Four-dimensional pp-wave Lie groups and harmonic curvature

We determine all four-dimensional Lie groups which have harmonic curvature. As a consequence, a description of four-dimensional pp-wave Lie groups is obtained.

💡 Research Summary

The paper investigates left‑invariant Lorentzian metrics on four‑dimensional Lie groups that are either pp‑wave metrics or have harmonic curvature (i.e., div R = 0, equivalently a divergence‑free Weyl tensor). After recalling that a pp‑wave is a Lorentzian manifold admitting a parallel null vector field U and locally expressed as
(g=2dx^{+}dx^{-}+H(x^{+},x^{1},x^{2})(dx^{+})^{2}+dx^{1,2}+dx^{2,2}),
the authors note that the Ricci tensor is rank‑one and that the metric is Einstein precisely when the spatial Laplacian of H vanishes. Moreover, a four‑dimensional pp‑wave has harmonic curvature exactly when (\Delta_{x}H=\phi(x^{+})) for some function (\phi). Plane waves correspond to the case where H is quadratic in the transverse coordinates, which automatically yields harmonic curvature.

The classification proceeds at the Lie‑algebra level. Any four‑dimensional Lie group is either non‑solvable (SU(2)×ℝ or SL(2,ℝ)×ℝ) or solvable, the latter being semi‑direct products of a three‑dimensional unimodular ideal ( \mathfrak{k}) with ℝ. The authors consider three possibilities for the restriction of the Lorentzian metric to (\mathfrak{k}): Riemannian, Lorentzian, or degenerate. Using Milnor’s description for the Riemannian case, Rahmani’s work for the Lorentzian case, and the analysis of derived algebras for the degenerate case, they write down the most general structure constants compatible with a left‑invariant metric.

To identify pp‑wave metrics, they impose the two‑step nilpotency of the Ricci operator and the Petrov type N condition. These translate into a system of polynomial equations in the structure constants. The authors solve these systems with Gröbner‑basis calculations (performed in Mathematica with a lexicographic order) and obtain a complete list of possibilities, presented as Theorem 1.1. There are five families:

1. (