Left-Invariant Pseudo-Riemannian Metrics on Lie Groups

Abstract

n differential geometry and mathematical physics, there is interest in left-invariant pseudo-Riemannian metrics on Lie groups. We learn and review Lie theory, representation theory, geometric invariant theory, and differential geometry. We apply this theory to find pseudo-Riemannian metrics for certain Lie groups such that all polynomial curvature invariants are identically zero. We find that the six nilpotent Lie algebras of dimension five can be equipped with pseudo-Riemannian metrics with non-zero curvature such that the Ricci tensor is zero, and all polynomial curvature invariants are identically zero as well. We also find that a class of Lie groups G that can be realized as products or semi-direct products of a Lie group Hand R^n in a certain way can be equipped with pseudo-Riemannian metrics in such a way that all polynomial curvature invariants are identically zero.