Left invariant Riemannian metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension $\leq6$

TL;DR

This paper proves that on solvable Lie groups and all Lie groups of dimension six or less, any left invariant metric with harmonic curvature must also be Ricci-parallel, revealing a strong geometric property.

Contribution

It establishes a new rigidity result linking harmonic curvature and Ricci-parallelism for low-dimensional and solvable Lie groups.

Findings

Left invariant harmonic curvature metrics are Ricci-parallel on solvable Lie groups.

The same Ricci-parallel property holds for Lie groups of dimension six or less.

Provides a classification result for these geometric structures.

Abstract

We show that any left invariant metric with harmonic curvature on a solvable Lie group is Ricci-parallel. We show the same result for any Lie group of dimension $\leq$ 6.