A Remark on Ricci Flow of Left Invariant Metrics

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) → UT , where UT is the group of upper triangular matrices. We decompose the matrix Rij of Ricci tensor coordinates with respect to an orthonormal frame field Ei into a sum 1 Rij + 2 Rij + 3 Rij + 4 Rij such that, for any Ei′ = U i i′Ei with ||U i i || ∈ O(n), α Ri′j′ = U i i α RijU 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.