arXiv research

Study of spatial curves in generalized Minkowski spaces.

Weaved helices form mechanically stable 3D structures.

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 …

Helical ribbons arise in many biological and engineered systems, often driven by anisotropic surface stress, residual strain, and geometric or elastic mismatch between layers of a laminated composite. A full mathematical analysis is developed to analytically predict the equilibrium deformed helical shape of an initiall…

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.

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.

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…

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

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

Characterizes concircular helices in space forms and ruled hypersurfaces.

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…

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.

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…

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…

We introduce semi-helix hyper surfaces of Euclidean spaces. We also provide a local characterization of how these semi-helices are constructed.

Study finds all helical surfaces with a constant ratio of principal curvatures.

Researchers create exact minimal surfaces with helical motifs in biological structures.

Study isotropic Riemannian maps and helices along them.

We prove that any regular Casimir in 3D magnetohydrodynamics is a function of the magnetic helicity and cross-helicity. In other words, these two helicities are the only independent regular integral invariants of the coadjoint action of the MHD group $\text{SDiff}(M)\ltimes\mathfrak X^*(M)$, which is the semidirect pro…

We obtain explicit, isometry-invariant integral formulas for twisting, writhing and helicity, and prove the theorem LINK = TWIST + WRITHE on the 3-sphere and in hyperbolic 3-space. We then use these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl op…

A topologically minimal surface may be isotoped into a normal form with respect to a fixed triangulation. If the intersection with each tetrahedron is simply connected, then the pieces of this normal form are triangles, quadrilaterals, and helicoids. Helical pieces can have any number of positive or negative twists. We…

We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional $\mathcal I$ defined on exact divergence-free vector fields of class $C^1$ on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that $\mat…

Proves helicity is the only regular Casimir for 3D hydrodynamics.

In this note, we consider generalizations of the asymptotic Hopf invariant, or helicity, for Hamiltonian systems with one-and-a-half degrees of freedom and symplectic diffeomorphisms of a two-disk to itself.

Study cylindrical symmetric Finsler metrics with vanishing Douglas curvature.

In this paper we consider the spherical slant helices in $R^3$. More- over, we show how could be obtained to a spherical slant helix and we give some spherical slant helix examples in Euclidean 3-space.

Asymptotically cylindrical Ricci-flat manifolds play a key role in constructing Topological Quantum Field Theories. It is particularly important to understand their behavior at the cylindrical ends and the natural restrictions on the geometry. In this paper we show that an orientable, connected, asymptotically cylindri…

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.

Study reveals failure of uniqueness in dynamical invariants for 3D volume-preserving diffeomorphisms.

Study resolves flow through cylindrical singularities, proving nonfattening.

We study gluings of asymptotically cylindrical special Lagrangian submanifolds in asymptotically cylindrical Calabi--Yau manifolds. We prove both that there is a well-defined gluing map, and, after reviewing the deformation theory for special Lagrangians, prove that this gluing map defines a local diffeomorphism from m…

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, …

Study cylindrical symmetric Finsler metrics that are projectively flat.

In this paper, we prove that the position vector of every space curve satisfies a vector differential equation of fourth order. Also, we determine the parametric representation of the position vector $ψ=\Big(ψ_1,ψ_2,ψ_3\Big)$ of general helices from the intrinsic equations $κ=κ(s)$ and $τ=τ(s)$ where $κ$ and $τ$ are th…

In this paper, we give the definitions and characterizations of quaternionic Salkowski, quaternionic anti-Salkowski and quaternionic similar curves in the Euclidean spaces E^3 and E^4. We obtain relationships between these curves and some special quaternionic curves such as quaternionic slant helices and quaternionic B…

This paper studies mean curvature flows near cylindrical singularities.