Negative Ricci curvature on some non-solvable Lie groups

Abstract

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