arXiv research

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…

The paper studies curves of constant-ratio in pseudo-Galilean space.

Curves with constant torsion can be deformed arbitrarily.

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 re…

Study constructs closed curves with constant curvature on cylinders and tori.

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

Flow deforms locally convex curves to curves of constant k-order width.

Curves with constant curvature are flexible and can be deformed.

The abstract finds conditions for creating curves of constant curvature.

The paper classifies curves in dual affine and Lorentz-Minkowski planes with constant curvature.

Study curves of constant breadth in a specific 3D manifold.

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

Ruled surfaces with Ricci metrics use curves of constant torsion.

We give an explicit construction of a closed curve with constant torsion and everywhere positive curvature. We also discuss the restrictions on closed curves of constant torsion when they are constrained to lie on convex surfaces.

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

Proves existence of curves with constant curvature in a sphere.

We prove the existence of embedded closed constant curvature curves on convex surfaces.

Unified description of aesthetic curves through self-affinities.

Square inscribed in a curve made of two graph functions.

We determine the asymptotic behavior of the optimal Lipschitz constant for the systole map from Teichmuller space to the curve complex.

We study the geometric properties of Darboux transforms of constant mean curvature (CMC) surfaces and use these transforms to obtain an algebro-geometric representation of constant mean curvature tori. We find that the space of all Darboux transforms of a CMC torus has a natural subset which is an algebraic curve (call…

We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of these curves being centered at a point of a regular curve and contained in its normal p…

The paper classifies and constructs rotational surfaces with constant astigmatism in space forms.

We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global behavior may be periodic or the curve may be dense in a Clifford torus embedded…

In this paper, we classify the class of constant weighted curvature curves in the plane with a log-linear density, or in other words, classify all traveling curved fronts with a constant forcing term in $\Bbb R^2.$ The classification gives some interesting phenomena and consequences including: the family of curves conv…

Survey on geodesics on tetrahedra in curved spaces.

Researchers classify special curved spheres in a complex space.

Study on polyharmonic curves on spheres and space forms.

Conditions ensure constant curvature in negatively curved manifolds.

J.Eells and L. Lemaire introduced k-harmonic maps, and T. Ichiyama, J. Inoguchi and H.Urakawa showed the first variation formula. In this paper, we describe the ordinary differential equations of $3$-harmonic curves into a Riemannian manifold with constant sectional curvature, and show biharmonic curve is k-harmonic cu…

A Steiner deltoid maintains constant area across all boundary points of an ellipse.

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

Curves in ${\mathbb R}^n$ for which the ratios between two consecutive curvatures are constant are characterized by the fact that their tangent indicatrix is a geodesic in a flat torus. For $n= 3,4$, spherical curves of this kind are also studied and compared with intrinsic helices in the sphere.

The paper classifies surfaces with constant skew curvature in 3-space forms.

General area-preserving motion of polygonal curves is formulated as a system of ODEs. Solution polygonal curves belong to a prescribed polygonal class, which is similar to the admissible class used in the crystalline curvature flow. The ODEs are discretized implicitly in time keeping a given constant area speed while s…

In this paper we study properties of the area evolute (AE) and the center symmetry set (CSS) of a convex planar curve $γ$. The main tool is to define a Minkowski plane where $γ$ becomes a constant width curve. In this Minkowski plane, the CSS is the evolute of $γ$ and the AE is an involute of the CSS. We prove that the…

The paper defines and classifies special curves in Riemannian manifolds.

Improved bounds on curve filling areas in Banach spaces, leading to rigidity of Pu's inequality.

We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting of geodesics. Furthermore, the method gives inequalities with a uniform constant…

Considering Riemannian submersions, we find necessary and sufficient conditions for when sub-Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvatures. We describe a canonical extension of the sub-Riemannian metric and study geometric p…

Paper explores folding patterns of curved creases preserving their geometric properties.

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

Constructs orthogonal coordinates in curved spaces.

Study biharmonic curves in warped product manifolds with curvature analysis.

ccc-Autoevolutes are closed curves congruent to their evolutes, constructed via symmetry.

Study geometric flow on curves with positive torsion, finding stationary solutions and their stability.

The paper tackles Kakeya and Nikodym sets on curved manifolds, reducing problems to Euclidean space.

The paper explores geometric properties of interception curves on planes and spheres.