arXiv research

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

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

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

Proves minimum number of normals to curves in 3D space.

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.

Distance, normals, and double normals for real plane curves with singularities

Study on algebraic curves' invariants and vanishing criteria.

Defines weak normals for irregular curves in high-dimensional spaces.

Study on curves around a Whitney umbrella focusing on geodesic and normal curvatures.

Simplified proof classifies surfaces using normal curves.

Study normal curves in sub-Finsler Lie groups with specific norms, focusing on branching and face stability.

In this paper, we study normal homogeneous Finsler spaces. We first define the notion of a normal homogeneous Finsler space, using the method of isometric submersion of Finsler metrics. Then we study the geometric properties. In particular, we establish a technique to reduce the classification of normal homogeneous Fin…

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

In this paper, we consider a regular curve on an oriented surface in Euclidean 3-space with the Darboux frame $\{\mathsf{T},\mathsf{V},\mathsf{U}\}$ along the curve, where $\mathsf{T}$ is the unit tangent vector field of the curve, $\mathsf{U}$ is the surface normal restricted to the curve and $\mathsf{V}=\mathsf{U}\ti…

The paper studies circular evolutes and involutes of framed curves in Euclidean space.

Algorithm counts intersections of normal curves efficiently.

We deal with a notion of weak binormal and weak principal normal for non-smooth curves of the Euclidean space with finite total curvature and total absolute torsion. By means of piecewise linear methods, we first introduce the analogous notation for polygonal curves, where the polarity property is exploited, and then m…

The presented paper is devoted to study the curvature and torsion of slant Frenet curves in 3-dimensional normal almost paracontact metric manifolds. Moreover, in this class of manifolds, properties of non- Frenet slant curves (with null tangents or null normals) are obtained. The achieved results are illustrated by ex…

Curves in Carnot groups avoid compact sets, growing at least $t^{1/s}$.

We defined normal and rectifying curves in Pseudo-Galilean Space G_3^1. Also we obtained some characterizations of this curves in G_3^1.

The paper studies magnetic curves in $C$-manifolds and their properties.

The paper proves every closed curve on a bouquet of circles lifts to finite covers.

A new probabilistic polygonal curve representation using Gaussian Mixture Models.

In this article, we study rectifying curves in arbitrary dimensional Euclidean space. A curve is said to be a rectifying curve if, in all points of the curve, the orthogonal complement of its normal vector contains a fixed point. We characterize rectifying curves in the $n$-dimensional Euclidean space in different ways…

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…

The normal map of curves is analyzed as a vector field on a cylinder.

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 study explores normal generators for mapping class groups and their properties.

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

The multiplier spectral curve of a conformal torus in the 4-sphere is essentially, see arXiv:0712.2311, given by all Darboux transforms of the conformal torus. In the particular case when the conformal immersion is a Hamiltonian stationary torus in Euclidean 4-space, the left normal of the immersion is harmonic, hence …

Normal forms of almost complex structures in a neighborhood of pseudoholomorphic curve are considered. We define normal bundles of such curves and study the properties of linear bundle almost complex structures. We describe 1-jet of the almost complex structure along a curve in terms of its Nijenhuis tensor. For pseudo…

Study of normal and tangent maps to frontals.

The concept of a normal surface in a triangulated, compact 3-manifold was generalised by Thurston to a spun-normal surface in a non-compact 3-manifold with ideal triangulation. This paper defines a boundary curve map which takes a spun-normal surface to an element of the direct sum of the first homology groups of the v…

In this paper we define and study pseudoholomorphic vector bundles structures, particular cases of which are tangent and normal bundle almost complex structures. These are intrinsically related to the Gromov D-operator. As an application we deduce normal forms of 1-jets of almost complex structures along a submanifold.…

In the present paper we study pseudo-Riemannian submanifolds which have 3-planar geodesic normal sections.We consider W-curves (helices) on pseudo-Riemannian submanifolds. Finally, we give neccessary and sufficient condition for a normal section to be a W-curve on pseudo-Riemannian submanifolds.

Study curves on a Whitney umbrella using geometric invariants.

In this paper we study slant null curves with respect to the original parameter on 3-dimensional normal almost contact B-metric manifolds with parallel Reeb vector field. We prove that for non-geodesic such curves there exists a unique Frenet frame for which the original parameter is distinguished. Moreover, we obtain …

We prove that a normal vector field along a curve in R3 is rotation minimizing (RM) if and only if it is parallel respect to the normal connection. This allows us to generalize all the results of RM vectors and frames to curves immersed in Riemannian manifolds.

We construct normal rationally connected varieties (of arbitrarily large dimension) not containing any smooth rational curves.

In this paper, we study the spherical indicatrices of W-direction curves in three dimensional Euclidean space which were defined by using the unit Darboux vector field W of a Frenet curve, in [11]. We obtain the Frenet apparatus of these spherical indicatrix curves and the characterizations of being general helix and s…

Study of null φ-slant curves in specific 3D manifolds.

We study the equilibrium positions of three points on a convex curve under influence of the Coulomb potential. We identify these positions as orthotripods, three points on the curve having concurrent normals. This relates the equilibrium positions to the caustic (evolute) of the curve. The concurrent normals can only m…

We construct normal forms for Lorentzian metrics on Engel distributions under the assumption that abnormal curves are timelike future directed Hamiltonian geodesics. Then we indicate some cases in which the abnormal timelike future directed curve initiating at the origin is geometrically optimal. We also give certain e…

This paper extends Jacobi field theory to Jacobi curves and their curvatures.

The article constructs Feynman propagators for normally hyperbolic operators on curved spacetimes.

Every normal subgroup of Cantor tree's mapping class group is geometric.

It is shown that various questions about the existence of simple closed curves in normal subgroups of surface groups are undecidable.

The main drawback of the Frenet frame is that it is undefined at those points where the curvature is zero. Further- more, in the case of planar curves, the Frenet frame does not agree with the standard framing of curves in the plane. The main drawback of the Bishop frame is that the principle normal vector N is not in …