Curvature property on Hermitian Lie algebras with abelian ideals of codimension two

Abstract

Let $(\mathfrak g,J,g)$ be a unimodular Hermitian Lie algebra containing an abelian ideal $\mathfrak a$ of real codimension two. We study the curvature behaviour of $g$ and show that, if $g$ has constant Chern holomorphic sectional curvature, then it must be Chern flat. We also show that, for every canonical metric connection $D_s^r$ of $g$ other than the Chern connection, if $D_s^r$ has constant holomorphic sectional curvature, then $g$ is K\"ahler flat, and in this case $\mathfrak g/\mathfrak a$ is abelian.