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Left-invariant Codazzi tensors and harmonic curvature on Lie groups endowed with a left invariant Lorentzian metric (2402.16381v1)

Published 26 Feb 2024 in math.DG

Abstract: A Lorentzian Lie group is a Lie group endowed with a left invariant Lorentzian metric. We study left-invariant Codazzi tensors on Lorentzian Lie groups. We obtain new results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator. In contrast to the Riemannian case, the Ricci operator of a let-invariant Lorentzian metric can be of four types: diagonal, of type ${n-2,z\bar{z}}$, of type ${n,a2}$ and of type ${n,a3}$. We first describe Lorentzian Lie algebras with a non-diagonal Codazzi operator and with these descriptions in mind, we study three classes of Lorentzian Lie groups with harmonic curvature. Namely, we give a complete description of the Lie algebra of Lorentzian Lie groups having harmonic curvature and where the Ricci operator is non-diagonal and its diagonal part consists of one real eigenvalue $\alpha$.

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