A remark on Ricci flow of left invariant metrics

Abstract

We prove that the Ricci flow equation for left invariant metrics on Lie groups reduces to a first order ordinary differential equation for a map $Q : (-a,a) \to UT$, where $UT$ is the group of upper triangular matrices. We decompose the matrix $R_{ij}$ of Ricci tensor coordinates with respect to an orthonormal frame field $E_{i}$ into a sum $\overset{1}{R}_{ij} + \overset{2}{R}_{ij} + \overset{3}{R}_{ij} + \overset{4}{R}_{ij}$ such that, for any $E_{i'} = U^i_{i'} E_i$ with $||U^i_{i'}|| \in O(n)$, $\overset{\alpha}{R}_{i'j'} = U_{i'}^i \overset{\alpha}{R}_{ij} U^j_{j'}$. This allows us to specify several cases when the differential equation can be simplified. As an example we consider three-dimensional unimodular Lie groups.