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 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$^3_1$ 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$^3_1$ are studied in terms of Frenet equations. First, we prove that position vector of every time-like space curve in Minkowski space E$^3_1$ 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

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 f(R,T)f\left(R,T\right) Gravity

In this paper, we search the existence of invariant solutions of Bianchi type I space-time in the context of $f\left(R,T\right)$ 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 $X^{(1)}$ and $X^{(3)}$. The model with symmetries $X^{(1)}$ begins with big bang singularity while the model with symmetries $X^{(3)}$ 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 $f\left(R,T\right)$.Comment: 14 Pages, 2 Figure panels, Textual changes and 01 reference adde

k−k-type partially null and pseudo null slant helices in Minkowski 4-space

We introduce the notion of $k$-type slant helix in Minkowski space \e_1^4. For partially null and pseudo null curves in \e_1^4, we express some characterizations in terms of their curvature and torsion functions.Comment: 12 pages and 2 figur

Concircular vector fields for Kantowski Sachs and Bianchi type III spacetimes

This paper intends to obtain concircular vector fields of Kantowski Sachs and Bianch type III spacetimes. For this purpose, ten conformal Killing equations and their general solution in the form of conformal Killing vector fields are derived along with their conformal factors. The obtained conformal Killing vector fields are then placed in Hessian equations to obtain the final form of concircular vector fields. The existence of concircular symmetry imposes restrictions on the metric functions. The conditions imposing restrictions on these metric functions are obtained as a set of integrability conditions. It is shown that Kantowski Sachs and Bianchi type III spacetimes admit four, six, or fifteen dimensional concircular vector fields. It is established that for Einstein spaces, every conformal Killing vector field is a concircular vector field. Moreover, it is explored that every concircular vector field obtained here is also a conformal Ricci collineation.Comment: 21 pages, 23 Reference

Constant Scalar Curvature of Three Dimensional Surfaces Obtained by the Equiform Motion of a helix

In this paper we consider the equiform motion of a helix in Euclidean space $\mathbf{E}^7$. We study and analyze the corresponding kinematic three dimensional surface under the hypothesis that its scalar curvature $\mathbf{K}$ is constant. Under this assumption, we prove that if the scalar curvature $\mathbf{K}$ is constant, then $\mathbf{K}$ must equal zero.Comment: 11 pages, 1 figur