A remark on Ricci flow of left invariant metrics

2005/07/22 by J. A. Arteaga, Arteaga, J. A., M. A. Malakhaltsev +1
Mathematics · #53C21 #53C25 #53C30 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C21 #msc:53C25 #msc:53C30

paper · pdf · doi:10.48550/arxiv.math/0507473

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

arxiv created 2005/07/27 · arxiv updated 2009/12/01

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) → 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 \overset1R_ij + \overset2R_ij + \overset3R_ij + \overset4R_ij such that, for any E_i' = Uⁱ_i' E_i with ||Uⁱ_i'|| ∈ O(n), \oversetαR_i'j' = U_i'ⁱ \oversetαR_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.

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