The study explores Bertrand and Mannheim curves in 4D Euclidean space for framed curves.
problem Exploring Bertrand and Mannheim curves in 4D Euclidean space for framed curves.
method Defining and investigating Bertrand and Mannheim curves of framed curves in 4D Euclidean space.
result Bertrand and Mannheim curves exist even for framed curves in 4D Euclidean space, contrary to regular curves.
The paper characterizes curves in pseudo-Galilean 4-space.
problem Characterizing curves in the pseudo-Galilean 4-space G14. method Investigation and characterisation of admissible curves in terms of curvature functions.
result Necessary and sufficient conditions for admissible rectifying curves in G14. The paper generalizes rectifying and normal curves in Lorentzian n-space.
problem Characterizing and classifying g−rectifying and g−normal curves in Lorentzian n-space. method Introducing a g−position vector field and defining g−rectifying and g−normal curves based on this field. result Comprehensive characterization and classification of g−rectifying and g−normal curves. Closed finite gap curves are dense in Sobolev spaces of curves in hyperbolic and Euclidean 3-spaces.
problem Density of closed finite gap curves in Sobolev spaces.
method Proof of density in Sobolev spaces for curves in hyperbolic and Euclidean 3-spaces.
result Closed finite gap curves are dense in Sobolev spaces of curves in hyperbolic and Euclidean 3-spaces.
Study finds formulas for special curves in complex spaces.
problem Understanding special curves in complex spaces.
method Obtained explicit formulas for Killing magnetic curves.
result Explicit formulas for Killing magnetic curves in non-flat Lorentzian-Heisenberg spaces.
The paper explores Bertrand and framed curves in 3D space.
problem Characterizing Bertrand and framed curves in Euclidean 3-space.
method Analyzing curves where tangent, normal, or binormal lines match another curve's lines.
result Conditions for the existence of Bertrand and framed curves are clarified.
Study uses isotropic projection to link plane and null curves in Minkowski space.
problem Understanding the geometry of null curves in Minkowski 3-space.
method Utilizes isotropic projection of Laguerre geometry to establish a correspondence.
result Alternative description of plane curves congruent to a given one.
Study generalizes Lebesgue curves to new space-filling and fractal sets.
problem Generating space-filling curves from planar substitutions.
method Generalized Lebesgue's construction to new curves and fractal sets.
result Some substitutions create relatively dense fractal-like sets.
Study on curve shortening flow in 3D space curves, showing convexity preservation and avoidance principle.
problem Analyzing the behavior of space curves under curve shortening flow in R3. method Analysis of properties of space curves evolved by the curve shortening flow, including convexity preservation and avoidance principle.
result Orthogonal projections of space curves remain convex, and the Avoidance principle is shown for spherical curves.
Study finds conditions for Legendre curves to be interpolating sesqui-harmonic in Sasakian space forms.
problem Characterizing Legendre curves in Sasakian space forms.
method Analyzes necessary and sufficient conditions for Legendre curves to be interpolating sesqui-harmonic.
result Obtains an example of an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.
Space curves with convex projections evolve smoothly until shrinking to a point.
problem Evolution of space curves with convex projections.
method Space Curve Shortening flow.
result Convex projections remain convex throughout the evolution.
Generalizes SRVF to curves in homogeneous spaces for efficient distance computation.
problem Distance on curves in homogeneous spaces.
method Generalizes SRVF to homogeneous spaces and proves existence of optimal reparametrizations.
result Efficient computation of quotient distance on curves in homogeneous spaces.
Study of curves in dual space with constant curvature and torsion.
problem Classifying curves in dual space with specific geometric properties.
method Defined curvature and torsion for curves in dual space, classified curves with constant properties, and proved existence theorems.
result Established fundamental theorem of existence for dual curves with prescribed 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 …
Study of spatial curves in generalized Minkowski spaces.
problem Characterizing and invariants of spatial curves in non-Euclidean spaces.
method Derive Frenet-type results and invariants for spatial curves in generalized Minkowski spaces.
result Characterization of cylindrical helices and rectifying curves in generalized Minkowski spaces.
New solitons found in curve metric space.
problem Elastic metric on curve spaces for shape analysis.
method Reparametrization-invariant Sobolev metric extension, geodesic equation analysis.
result Geodesics are soliton solutions for elastic metric.
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 S12. In addition, we investigate the invariants and geometric properties of these curves. Afterwards, we show that space-like Bertrand curves and time-…
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…
Study on Mannheim curves in 3D space with modified frame.
problem Characterizing Mannheim curves in modified orthogonal frames.
method Investigation of Mannheim pairs, Frenet-Mannheim, and Weakened Mannheim curves.
result Characterizations of Mannheim curves in modified frames.
Study on Bertrand lightcone framed curves in Lorentz-Minkowski 3-space.
problem Analyzing mixed types of curves with singular points in Lorentz-Minkowski 3-space.
method Using lightcone frame to consider Bertrand types for lightcone framed curves.
result Existence conditions of Bertrand lightcone framed curves in all cases.
The study characterizes rectifying curves in n-dimensional space.
problem Understanding rectifying curves in arbitrary dimensions.
method Characterization through various conditions and constructions.
result Different ways to characterize rectifying curves in n-dimensional Euclidean space.
Defining Lorentzian Sabban frame of the unit speed time-like curves on de Sitter 2-space S12 and introducing space-like height function on the unit speed time-like curves on S12, the invariants of the unit speed time-like curves on S12 and geometric properties of de Si…
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.
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 …
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 …
Study classifies magnetic curves in Galilean space tied to Killing vectors.
problem Classifying magnetic curves in Galilean space.
method Complete classification of magnetic curves associated to Killing vector fields.
result Completely classified magnetic curves in Galilean 3-space.
Study geodesics in constrained curve spaces, including elastic curves and concentric circles.
problem Geodesics in constrained curve spaces with Sobolev metrics.
method Intrinsic and constructive approaches.
result Construct geodesics in elastic curve and concentric circle spaces.
Defines Vassiliev complexity measures for open and closed curves in 3D space.
problem Measuring complexity of curves in 3D space.
method Using enhanced Jones polynomial coefficients and Gauss code diagrams.
result Second Vassiliev measure converges to knot invariants as curve ends coincide.
Same cohomology for curves with levels, proving stable range.
problem Stability of cohomology in moduli spaces with level structures.
method Proving cohomology equality through stable range analysis.
result Rational cohomology of moduli space with levels matches ordinary space.
New discrete curves defined in space forms with geometric properties.
problem Defining discrete elastic and constrained elastic curves in space forms.
method Extending discrete Euclidean curvature to space forms and using Bäcklund transformations.
result Discrete elastic and constrained elastic curves are elements of a curve hierarchy.
Study on triharmonic curves in 3D spaces, proving their existence and classification.
problem Characterizing triharmonic curves in 3D homogeneous spaces.
method Analyzing curves with constant curvature in Riemannian manifolds, focusing on Frenet helices and space forms.
result Classification of triharmonic Frenet helices in space forms and Bianchi-Cartan-Vranceanu spaces.
Defines non-parabolic curves in spatial hybrid space with applications.
problem Defining and analyzing non-parabolic spatial hybrid framed curves.
method Definition and proof of existence and uniqueness theorem for non-parabolic spatial hybrid framed curves.
result Existence and uniqueness theorem for non-parabolic spatial hybrid framed curves.
Characterizes curves in totally umbilical surfaces of space forms.
problem Identifying curves in totally umbilical surfaces of space forms.
method Analyzes curvature and torsion conditions for curves in $\h^3$ and $\s^3$.
result Necessary and sufficient conditions for curves in totally umbilical surfaces.
Study of helices and Bertrand curves with modified orthogonal frame.
problem Classification of curves with modified orthogonal frame.
method Analysis of helices and Bertrand curves with modified orthogonal frame.
result Classification results of helices and Bertrand curves.
Study pseudo-hyperkähler geometry of curves in hyperkähler twistor spaces.
problem Understanding the geometry of rational curves in twistor spaces.
method Investigate pseudo-hyperkähler geometry of higher degree rational curves.
result Characterize the pseudo-hyperkähler structure of rational curves.
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 characterizes timelike rectifying curves in De Sitter 3-space.
problem Characterizing timelike rectifying curves in De Sitter 3-space.
method Defining timelike rectifying curves and conical surfaces, providing characterizations and results.
result Characterizations and results of timelike rectifying curves in De Sitter 3-space.
New method to construct partner curves of non-lightlike curves.
problem Constructing partner curves for non-lightlike curves.
method Using integral curves in Minkowski 3-space, direction curve, and donor curve.
result New methods to construct partner curves of a unit speed non-lightlike curve.
The paper finds two types of metric lines in curve spaces.
problem Classifying metric lines in jet spaces of curves.
method Established the existence of two families of metric lines in the 2-jet space of plane curves.
result Found precise criteria for identifying metric lines in sub-Riemannian geodesics.
Study on triharmonic curves in Sol space with constant curvature and torsion.
problem Characterizing triharmonic curves in the Sol space.
method Complete classification of proper triharmonic curves with constant geodesic curvature and torsion.
result Triharmonic curves form a constant angle with a Killing field of constant length.
Study characterizes involutes and evolutes of curves in n-dimensional space.
problem Characterizing involutes and evolutes of curves in n-dimensional Euclidean space.
method Analyzes orthogonal trajectories and osculating hyperspheres to define involutes and evolutes.
result Characterizes involute curves of order k and evolute curves in n-dimensional Euclidean space.
Proves connection between curve moduli and Feynman diagrams.
problem Understanding the relationship between symplectic and elliptic structures.
method Establishes homeomorphism between moduli spaces of curves and Feynman diagrams.
result Moduli spaces determine A∞ structures in both models. 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.
problem Classifying polyharmonic curves of constant curvature.
method Analyzing curves on spheres and space forms, deriving explicit families.
result New insights into higher order variational problems.
Paper studies geometric properties of nonlinear Lebesgue spaces.
problem Geometric and analytic properties of nonlinear Lebesgue spaces.
method Formalizes pointwise description of geometric properties using a nonlinear Fubini-Lebesgue theorem.
result Definition of length structure, Alexandrov curvature bounds, and speed for absolutely continuous curves in nonlinear Lebesgue spaces.
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.
The paper classifies maximal translation surfaces in Lorentz-Minkowski space.
problem Classifying maximal translation surfaces in Lorentz-Minkowski space.
method Analyzing surfaces defined as the sum of two spatial curves, proving properties of generating curves, and classifying surfaces based on curve types.
result A full description of maximal translation surfaces, including new examples not found in Euclidean space.
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…