scholarly journals SLANT CURVES AND PARTICLES IN THREE-DIMENSIONAL WARPED PRODUCTS AND THEIR LANCRET INVARIANTS

2012 ◽  
Vol 88 (1) ◽  
pp. 128-142 ◽  
Author(s):  
CONSTANTIN CĂLIN ◽  
MIRCEA CRASMAREANU
Keyword(s):  
Vector Field ◽  
Three Dimensional ◽  
Curvature Vector ◽  
Warping Function ◽  
Warped Products ◽  
Slant Curve ◽  
Lancret Invariant ◽  
Legendre Curves ◽  
Frenet Curvatures ◽  
Slant Curves

AbstractSlant curves are introduced in three-dimensional warped products with Euclidean factors. These curves are characterised by the scalar product between the normal at the curve and the vertical vector field, and an important feature is that the case of constant Frenet curvatures implies a proper mean curvature vector field. A Lancret invariant is obtained and the Legendre curves are analysed as a particular case. An example of a slant curve is given for the exponential warping function; our example illustrates a proper (that is, not reducible to the two-dimensional) case of the Lancret theorem of three-dimensional hyperbolic geometry. We point out an eventuality relationship with the geometry of relativistic models.

Unitary vector fields are Fermi–Walker transported along Rytov–Legendre curves

2015 ◽  
Vol 12 (10) ◽  
pp. 1550111 ◽  
Author(s):  
Mircea Crasmareanu ◽  
Camelia Frigioiu
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Conformal Killing ◽  
Space Forms ◽  
Potential Vector ◽  
Legendre Curves ◽  
Complex Space Forms

Fix ξ a unitary vector field on a Riemannian manifold M and γ a non-geodesic Frenet curve on M satisfying the Rytov law of polarization optics. We prove in these conditions that γ is a Legendre curve for ξ if and only if the γ-Fermi–Walker covariant derivative of ξ vanishes. The cases when γ is circle or helix as well as ξ is (conformal) Killing vector filed or potential vector field of a Ricci soliton are analyzed and an example involving a three-dimensional warped metric is provided. We discuss also K-(para)contact, particularly (para)Sasakian, manifolds and hypersurfaces in complex space forms.

scholarly journals Slant Curves and Contact Magnetic Curves in Sasakian Lorentzian 3-Manifolds

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 784 ◽  
Author(s):  
Ji-Eun Lee
Keyword(s):  
Three Dimensional ◽  
Lorentzian Manifold ◽  
Cross Product ◽  
Slant Curve ◽  
Magnetic Curve ◽  
Three Space ◽  
Slant Curves

In this article, we define Lorentzian cross product in a three-dimensional almost contact Lorentzian manifold. Using a Lorentzian cross product, we prove that the ratio of κ and τ − 1 is constant along a Frenet slant curve in a Sasakian Lorentzian three-manifold. Moreover, we prove that γ is a slant curve if and only if M is Sasakian for a contact magnetic curve γ in contact Lorentzian three-manifold M. As an example, we find contact magnetic curves in Lorentzian Heisenberg three-space.

scholarly journals LEGENDRE CURVES ON 3-DIMENSIONAL f-KENMOTSU MANIFOLDS ADMITTING SCHOUTEN-VAN KAMPEN CONNECTION

2020 ◽  
pp. 357
Author(s):  
Ashis Mondal
Keyword(s):  
Three Dimensional ◽  
Kenmotsu Manifold ◽  
3 Dimensional ◽  
Legendre Curve ◽  
Kenmotsu Manifolds ◽  
Legendre Curves ◽  
Slant Curves

In the present paper, biharmonic Legendre curves with respect to Schouten-Van Kampen connection have been studied on three-dimensional f-Kenmotsu manifolds. Locally $\phi $-symmetric Legendre curves on three-dimensional f-Kenmotsu manifolds with respect to Schouten-Van Kampen Connection have been introduced.Also slant curves have been studied on three-dimensional f-Kenmotsu manifolds with respect to Schouten-Van Kampen connection. Finally, we constract an example of a Legendre curve in a 3-dimensional f-Kenmotsu manifold.

scholarly journals ON LEGENDRE CURVES IN CONTACT PSEUDO-HERMITIAN 3-MANIFOLDS

2009 ◽  
Vol 81 (1) ◽  
pp. 156-164 ◽  
Author(s):  
JI-EUN LEE
Keyword(s):  
Vector Field ◽  
Mean Curvature ◽  
Sufficient Conditions ◽  
Curvature Vector ◽  
Sasakian Manifold ◽  
Mean Curvature Vector ◽  
Parallel Mean Curvature Vector ◽  
Parallel Mean Curvature ◽  
Legendre Curves ◽  
Necessary And Sufficient

AbstractWe find necessary and sufficient conditions for a Legendre curve in a Sasakian manifold to have: (i) a pseudo-Hermitian parallel mean curvature vector field; (ii) a pseudo-Hermitian proper mean curvature vector field in the normal bundle.

scholarly journals ARBTools: A Tricubic Spline Interpolator for Three-Dimensional Scalar or Vector Fields

2019 ◽  
Author(s):  
Paul Walker ◽  
Ulrich Krohn ◽  
Carty David
Keyword(s):  
Magnetic Field ◽  
Vector Field ◽  
Vector Fields ◽  
Three Dimensional ◽  
Spline Method ◽  
Particle Trajectories ◽  
The Core ◽  
Numerical Integrators ◽  
Data Points ◽  
System Requirements

ARBTools is a Python library containing a Lekien-Marsden type tricubic spline method for interpolating three-dimensional scalar or vector fields presented as a set of discrete data points on a regular cuboid grid. ARBTools was developed for simulations of magnetic molecular traps, in which the magnitude, gradient and vector components of a magnetic field are required. Numerical integrators for solving particle trajectories are included, but the core interpolator can be used for any scalar or vector field. The only additional system requirements are NumPy.

DYNAMICS AT INFINITY OF A CUBIC CHUA'S SYSTEM

2011 ◽  
Vol 21 (01) ◽  
pp. 333-340 ◽  
Author(s):  
MARCELO MESSIAS
Keyword(s):  
Dynamical System ◽  
Vector Field ◽  
Numerical Study ◽  
Three Dimensional ◽  
Polynomial Vector ◽  
Cubic Nonlinearity ◽  
Poincaré Compactification ◽  
Global Study ◽  
Dimensional Dynamical System ◽  
Parameter Values

We use the Poincaré compactification for a polynomial vector field in ℝ3 to study the dynamics near and at infinity of the classical Chua's system with a cubic nonlinearity. We give a complete description of the phase portrait of this system at infinity, which is identified with the sphere 𝕊2 in ℝ3 after compactification, and perform a numerical study on how the solutions reach infinity, depending on the parameter values. With this global study we intend to give a contribution in the understanding of this well known and extensively studied complex three-dimensional dynamical system.

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