗
arXiv2015

A New Approach on Curves of Constant Precession

In this paper, we investigate a curve whose spherical image the tangent indicatrix and binormal indicatrix is slant helix and called it as a slant helix. We obtain that the spherical images are spherical slant helices defined by [3]. This notation is a generalization of a slant helix. Furthermore, we have given some characterizations for the slant helix and we show that a curve of constant precession is a slant helix.

math.DG↗
arXiv2014

Geometric Interpretation of det(C^(3),C^(4),C^(5))=0 in E13

In this paper, we investigate the tangent indicatrix of the curve C with constant curvature. Tangent indicatrix of the curve C is characterized with det(C^(3),C^(4),C^(5))=0 in Minkowski 3-space E13. Moreover, we study null slant helices using the determinant approach and give the following characterization: A curve C is a null slant helix in E13 if and only if det(C^(3),C^(4),C^(5))=0. Then similar results are obtained for non-null curves with the condition k=1.

math.DG↗
arXiv2014

On space-like constant slope surfaces and Bertrand curves in Minkowski 3-space

In the present paper, we define the notions of Lorentzian Sabban frames and de Sitter evolutes of the unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$. In addition, we investigate the invariants and geometric properties of these curves. Afterwards, we show that space-like Bertrand curves and time-like Bertrand curves can be constructed from unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$ and hyperbolic space $\mathbb{H}^{2}$, respectively. We obtain the relations between Bertrand curves and helices. Also we show that pseudo-spherical Darboux images of Bertrand curves are equal to pseudo-spherical evolutes in Minkowski 3-space $\mathbb{R}^{3}_{1}$. Moreover we investigate the relations between Bertrand curves and space-like constant slope surfaces in $\mathbb{R}^{3}_{1}$. Finally, we give some examples to illustrate our main results.

math.DG↗
arXiv2014

Time-like constant slope surfaces and space-like Bertrand curves in Minkowski 3-space

Defining Lorentzian Sabban frame of the unit speed time-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$ and introducing space-like height function on the unit speed time-like curves on $\mathbb{S}^{2}_{1}$, the invariants of the unit speed time-like curves on $\mathbb{S}^{2}_{1}$ and geometric properties of de Sitter evolutes of the unit speed time-like curves on $\mathbb{S}^{2}_{1}$ are studied. A relation between space-like Bertrand curves and helices is obtained. De Sitter Darboux images of space-like Bertrand curves are equal to de Sitter evolutes. The relations between time-like constant slope surfaces lying in the space-like cone and space-like Bertrand curves in Minkowski 3-space $\mathbb{R}^{3}_{1}$ are obtained.

math.DG↗
arXiv2014

New Associated Curves k-Principle Direction Curves and N_k Slant Helix

The Frenet frame is generally known an orthonormal vector frame for curves. But, it does not always meet the needs of curve characterizations. In this study, with the help of associated curves of any spatial curve we obtained a new orthonormal frame which has the property that the second vector makes a constant angle with a fixed direction for the spatial curve. Then, the curve is named as the N_k slant helix and the special conditions are obtained effectively.

math.DG↗
arXiv2014

Generalized Similar Frenet Curves

The paper is devoted to differential geometric invariants determining a Frenet curve in up to a direct similarity These invariants can be presented by the Euclidean curvatures in terms of an arc lengths of the spherical indicatrices. Then, these invariants expressed by focal curvatures of the curve. And then, we give the relationship between curvatures of evolute curve and shape curvatures. Morever, these invariants is given the geometric interpretation.

math.DG↗
arXiv2013

Prolongations of Lie Algebra Representations

In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we define prolongations of representations of Lie algebras. We show that if a Lie algebra representation corresponds to a Lie group representation, then prolongation of Lie algebra representation corresponds to the prolonged Lie group representation.

math.DG↗
arXiv2013

Boost Invariant Surfaces with Pointwise 1-Type Gauss Map in Minkowski 4-Space E4-1

In this paper, we study spacelike rotational surfaces which are called boost invariant surfaces in Minkowski 4-space E41. We give necessary and sufficient condition for flat spacelike rotational surface to have pointwise 1-type Gauss map. Also, we obtain a characterization for boost invariant marginally trapped surface with pointwise 1-type Gauss map.

math.DG↗
arXiv2013

On Inclined Curves According to Parallel Transport Frame in E4

In this paper, we introduce an inclined curves according to parallel transport frame. Also, we define a vector field called Darboux vector field of an inclined curve in and we give a new characterization such as: "α: I \subset R \rightarrow E^4 is an inclined curve \Leftrightarrow k_1 \int k_1ds + k_2 \int \k_2 +k_3ds = 0" where k_1, k_2, K_3 are the principal curvature functions according to parallel transport frame of the curve and we give the similar characterizations such as "α: I \subset R \rightarrow E^3 is a generalized helix \Leftrightarrow k_1 \int k_1ds + k_2 \int k_2ds = 0" where k_1, k_2 are the principal curvature functions according to Bishop frame of the curve α. Moreover, we illustrate some examples and draw their figures with Mathematica Programme.

math.DG↗
arXiv2012

f-Eikonal helix submanifolds and f-Eikonal helix curves

Let M{\subset}\mathbb{R}^{n} be a Riemannian helix submanifold with respect to the unit direction d{\in}\mathbb{R}^{n} and f:M{\to}\mathbb{R} be a eikonal function. We say that M is a f-eikonal helix submanifold if for each q{\in}M the angle between {\nabla}f and d is constant.Let M{\subset}\mathbb{R}^{n} be a Riemannian submanifold and α:I{\to}M be a curve with unit tangent T. Let f:M{\to}\mathbb{R} be a eikonal function along the curve α. We say that α is a f-eikonal helix curve if the angle between {\nabla}f and T is constant along the curve α. {\nabla}f will be called as the axis of the f-eikonal helix curve.The aim of this article is to give that the relations between f-eikonal helix submanifolds and f-eikonal helix curves, and to investigate f-eikonal helix curves on Riemannian manifolds.

math.DG↗