On flat metric nilpotent Lie algebras

Abstract

We study flat left-invariant pseudo-Riemannian metrics on nilpotent Lie groups. Associated with a flat metric Lie algebra $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ there is a canonical affine action of the corresponding group on the vector space $\mathfrak{g}$, whose linear part is the Levi-Civita representation $L$. Combining this action with Scheuneman's theorem on pure translations in nilpotent affine groups, we prove that a flat metric nonabelian nilpotent Lie algebra has a nontrivial kernel of $L$, that is, $\ker L\neq 0$. As a consequence, every $n$-dimensional flat metric nonabelian nilpotent Lie algebra is a double extension of an $(n-2)$-dimensional flat metric nilpotent Lie algebra.