Parabolic transformation and solution of 3D Ricci flow equations using killing vector fields

Subject Areas : Differential geometry

M. Jafari ^{1 *}

1 - Department of Mathematics, Payame Noor University, PO BOX 19395-4697, Tehran, Iran.

Received: 2025-02-12 Accepted : 2025-04-07 Published : 2025-04-17

Keywords: Riemann solitons, killing vector field, general relativity,

Abstract :

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.‎

 

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