The moduli space of left-invariant metrics of a class of six-dimensional nilpotent Lie groups

Abstract

In this paper we determine the moduli space, up to isometric automorphism, of left-invariant metrics on a $6$-dimensional Lie group $H$, such that its Lie algebra $\mathfrak{h}$ admits a complex structure and has first Betti number equal to four. We also investigate which of these metrics are Hermitian and classify the corresponding complex structures.