No abstract

In this paper, Lie symmetry group method is applied to find the lie point symmetries group of a PDE system that is determined general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant solutions.

This paper is devoted to obtain the one-dimensional group invariant solutions of the two-dimensional Ricci flow ((2D) Rf) equation. By classifying the orbits of the adjoint representation, the optimal system of one-dimensional subalgebras of the ((2D) Rf) equation is obtained. For each class, we will find the reduced equation by method of similarity reduction. By solving these reduced equations we will obtain new sets of group invariant solutions for the ((2D) Rf) equation.In this paper we want to obtain new solutions of the ((2D) Rf) equation by method of Lie symmetry group. As it is well known, the Lie symmetry group method has an important role in the analysis of differential equations. The theory of Lie symmetry groups of differential equations was developed by Sophus Lie [8]. By this method we can reduce the order of ODEs and investigate the invariant solutions. Also we can construct new solutions from known ones (for more details about the applications of Lie symmetries, see [10,1,3]). Lei's method led to an algorithmic approach to find special solution of differential equation by its symmetry group. These solutions are called group invariant solutions and obtain by solving the reduced system of differential equation having fewer independent variables than the original system. Bluman and Cole generalized the Lie's symmetry method for finding the group-invariant solutions [2]. In this paper we apply this method to obtain the invariant solutions of ((2D) Rf) equation and classify them.This paper is organized as follows. In section 2, by using the mechanical model of Ricci flow, Lie symmetries of ((2D) Rf) equation will be state and some results yield from the structure of the Lie algebra of the Lie symmetry group. In section 3, we will construct an optimal system of one-dimensional subalgebras of the ((2D) Rf) equation which is useful for classifying of group invariant solutions. In section 4, the reduced equation for each element of optimal system is obtained. In section 5, we will solve the reduced equations by method of Lie symmetry group and obtain the group invariant solutions of ((2D) Rf) equation.

A complete treatment of one parameter homothetic motions in three and four dimensional Euclidean spaces is provided in the Yayli's PhD thesis [15]. Here we follow his idea to define one parameter homothetic motion in generalized 3-space 3 E.  By means of the generalized Hamilton operators, we also define a Hamilton motion and show that it is a homothetic motion. We investigate some properties of this motion and show that Darboux vector of the motion can be written as multiplication of two generalized quaternions.

Metrics which are harmonic with respect to Gödel-type metrics, as well as harmonicity of lifted metrics to the tangent bundle are studied.