parabolic transformation and solution of 3d ricci flow equations using killing vector fields

Ricci flow equations are among the most fundamental equations in riemannian geometry and classical field theory, playing a crucial role in modeling physical phenomena such as relativistic gravity and quantum field theory. in this paper, we transform the ricci flow equations for three-dimensional manifolds into a parabolic form by applying appropriate coordinate changes and solve them using invariant geometric structures, particularly the killing vector field. additionally, we propose a method for diagonalizing metrics on three-dimensional manifolds, which simplifies the dynamical analysis of these equations. this approach extends known results on two-dimensional ricci flow equations and, by leveraging algebraic structures related to toda equations, provides a more precise examination of possible solutions.