Critical metrics for quadratic curvature functionals on some solvmanifolds

Abstract

AbstractWe 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 $$G_{\mathcal S}=H \rtimes \exp ({\mathbb {R}}S)$$ G S = H ⋊ exp ( R S ) of the Heisenberg Lie group H, for any $$\mathcal S \in {\mathfrak {s}}{\mathfrak {p}}(1,\mathbb R)$$ S ∈ s p ( 1 , R ) , equipped with a family $$g_a$$ g a of left-invariant metrics. After showing the existence of lattices in all these four-dimensional solvable Lie groups, we completely determine when $$g_a$$ 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.