arXiv research

The paper investigates polyharmonic helices in 3D solvable Lie group Sol_3 and Euclidean spheres.

Study on triharmonic curves in f-Kenmotsu manifolds.

Study on triharmonic curves in 3D spaces, proving their existence and classification.

Study on null helices in semi-Riemannian manifolds with special submanifolds.

In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, inc…

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.

We consider a unit speed timelike curve $α$ in Minkowski 4-space $E_1^4$ and denote the Frenet frame of $α$ by $\{T,N,B_1,B_2\}$. We say that $α$ is a generalized helix if one of the unit vector fields of the Frenet frame has constant scalar product with a fixed direction $U$ of $E_1^4$. In this work we study those hel…

In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, inc…

Study of spatial curves in generalized Minkowski spaces.

In this paper, we define some new associated curves as integral curves of a vector field generated by Frenet vectors of tangent indicatrix of a curve in Euclidean 3-space. We give some relationships between curvatures of these curves. By using these associated curves, we give some methods to construct helices and slant…

We consider a curve $α=α(s)$ in Minkowski 3-space $E_1^3$ and denote by $\{T,N,B}$ the Frenet frame of $α$. We say that $α$ is a slant helix if there exists a fixed direction $U$ of $E_1^3$ such that the function $<N(s),U>$ is constant. In this work we give characterizations of slant helices in terms of the curvature a…

In this paper, position vector of a spacelike general helix with respect to standard frame in Minkowski space E$^3_1$ are studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine position vector of an arbitrary spacelike general helix. In terms of solution, …

In this paper, position vectors of a time-like curve with respect to standard frame of Minkowski space E$^3_1$ are studied in terms of Frenet equations. First, we prove that position vector of every time-like space curve in Minkowski space E$^3_1$ satisfies a vector differential equation of fourth order. The general so…

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.

We consider a unit speed curve $α$ in Euclidean $n$-dimensional space $E^n$ and denote the Frenet frame by $\{v_1,...,v_n\}$. We say that $α$ is a cylindrical helix if its tangent vector $v_1$ makes a constant angle with a fixed direction $U$. In this work we give different characterizations of such curves in terms of …

We consider a unit speed curve $α$ in Euclidean four-dimensional space $E^4$ and denote the Frenet frame by $\{T,N,B_1,B_2\}$. We say that $α$ is a slant helix if its principal normal vector $N$ makes a constant angle with a fixed direction $U$. In this work we give different characterizations of such curves in terms o…

The study examines null curves in specific geometric manifolds and their properties.

We present the general theory of curves in conformal geometry using tractor calculus. This primarily involves a tractorial determination of distinguished parametrizations and relative and absolute conformal invariants of generic curves. The absolute conformal invariants are defined via a tractor analogue of the classic…

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 cur…

Defines and analyzes generalized normal ruled surfaces of curves in 3D space.

Study constructs Frenet curves using semi-symmetric metric connection.

In this paper, in Euclidean n -space, we investigate the relation between slant helices and spherical helices. Moreover, in E n, we show that a slant helix and the tangent indicatrix of the slant helix have the same axis (or direction). Also, we give the important relations between slant helices, spherical helices in E…

The study characterizes helices in Euclidean and hyperbolic spaces.

In this study, we give definitions and characterizations of eikonal slant helices, eikonal Darboux helices and non-normed eikonal Darboux helices in 3-dimensional pseudo- Riemannian manifold M . We show that every eikonal slant helix is also an eikonal Darboux helix for timelike and spacelike curves. Furthermore, we ob…

The paper defines Vn-slant helices in a lightlike cone and their curvature functions.

Study classifies polyharmonic helices in various space forms.

Study natural and conjugate mates of Frenet curves in Lie groups.

In this paper, we define slant helices in three dimensional Lie Groups with a bi-invariant metric and obtain a characterization of slant helices. Moreover, we give some relations between slant helices and their involutes, spherical images.

A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. We later use this association to characterize rectifying curve…

Generalized Frenet frames for singular space curves

In this paper, we study spinor Frenet equations in three dimensional Lie Groups with a bi-invariant metric. Also, we obtain spinor Frenet equations for special cases of three dimensional Lie groups.

Building on previous results on the quadratic helicity in magnetohydrodynamics (MHD) we investigate particular minimum helicity states. Those are eigenfunctions of the curl operator and are shown to constitute solutions of the quasi-stationary incompressible ideal MHD equations. We then show that these states have inde…

Characterizes concircular helices in space forms and ruled hypersurfaces.

We consider the Frenet-Serret geometry of null curves in a three and a four-dimensional Minkowski background. We develop a theory of deformations adapted to the Frenet-Serret frame. We exploit it to provide a Lagrangian description of the dynamics of geometric models for null curves.

Optimized concentric helices minimize the ropelength of non-alternating torus knots.

The helicity of a vector field is a measure of the average linking of pairs of integral curves of the field. Computed by a six-dimensional integral, it is widely useful in the physics of fluids. For a divergence-free field tangent to the boundary of a domain in 3-space, helicity is known to be invariant under volume-pr…

Solvents can induce helical knots in simulated biopolymer tubes.

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 representatio…

Weaved helices form mechanically stable 3D structures.

The Frenet frame generalizes the Park transform for multi-phase circuits.

The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this t…

In this paper, we give some characterizations for spacelike helices in Minkowski space-time. We find the differential equations characterizing the spacelike helices and also give the integral characterizations for these curves in Minkowski space-time.

In this work, we give some new characterizations for inclined curves and slant helices in n-dimensional Euclidean space E^{n}. Morever, we consider the pre-characterizations about inclined curves and slant helices and reconfigure them.

Characterizes concircular helices and surfaces in 3D space.

A new algebraic method for computing helicity is developed, by discovering a relationship between helicity of fluid mechanics and algebraic polynomial invariants of knot theory. We have constructed a topological invariant $t^{H\left(\mathcal{L}\right)}$ for a link $\mathcal{L}$ of knots, where $H$ is the helicity of a …

New constraints rule out some optimal domains for helicity maximisation.

In this study, we give definitions and characterizations of eikonal slant helix curves, eikonal Darboux helices and non-normed eikonal Darboux helices in three dimensional Riemannian manifold 3 M . We show that every eikonal slant helix is also an eikonal Darboux helix. Furthermore, we obtain that if the curve a is a n…

Study inextensible flows of curves in 4D pseudo-Galilean space and defines energy functions.