arXiv research

The study explores Bertrand and Mannheim curves in 4D Euclidean space for framed curves.

The paper characterizes curves in pseudo-Galilean 4-space.

The paper generalizes rectifying and normal curves in Lorentzian n-space.

Study finds formulas for special curves in complex spaces.

The paper explores Bertrand and framed curves in 3D space.

Study generalizes Lebesgue curves to new space-filling and fractal sets.

Study on curve shortening flow in 3D space curves, showing convexity preservation and avoidance principle.

Space curves with convex projections evolve smoothly until shrinking to a point.

Study of curves in dual space with constant curvature and torsion.

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 …

Salkowski \cite{salkow}, one century ago, introduced a family of curves with constant curvature but non-constant torsion (Salkowski curves) and a family of curves with constant torsion but non-constant curvature (anti-Salkowski curves) in Euclidean 3-space $\e^3$. In this paper, we adapt definition of such curves to ti…

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-…

Study on Bertrand lightcone framed curves in Lorentz-Minkowski 3-space.

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.

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 Si…

In this paper, space and timelike admissible Smarandache curves in the pseudo- Galilean 3-space are investigated. Also, Smarandache curves of the position vector of space and timelike arbitrary curve and some of its special curves in the pseudo- Galilean 3-space are obtained. To confirm our main results, some examples …

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 …

Study geodesics in constrained curve spaces, including elastic curves and concentric circles.

Same cohomology for curves with levels, proving stable range.

Defines Vassiliev complexity measures for open and closed curves in 3D space.

New discrete curves defined in space forms with geometric properties.

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

Defines non-parabolic curves in spatial hybrid space with applications.

Study pseudo-hyperkähler geometry of curves in hyperkähler twistor spaces.

Characterizes curves in totally umbilical surfaces of space forms.

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 t…

The paper finds two types of metric lines in curve spaces.

Study on triharmonic curves in Sol space with constant curvature and torsion.

Lagrangian curves in 4-space entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangrian curves with cons…

Study on polyharmonic curves on spheres and space forms.

In this paper, we give the generalization of the criterion for a 3-space curve to be closed given by [3] to an n-space curve in Minkowski space-time E_v^n. Furthermore, we apply this criterion for a curve lying on an oriented surface in the Minkowski space E_v^n as an application.

We show that the space of nonpositively curved metrics of a negatively curved manifold is highly non connected.

Paper studies geometric properties of nonlinear Lebesgue spaces.

In this paper we deal with curves with degeneration degree two in pseudo-Euclidean spaces of index two. We characterize Bertrand curves. We show a correspondence between the evolute of a null curve and the involute of a certain spacelike curve in the $6-$dimensional pseudo-Euclidean space of index two. Also we characte…

The paper classifies maximal translation surfaces in Lorentz-Minkowski space.

The paper studies curves in Finsler-like spaces and their properties.

Classifies self-similar curve shortening flows in hyperbolic 2-space.

New Einstein-Weyl spaces derived from hyperelliptic curves.

Of concern is the study of the space of curves in homogeneous spaces. Motivated by applications in shape analysis we identify two curves if they only differ by their parametrization and/or a rigid motion. For curves in Euclidean space the Square-Root-Velocity-Function (SRVF) allows to define and efficiently compute a d…

New condition prevents hyperbolic spaces from matching curve complexes.

Paper proves unique energy-minimizing curves in constrained spaces.

We consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We find the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. Finally, we obtain an example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.

In this article we investigate a first order reparametrization-invariant Sobolev metric on the space of immersed curves. Motivated by applications in shape analysis where discretizations of this infinite-dimensional space are needed, we extend this metric to the space of Lipschitz curves, establish the wellposedness of…

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…

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…

Backlund transformations of admissible curves in the Galilean 3-space and pseudo-Galilean 3-space and also spatial Backlund transformations of space curves in Galilean 4-space preserve the torsions under certain assumptions.

A space curve is determined by conformal arc-length, conformal curvature, and conformal torsion, up to Möbius transformations. We use the spaces of osculating circles and spheres to give a conformally defined moving frame of a curve in the Minkowski space, which can naturally produce the conformal invariants and the no…

In this paper, we give definitions and characterizations of normal and spherical curves in the dual space. We show that normal curves are also spherical curves in D^3.