Left invariant Ricci flat metrics on Lie groups

Abstract

Abstract In this paper, we apply the double extension process to study left invariant Ricci flat metrics on solvable and non-solvable Lie groups. An inductive method to produce new Ricci flat metrics from the old ones is established. As applications, we prove the following two results: (i) Every nilpotent Lie group with dim ⁡ C ⁢ ( G ) ≥ 1 2 ⁢ ( dim ⁡ G - 1 ) {\dim\mathrm{C}(G)\geq\frac{1}{2}(\dim G-1)} admits a left invariant Ricci flat metric. (ii) Given a Lie group G, there exists a nilpotent Lie group N with nilpotent index at most 2 such that G × N {G\times N} admits a left invariant Ricci flat metric. We also construct infinitely many new explicit examples of left invariant Ricci flat metrics on nilpotent Lie groups.