Critical metrics for quadratic curvature functionals on some solvmanifolds

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(S) = H (sic) exp(RS) of the Heisenberg Lie group H, for any S is an element of sp(1, 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 fourdimensional solvmanifolds raising from these solvable Lie groups admit non-Einstein Lorentzian metrics, which are critical for all quadratic curvature functionals.