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 ${\stackrel{1}{R}}_{ij}+{\stackrel{2}{R}}_{ij}+{\stackrel{3}{R}}_{ij}+{\stackrel{4}{R}}_{ij}$ such that, for any $E_{i^{\prime }}=U_{i^{\prime }}^i{E}_i$ with $||U_{i^{\prime }}^i||\in O(n)$, ${\stackrel{\alpha }{R}}_{i^{\prime }{j}^{\prime }}=U_{i^{\prime }}^i{\stackrel{\alpha }{R}}_{ij}{U}_{j^{\prime }}^j$. This allows us to specify several cases when the differential equation can be simplified. As an example we consider three-dimensional unimodular Lie groups.

Paper is replaced because of some typos in formulas (especially in part II)