Critical metrics for quadratic curvature functionals on some solvmanifolds

Abstract

We prove the existence of four-dimensional compact manifolds admitting some non-Einstein Lorentzian metrics, which are critical points for all quadratic curvature functionals. For this purpose, we consider left-invariant semi-direct extensions G_{mathcal S}=H rtimes exp ({mathbb {R}}S) of the Heisenberg Lie group H, for any mathcal S in {mathfrak {s}}{mathfrak {p}}(1,mathbb R), equipped with a family g_a of left-invariant metrics. After showing the existence of lattices in all these four-dimensional solvable Lie groups, we completely determine when g_a is a critical point for some quadratic curvature functionals. In particular, some four-dimensional solvmanifolds raising from these solvable Lie groups admit non-Einstein Lorentzian metrics, which are critical for all quadratic curvature functionals.