Codazzi tensors and harmonic curvature for left invariant metrics (Q1079827)

Let A be a left-invariant Codazzi tensor on a Lie group G with a left- invariant metric. The eigenspaces of A then correspond to certain subalgebras of the Lie algebra of G. It is shown that, if none of these subalgebras is an ideal and \(\nabla A\neq 0\), then G is not nilpotent or split solvable. Another result of the paper states that, if \(\nabla A\neq 0\), the sectional curvature of G must assume both positive and negative values. In the case where A is the Ricci tensor (so the metric has harmonic curvature) and \(A\neq 0\), the author proves that not all eigenspace subalgebras are Abelian.