scholarly journals Null Curve Evolution in Four-Dimensional Pseudo-Euclidean Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
José del Amor ◽  
Ángel Giménez ◽  
Pascual Lucas
Keyword(s):  
Vector Fields ◽  
Evolution Equations ◽  
Space Form ◽  
Curve Evolution ◽  
Lie Bracket ◽  
Null Curve ◽  
Induction Equation ◽  
Null Curves ◽  
Local Vector ◽  
Local Symmetries

We define a Lie bracket on a certain set of local vector fields along a null curve in a 4-dimensional semi-Riemannian space form. This Lie bracket will be employed to study integrability properties of evolution equations for null curves in a pseudo-Euclidean space. In particular, a geometric recursion operator generating infinitely many local symmetries for the null localized induction equation is provided.

Pseudo null curve variations for Bishop frame in 3D semi-Riemannian manifold

2019 ◽  
Vol 16 (03) ◽  
pp. 1950043 ◽  
Author(s):  
Zehra Özdemi̇r
Keyword(s):  
Magnetic Field ◽  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Space Form ◽  
Charged Particles ◽  
Null Curve ◽  
Null Curves ◽  
The Magnetic Field ◽  
Killing Equations

In the present paper, the relation between invariants of the pseudo null curves and the variational vector fields of semi-Riemannian manifolds is introduced. After that, the Killing equations are written in terms of the Bishop curvatures along the pseudo null curve. By means of this approach, Killing equations make allow to interpret the movement of charged particles within the magnetic field. Afterwards, as an application, pseudo null magnetic curves are defined using the Killing variational vector field. The parametric representations of all pseudo null magnetic curves are determined in semi-Riemannian space form. Moreover, various examples of pseudo null magnetic curves are illustrated.

scholarly journals Integrability aspects of the vortex filament equation for pseudo-null curves

2017 ◽  
Vol 14 (06) ◽  
pp. 1750090 ◽  
Author(s):  
José del Amor ◽  
Ángel Giménez ◽  
Pascual Lucas
Keyword(s):  
Space Form ◽  
Burgers Equation ◽  
Three Dimensional ◽  
Recursion Operator ◽  
Vortex Filament ◽  
Lorentzian Space ◽  
Null Curve ◽  
Vortex Filament Equation ◽  
Null Curves ◽  
The Relationship

An algebraic background in order to study the integrability properties of pseudo-null curve motions in a three-dimensional Lorentzian space form is developed. As an application, we delve into the relationship between the Burgers’ equation and the pseudo-null vortex filament equation. A recursion operator for the pseudo-null vortex filament equation is also provided.

scholarly journals Special Curves According to Bishop Frame in Minkowski 3-Space

2020 ◽  
Vol 5 (1) ◽  
pp. 237-248
Author(s):  
Muhammad Abubakar Isah ◽  
Mihriban Alyamaç Külahçı
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bishop Frame ◽  
Null Curve ◽  
Minkowski Spaces ◽  
Slant Helix ◽  
Null Curves ◽  
Necessary And Sufficient

AbstractPseudo null curves were studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curve are not considered. In this paper, we study weak AW (k) – type and AW (k) – type pseudo null curve in Minkowski 3-space [E_1^3 . We define helix and slant helix according to Bishop frame in [E_1^3 . Furthermore, the necessary and sufficient conditions for the slant helix and helix in Minkowski 3-space are obtained.

scholarly journals Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1547
Author(s):  
Stephen C. Anco ◽  
Bao Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Fields ◽  
Evolution Equations ◽  
Applied Mathematics ◽  
Tangent Vector ◽  
Solution Space ◽  
Partial Differential ◽  
Geometrical Meaning ◽  
Geometrical Formulation

A geometrical formulation for adjoint-symmetries as one-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are presented. Additionally, for systems of evolution equations, adjoint-symmetries are shown to have another geometrical formulation given by one-forms that are invariant under the flow generated by the system on the solution space. This result is generalized to systems of evolution equations with spatial constraints, where adjoint-symmetry one-forms are shown to be invariant up to a functional multiplier of a normal one-form associated with the constraint equations. All of the results are applicable to the PDE systems of interest in applied mathematics and mathematical physics.

RELATIVISTIC PARTICLES ALONG NULL CURVES IN 3D LORENTZIAN SPACE FORMS

2010 ◽  
Vol 20 (09) ◽  
pp. 2851-2859 ◽  
Author(s):  
ÁNGEL GIMÉNEZ
Keyword(s):  
Vector Fields ◽  
Lagrange Equation ◽  
Killing Vector ◽  
Space Forms ◽  
Lorentzian Space ◽  
Motion Equations ◽  
Critical Curves ◽  
Null Curves ◽  
Cartan Curvature ◽  
Relativistic Particles

We study relativistic particles modeled by actions whose Lagrangians are arbitrary functions on the curvature of null paths in (2 + 1)-dimensions backgrounds with constant curvature. We obtain first integrals of the Euler–Lagrange equation by using geometrical methods involving the search for Killing vector fields along critical curves of the action. In the case in which Lagrangian density depends quadratically on Cartan curvature, it is shown that the mechanical system is governed by a stationary Korteweg–De Vries system. Motion equations are completely integrated by quadratures in terms of elliptic and hyperelliptic functions.

SPECIAL PROJECTIVE ALGEBRA OF RANDERS METRICS OF CONSTANT S-CURVATURE

2012 ◽  
Vol 09 (04) ◽  
pp. 1250034 ◽  
Author(s):  
M. RAFIE-RAD
Keyword(s):  
Vector Fields ◽  
Finsler Space ◽  
Finsler Geometry ◽  
Lie Bracket ◽  
Finite Dimensional ◽  
Projective Vector ◽  
Projective Algebra ◽  
Local Characterization ◽  
Randers Metrics

The collection of all projective vector fields on a Finsler space (M, F) is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra. A specific Lie sub-algebra of projective algebra of Randers spaces (called the special projective algebra) of non-zero constant S-curvature is studied and it is proved that its dimension is at most [Formula: see text]. Moreover, a local characterization of Randers spaces whose special projective algebra has maximum dimension is established. The results uncover somehow the complexity of projective Finsler geometry versus Riemannian geometry.

On Bäcklund transformation and vortex filament equation for pseudo null curves in Minkowski 3-space

2016 ◽  
Vol 13 (06) ◽  
pp. 1650077 ◽  
Author(s):  
Milica Grbović ◽  
Emilija Nešović
Keyword(s):  
Evolution Equation ◽  
Bäcklund Transformation ◽  
Sufficient Conditions ◽  
Vortex Filament ◽  
Backlund Transformation ◽  
Burger's Equation ◽  
Null Curve ◽  
Vortex Filament Equation ◽  
Null Curves ◽  
The Given

In this paper, we introduce Bäcklund transformation of a pseudo null curve in Minkowski 3-space as a transformation mapping a pseudo null helix to another pseudo null helix congruent to the given one. We also give the sufficient conditions for a transformation between two pseudo null curves in the Minkowski 3-space such that these curves have equal constant torsions. By using the Da Rios vortex filament equation, based on localized induction approximation (LIA), we derive the vortex filament equation for a pseudo null curve and prove that the evolution equation for the torsion is the viscous Burger’s equation. As an application, we show that pseudo null curves and their Frenet frames generate solutions of the Da Rios vortex filament equation.

Duals of Vector Fields and of Null Curves

2007 ◽  
Vol 50 (1-2) ◽  
pp. 53-79
Author(s):  
Hubert Gollek
Keyword(s):  
Vector Fields ◽  
Null Curves

scholarly journals Equality of the algebraic and geometric ranks of Cartan subalgebras and applications to linearization of a system of ordinary differential equations

2017 ◽  
Vol 28 (11) ◽  
pp. 1750080
Author(s):  
Hassan Azad ◽  
Indranil Biswas ◽  
Fazal M. Mahomed
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vector Fields ◽  
Symmetry Algebra ◽  
Second Order ◽  
Semisimple Lie Algebra ◽  
Semisimple Algebras ◽  
Generic Orbit ◽  
Cartan Subalgebras ◽  
Local Vector

If [Formula: see text] is a semisimple Lie algebra of vector fields on [Formula: see text] with a split Cartan subalgebra [Formula: see text], then it is proved here that the dimension of the generic orbit of [Formula: see text] coincides with the dimension of [Formula: see text]. As a consequence one obtains a local canonical form of [Formula: see text] in terms of exponentials of coordinate functions and vector fields that are independent of these coordinates — for a suitable choice of coordinate system. This result is used to classify semisimple algebras of local vector fields on [Formula: see text] and to determine all representations of [Formula: see text] as local vector fields on [Formula: see text]. These representations are in turn used to find linearizing coordinates for any second-order ordinary differential equation that admits [Formula: see text] as its symmetry algebra and for a system of two second-order ordinary differential equations that admits [Formula: see text] as its symmetry algebra.

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