Negative Ricci curvature on some non-solvable Lie groups

Abstract

We show that for any non-trivial representation $$(V, \pi )$$ of $$\mathfrak {u}(2)$$ with the center acting as multiples of the identity, the semidirect product $$\mathfrak {u}(2) \ltimes _\pi V$$ admits a metric with negative Ricci curvature that can be explicitly obtained. It is proved that $$\mathfrak {u}(2) \ltimes _\pi V$$ degenerates to a solvable Lie algebra that admits a metric with negative Ricci curvature. An n-dimensional Lie group with compact Levi factor $$\mathrm {SU}(2)$$ admitting a left invariant metric with negative Ricci is therefore obtained for any $$n \ge 7$$ .