A remark on Ricci flow of left invariant metrics

Details A remark on Ricci flow of left invariant metrics A remark on Ricci flow of left invariant metrics

2005-07-22

arxiv.org/abs/math/0507473v2

Mathematics

Differential Geometry

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

Scientific paper

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.

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