35,330 research outputs found
Time-like Salkowski and anti-Salkowski curves in Minkowski space \e_1^3
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 time-like curves in Minkowski 3-space \e_1^3. Thereafter, we introduce an
explicit parametrization of a time-like Salkowski curves and a time-like
Anti-Salkowski curves in Minkowski space \e_1^3. Also, we characterize them
as space curve with constant curvature or constant torsion and whose normal
vector makes a constant angle with a fixed line.Comment: 9 pages onl
Conformal Ricci Collineations of Plane Symmetric Static Spacetimes
This article explores the Conformal Ricci Collineations (CRCs) for the
plane-symmetric static spacetime. The non-linear coupled CRC equations are
solved to get the general form of conformal Ricci symmetries. In the
non-degenerate case, it turns out that the dimension of the Lie algebra of CRCs
is finite. In the case were the Ricci tensor is degenerate, it found that the
algebra of CRCs for the plane-symmetric static spacetime is mostly, but not
always, infinite dimensional. In one case of degenerate Ricci tensor, we solved
the differential constraints completely and a spacetime metric is obtained
along with CRCs. We found ten possible cases of finite and infinite dimensional
Lie algebras of CRCs for the considered spacetime.Comment: 17 page
Position vectors of slant helices in Euclidean space E
In classical differential geometry, the problem of the determination of the
position vector of an arbitrary space curve according to the intrinsic
equations and (where and are
the curvature and torsion of the space curve , respectively) is still
open \cite{eisenh, lips}. However, in the case of a plane curve, helix and
general helix, this problem is solved. In this paper, we solved this problem in
the case of a slant helix. Also, we applied this method to find the
representation of a Salkowski, anti-Salkowski curves and a curve of constant
precession, as examples of a slant helices, by means of intrinsic equations.Comment: 14 pages, 3 figure
Position vectors of a spacelike general helices in Minkowski Space \e_1^3
In this paper, position vector of a spacelike general helix with respect to
standard frame in Minkowski space E 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, we determine the parametric representation of the general helices
from the intrinsic equations. Moreover, we give some examples to illustrate how
to find the position vectors of a spacelike general helices with a spacelike
and timelike principal normal vector from the intrinsic equations.Comment: 15 pages, 3 figure
Determination of time-like helices from intrinsic equations in Minkowski 3-Space
In this paper, position vectors of a time-like curve with respect to standard
frame of Minkowski space E are studied in terms of Frenet equations.
First, we prove that position vector of every time-like space curve in
Minkowski space E satisfies a vector differential equation of fourth
order. The general solution of mentioned vector differential equation has not
yet been found. By special cases, we determine the parametric representation of
the general helices from the intrinsic equations (i.e. curvature and torsion
are functions of arc-length) of the time-like curve. Moreover, we give some
examples to illustrate how to find the position vector from the intrinsic
equations of general helices.Comment: p pages 3 figure
Some Characterizations of Special Curves in the Euclidean Space
In this work, first, we express some characterizations of helices and ccr
curves in the Euclidean 4-space. Thereafter, relations among Frenet-Serret
invariants of Bertrand curve of a helix are presented. Moreover, in the same
space, some new characterizations of involute of a helix are presented.Comment: 11 pages onl
New Class of Magnetized Inhomogeneous Bianchi Type-I Cosmological Model with Variable Magnetic Permeability in Lyra Geometry
Inhomogeneous Bianchi type-I cosmological model with electro-magnetic field
based on Lyra geometry is investigated. Using separated method, the Einstein
field equations have been solved analytically with the aid of Mathematica
programm. A new class of exact solutions have been obtained by considering the
potentials of metric and displacement field are functions of coordinates t and
x. We have assumed that F(12) is the only non-vanishing component of
electro-magnetic field tensor F(ij). The Maxwells equations show that F(12) is
the function of x alone whereas the magnetic permeability is the function of x
and t both. To get the deterministic solution, it has been assumed that the
expansion scaler Theta in the model is proportional to the value sigma(11) of
the shear tensor sigma(ij). Some physical and geometric properties of the model
are also discussed and graphed.Comment: Int. J. Theor. Phys. 52, 4055 (2013
Symmetry Group Analysis for perfect fluid Inhomogeneous Cosmological Models in General Relativity
In this paper, we have searched the existence of the similarity solution for
plane symmetric inhomogeneous cosmological models in general relativity. The
matter source consists of perfect fluid with proportionality relation between
expansion scalar and shear scalar. The isovector fields of Einstein's field
equation for the models under consideration are derived. A new class of exact
solutions of Einstein's field equation have been obtained for inhomogeneous
space-time. The physical behaviors and geometric aspects of the derived models
have been discussed in detail.Comment: 13 page
An optimal system and invariant solutions of dark energy Models in cylindrically symmetric space-time
In this paper, we derive some new invariant solutions of dark energy models
in cylindrically symmetric space-time. To quantify the deviation of pressure
from isotropy, we introduce three different time dependent skewness parameters
along the spatial directions. The matter source consists of dark energy which
is minimally interact with perfect fluid. We use symmetry analysis method for
solving the non-linear partial differential equations (NLPDEs) which is more
powerful than the classical methods of solving NLPDEs. The geometrical and
kinematical features of the models and the behaviour of the anisotropy of dark
energy, are examined in detail.Comment: 13 page
Invariant Bianchi type I models in Gravity
In this paper, we search the existence of invariant solutions of Bianchi type
I space-time in the context of gravity. The exact solution
of the Einstein's field equations are derived by using Lie point symmetry
analysis method that yield two models of invariant universe for symmetries
and . The model with symmetries begins with big
bang singularity while the model with symmetries does not favour the
big bang singularity. Under this specification, we find out at set of singular
and non singular solution of Bianchi type I model which present several other
physically valid features within the framework of .Comment: 14 Pages, 2 Figure panels, Textual changes and 01 reference adde
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