Symmetry Reduction of the Two-Dimensional Ricci Flow Equation

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 of the symmetry group on its Lie algebra, the optimal system of one-dimensional subalgebras of the ((2D) Rf) equation is obtained. For each class, we will find the reduced equation by the method of similarity reduction. By solving these reduced equations, we will obtain new sets of group invariant solutions for the ((2D) Rf) equation. 1. Introduction The Ricci flow was introduced by Hamilton in his seminal paper, “Three-manifolds with positive Ricci curvature” in 1982 [1]. Since then, Ricci flow has been a very useful tool for studying the special geometries which a manifold admits. Ricci flow is an evolution equation for a Riemannian metric which sometimes can be used in order to deform an arbitrary metric into a suitable metric that can specify the topology of the underlying manifold. If is a smooth Riemannian manifold, Ricci flow is defined by where Ric denotes the Ricci tensor of the metric . By using the concept of Ricci flow, Grisha Perelman completely proved the Poincaré conjecture around 2003 [2–4]. The Ricci flow also is used as an approximation to the renormalization group flow for the two-dimensional nonlinear -model, in quantum field theory; see [5] and references therein. The ricci flow equation is related to one of the models used in obtaining the quantum theory of gravity [6]. Because some difficulties appear when a quantum field theory is formulated, the studies focus on less dimensional models which are called mechanical models. In this paper, we want to obtain new solutions of ((2D) Rf) equation by method of Lie symmetry group. As it is well known, 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 Lie at the end of nineteenth century [7]. 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 [8–10]). Lie'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 obtained by solving the reduced system of differential equation having fewer independent variables than the original system. This fact that for some PDEs, the symmetry reductions are unobtainable by the Lie symmetry