Yusuf Yayli, Semra Saracoglu
In this study, the new characterizations of special curves are investigated without using the curvatures of these special curves: general helices, slant helices, Bertrand curves, Mannheim curves. The curvatures are given by the help of the norms of the derivatives of Frenet vectors.
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In this study, we investigate Bertrand curves in three dimensional dual space D3 and we obtain the characterizations of these curves in dual space D3. Also we show that involutes of a curve constitute Bertrand pair curves.
In this paper, new representations of a Bertrand curve pair in three dimensional Lie groups with bi-invariant metric are given. Besides, the spherical indicatrices of a Bertrand curve pair are obtain and the relations between the spherical indicatrices and new representations of Bertrand curve pair are shown.
In this paper we study null Bertrand curves in $R_{1}^{4}$ under the assumption the curve has a Cartan frame. We show that if the derivative vectors of the null Cartan curve in $R_{1}^{4}$ is linearly independent, then this curve is not a Bertrand curve. Since then the already known notion of null Bertrand curves in $R_{1}^{4}$ occurs only if the derivative vectors of the curve is linearly dependent. We will introduce an idea of Bertrand curves and abiding by this idea we bring to light under which conditions a null Cartan curve in $R_{1}^{4}$ is a Bertrand curve.
In this paper we consider the idea of Bertrand curves for curves lying on surfaces and by considering the Darboux frames of them we define these curves as Bertrand D-curves and give the characterizations for these curves. We also find the relations between the geodesic curvatures, the normal curvatures and the geodesic torsions of these associated curves. Furthermore, we show that the definition and the characterizations of Bertrand D-curves include those of Bertrand curves in some special cases.
Bertrand Legendre curves in the unit tangent bundle over Euclidean plane
本文研究了正则平面曲线及其勒内曲线的关联曲线，探讨了伯特兰正则曲线和伯特兰勒内曲线的性质，提出存在条件及逆操作，并定义了勒内曲线集之间的映射，证明其在等价关系下的双射性。
In this paper, it is proved that, no special timelike Frenet curve is a Bertrand curve in $\mathbb{E}_2^4$ and also, in $\mathbb{E}_ν^{n+1}$ $ ({n \ge 3})$, such that the notion of Bertrand curve is definite only in $\mathbb{E}_1^2$ and $\mathbb{E}_1^3$. Therefore, a generalization of timelike Bertrand curve is defined and called as timelike (1,3)-Bertrand curve in $\mathbb{E}_2^4$. Moreover, the characterization of timelike (1,3)-Bertrand curve is given in $\mathbb{E}_2^4$.
In this paper, we consider the idea of Bertrand curves for curves lying on surfaces in Minkowski 3-space. By considering the Darboux frame, we define these curves as Bertrand D-curves and give the characterizations for those curves. We also find the relations between the geodesic curvatures, the normal curvatures and the geodesic torsions of these associated curves. Furthermore, we show that in Minkowski 3-space, the definition and the characterizations of Bertrand D-curves include those of Bertrand curves in some special cases.
In [18], L. R. Pears proved that Bertrand curves in E-n(n > 3) are degenerate curves. This result restate in [16] by Matsuda and Yorozu. They proved that there is no special Bertrand curves in E-n(n > 3) and they define new kind of Bertrand curves called (1, 3)-type Bertrand curves in 4-dimensional Euclidean space. In this study, we define a quaternionic Bertrand curve ?(4) in Euclidean space E4 and investigate its properties for two cases. In the first case; we consider quaternionic Bertrand curve in the Euclidean space E4 for r-K = 0 where r is the torsion of the spatial quaternionic curve ?…
Bertrand types of regular curves and Bertrand framed curves in the Euclidean 3-space
本文研究欧几里得3维空间中切线、主法线或副法线重合的 Bertrand 型曲线，并推广到带奇点的 Bertrand 标架曲线，给出各类存在条件。
If there exists a quaternionic Bertrand curve in E4, then its torsion or bitorsion vanishes. So we can say that there is no quaternionic Bertrand curves whose torsion and bitorsion are non-zero. Hence by using the method which is given by Matsuda and Yorozu [13], we give the definition of quaternionic (1,3)-Bertrand curve according to Type 2-Quaternionic Frame and obtain some results about these curves.
A Bertrand curve in the 4-dimensional Euclidean space is a space curve whose first normal line is the same as the first normal line of another curve. On the other hand, a Mannheim curve in the 4-dimensional Euclidean space is a space curve whose first normal line is the same as the second or third normal line of another curve. By definitions, another curve is a parallel curve with respect to the direction of the first normal vector. As smooth curves with singular points, we consider framed curves in the Euclidean space. Then we define and investigate Bertrand and Mannheim curves of framed curv…
Bertrand lightcone framed curves in the Lorentz-Minkowski 3-space
本文研究Lorentz-Minkowski三维空间中混合类型曲线，引入光锥标架定义Bertrand光锥标架曲线，阐明其存在条件，并发现混合类型曲线的伪圆渐伸线和渐屈线。
In the present paper, we give new Frenet formulas for the Bertrand partner curve by taking the advantage of relations between curvatures and a curve itself. Then making use of these formulas we write the differential equations and sufficient conditions of harmonicity of the Bertrand partner curve in terms of the main curve. Finally, we exemplify our assertions on the curve helix to see how the formulas we developed work.
In this work, we studied the properties of the spherical indicatrices of a Bertrand curve and its mate curve and presented some characteristic properties in the cases that Bertrand curve and its mate curve are slant helices, spherical indicatrices are slant helices and we also researched that whether the spherical indicatrices made new curve pairs in the means of Mannheim, involte-evolute and Bertrand pairs. Further more, we investigated the relations between the spherical images and introduced new representations of spherical indicatrices.
In this paper, we give the defination of harmonic curvature function some special curves such as helix, slant curves, Mannheim curves and Bertrand curves. Then, we recall the characterizations of helices [8], slant curves (see [19]) and Mannheim curves (see [12]) in three dimensional Lie groups using their harmonic curvature function. Moreover, we define Bertrand curves in a three dimensional Lie group G with a bi-invariant metric and the main result in this paper is given as (Theorem 3.4): A curve ?α with the Frenet apparatus {T,N,B,κ,τ} in G is a Bertrand curve if and only if λκ+μκH=1 where …
This paper deals with the natural lift curves and the geodesic sprays for the spherical indicatrices of the timelike-spacelike Bertrand couple on the tangent bundle or in Minkowski 3-space and then give some new characterizations for these curves. Additionally we illustrate an example of our main results.
In this paper we define a new type of 2-degenerate Cartan curves in Minkowski spacetime $R_{1}^{4}$. We prove that this type of curves contain only the polynomial functions as its components whose third derivative vanish completely. No curve with acceleration zero in $R_{1}^{4}$ is a 2-degenerate Cartan curve, therefore we show that the type of curves that we search for must contain polynomials of degree two among its components.
Time-like constant slope surfaces and space-like Bertrand curves in Minkowski 3-space
研究了德西特空间中类时间曲线的几何性质，并揭示了类空间伯兰德曲线与螺旋线之间的关系，以及类时间常斜面与类空间伯兰德曲线的联系。
In [1], we gave a method for constructing Bertrand curves from the spherical curves in 3 dimensional Minkowski space. In this work, we construct the Bertrand curves corresponding to a spacelike geodesic and a null helix in Minkowski 4 spacetime.
In this article, we investigate Bertrand curves corresponding to the spherical images of the tangent, binormal, principal normal and Darboux indicatrices of a space curve in Euclidean 3-space. As a result, in case of a space curve is a general helix, we show that the curves corresponding to the spherical images of its the tangent indicatrix and binormal indicatrix are both Bertrand curves and circular helices. Similarly, in case of a space curve is a slant helix, we demonstrate that the curve corresponding to the spherical image of its the principal normal indicatrix is both a Bertrand curve a…
On space-like constant slope surfaces and Bertrand curves in Minkowski 3-space
本文研究了德西特二维空间和双曲空间中的空间样曲线与伯恩特曲线的几何性质及其与常斜曲面的关系。
This paper is devoted to the study of AW(k)-type curves according to the equiform differential geometry of the pseudo-Galilean space. We show that equiform Bertrand curves are circular helices or isotropic circles of the pseudo-Galilean space. Also, there are equiform Bertrand curves of AW(3) and weak AW(3)-types. Moreover, we give the relations between the equiform curvatures of these curves. Finally, examples of some special curves are given and plotted.
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 study, we introduce a new approach to curve pairs by using integral curves. We consider the direction curve and donor curve to study curve couples such as involute-evolute curves, Mannheim partner curves and Bertrand partner curves. We obtain new methods to construct partner curves of a unit speed curve and give some applications related to helices, slant helices and plane curves.
In this work, first, we express some characterizations of helices and ccr curves in the Euclidean 4-space. Thereafter, relations among Frenet-Serret invariants of Bertrand curve of a helix are presented. Moreover, in the same space, some new characterizations of involute of a helix are presented.
In this paper, we introduce a new approach to non-lightlike curve pairs by using integral curves in Minkowski 3-space. We consider direction curve and donor curve to study non-lightlike curve couples such as involute-evolute curves, Mannheim partner curves and Bertrand partner curves. We obtain new methods to construct partner curves of a unit speed non-lightlike curve and give some applications related to helices and slant helices.
In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. Later, we characterize plane and spherical curves as curves whose position vector lies, up to a translation, on a moving plane spanned by their unit tangent and an RM vector field. Finally, as an application, we characterize Bertrand curves and slant helices as curves whose so-cal…
A Note on the Cesàro Condition for Curves in the Three-Dimensional Heisenberg Group
该研究针对三维Heisenberg群中水平正则曲线，基于适配Frenet-Serret公式推导三类相关曲线的性质，分离了固定Reeb方向对欧氏空间曲线经典构造的影响。
In the present paper, we consider a position vector of an arbitrary curve in the three-dimensional Galilean 3-space. Furthermore, we give some conditions on the curvatures of this arbitrary curve to study special curves and their Smarandache curves. Finally, in the light of this study, some related examples of these curves are provided and plotted
Special curves and their characterizations are one of the main area of mathematicians and physicians. As a special curve we will mainly focus on Mannheim curve which has the following relation: k1=β(k1^2+k2^) where k1 and k2 are curvature and torsion, respectively. In the present paper we define Mannheim curves for 4-dimensional Galilean space and investigate some characterization of it.
We characterize the biharmonic curves in the special linear group $SL(2,R)$. In particular, we show that all proper biharmonic curves in $SL(2,R)$are helices and we give their explicit parametrizations as curves in the pseudo-Euclidean space $R^4_2$.
In this paper, we investigate special curves on a strong r-helix submanifold in Euclidean n-space E n. Also, we give the important relations between strong r-helix submanifolds and the special curves such as line of curvature, geodesic and slant helix.
In this paper, we define a new special curve in Euclidean 3-space which we call {\it $k-$slant helix} and introduce some characterizations for this curve. This notation is generalization of a general helix and slant helix. Furthermore, we have given some necessary and sufficient conditions for the $k-$slant helix.
In this paper, we take into account the opinion of involute-evolute curves which lie on fully surfaces and by taking into account the Darboux frames of them we illustrate these curves as special involute-evolute partner D-curves in E3. Besides, we find the relations between the normal curvatures, the geodesic curvatures and the geodesic torsions of these curves. Finally, some consequences and examples are given.
The aim of this paper is to give some characterizations for N-Legendre and N-slant curves in the unit tangent bundles of surfaces endowed with natural diagonal lifted structures.
We investigate contact magnetic curves in the real special linear group of degree 2. They are geodesics of the Hopf tubes over the projection curve. We prove that periodic contact magnetic curves in SL(2,R) can be quantized in the set of rational numbers. Finally, we study contact homogeneous magnetic trajectories in SL(2,R) and show that they project to horocycles in H2(-4).
In this study, we introduce a new type of surface curves called D-type curve. This curve is defined by the property that the unit Darboux vector W0 of a space curve r(s) and unit surface normal n along the curve r(s) satisfy the condition <n,W0>=constant. We point out that a D-type curve is a geodesic curve or an asymptotic curve in some special cases. Then, by using the Frenet vectors and parametric representation of a surface pencil as a linear combination of the Frenet vectors, we investigate necessary and sufficient condition for a curve to be a D-type curve on a surface pencil. Moreover, …
From tropical curves to special Lagrangians
该研究通过粘合构造匹配特殊拉格朗日局部模型与热带曲线的组合结构，证明单位边权的局部平面热带曲线可实现为$T^*T^n$中特殊拉格朗日子流形族的缩放矩映射像极限，建立了两类几何的直接联系。
We describe the Special Kähler structure on the base of the so-called Hitchin system in terms of the geometry of the space of spectral curves. It yields a simple formula for the Kähler potential. This extends to the case of a singular spectral curve and we show that this defines the Special Kähler structure on certain natural integrable subsystems. Examples include the extreme case where the metric is flat.
In this study, we determine some special Smarandache curves in E1 3.We give some characterizations consequences of Smarandache curves.
In this study, we determine some special Smarandache curves according to Darboux frame in E3. We give some characterizations and consequences of Smarandache curves.
We give a different formulation for describing maximal surfaces in Lorentz-Minkowski space, $\mathbb{L}^3$, using the identification of $\mathbb L^3$ with $\mathbb C\times \mathbb R$. Further we give a different proof for the singular Björling problem for the case of closed real analytic null curve. As an application, we show the existence of maximal surface which contains a given curve and has a special singularity.
In this study, by using the facts that det(α^{(1)}, α^{(2)}, α^{(3)}) = 0 characterizes plane curve, and det(α^{(2)}, α^{(3)}, α^{(4)}) = 0 does a curve of constant slope, we give the special space curves that are characterized by det(α^{(3)}, α^{(4)}, α^{(5)}) = 0, in different approaches. We find that the space curve is Salkowski if and only if det(α^{(3)}, α^{(4)}, α^{(5)}) = 0. The approach we used in this paper is useful in understanding the role of the curves that are characterized by det(α^{(3)}, α^{(4)}, α^{(5)})=0 in differential geometry.
The investigation of some special curves in the pseudo-Galilean 4-space $G_{1}^{4} $
本文依据曲率函数刻画伪伽利略4-空间 $G_{1}^{4}$ 中的任意容许曲线，并给出位置曲线、从切曲线、斜螺旋线和球面曲线等特殊曲线的分类及充要条件。
We classify all special homogeneous curves. A special homogeneous curve $\mathcal{H}$ consists of connected components of the hyperbolic points in the level set $\{h=1\}$ of a homogeneous polynomial $h$ in two real variables of degree at least three, and admits a transitive group action of a subgroup $G\subset\mathrm{GL}(2)$ on $\mathcal{H}$ that acts via linear coordinate change.
In this paper, we get the time evolution equations of the curvature and torsion of the evolving spacelike curves in the Minkowski space. Also, we give inextensible evolutions of timelike ruled surfaces that are produced by the timelike normal and spacelike binormal vector fields of spacelike curve and derive the necessary conditions for an inelastic surface evolution. Then, we compute the coefficients of the first and second fundamental forms, the Gauss and mean curvatures for timelike special ruled surfaces. As a result, we give applications of the evolution equations for the curvatures of th…
We classify maximal quartic generalised projective special real curves up to equivalence. A maximal quartic generalised projective special real curve consists of connected components of the intersection of the hyperbolic points of a quartic homogeneous real polynomial $h:\mathbb{R}^2\to\mathbb{R}$ and its level set $\{h=1\}$. Two such curves are called equivalent if they are related by a linear coordinate transformation. As an application of our results we prove that quartic generalised projective special real manifolds that are homogeneous spaces have non-regular boundary behaviour, meaning t…
Holomorphic null curves in the special linear group
本文建立 ${\rm SL}_2(\mathbb{C})$ 中全纯零浸入的多种逼近与插值定理，并证明任意开黎曼曲面均可真全纯零嵌入该群，从而解决相关公开问题。
A Hermitian metric $ω$ on a complex manifold is called SKT or pluriclosed if $dd^cω=0$. Let M be a twistor space of a compact, anti-selfdual Riemannian manifold, admitting a pluriclosed Hermitian metric. We prove that in this case M is Kähler, hence isomorphic to $\C P^3$ or a flag space. This result is obtained from rational connectedness of the twistor space, due to F. Campana. As an aside, we prove that the moduli space of rational curves on the twistor space of a K3 surface is Stein.
We discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The representation of the shape as a path in the special Euclidean algebra is intrinsic to the description of the mechanical properties of a rod, since it is given directly in terms of the strain fields that stimulate the elastic response of special Cosserat rods. Moreover, such a representation leads naturally to discretization schemes that avoid the need for t…
Weakly Special Manifolds with no rational curves
本文在假设丰盛猜想和终端卡拉比-约当簇上存在稠密有理曲线的前提下，证明了无有理曲线的复投影弱特殊流形是阿贝尔簇的 étale 商。若流形包含稠密整个曲线，则在假设朗猜想下同样成立，表明非超广复投影流形包含阿贝尔簇的像。
We propose a notion of distance between two parametrized planar curves, called their discrepancy, and defined intuitively as the minimal amount of deformation needed to deform the source curve into the target curve. A precise definition of discrepancy is given as follows. A curve of transformations in the special Euclidean group SE(2) is said to be admissible if it maps the source curve to the target curve under the point-wise action of SE(2) on the plane. After endowing the group SE(2) with a left-invariant metric, we define a relative geodesic in SE(2) to be a critical point of the energy fu…
Rank-2 wobbly bundles from special divisors on spectral curves
本文研究了从谱曲上的特殊除子出发的秩2摇摆丛的构造与性质，提出半稳定丛为摇摆丛的充分条件，并探讨了摇摆定位的奇点来源。
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