Geometric structures on Lie groups with flat bi-invariant metric

TL;DR

This paper classifies simply connected Lie groups with flat bi-invariant pseudo-Riemannian metrics, linking their structure to special 3-forms, and constructs all complete flat nearly (para-)K"ahler manifolds, including compact examples.

Contribution

It provides a complete classification of flat bi-invariant pseudo-Riemannian Lie groups and constructs all complete flat nearly (para-)K"ahler manifolds, including compact inhomogeneous examples.

Findings

Lie groups are 2-step nilpotent and determined by a 3-form ta.

Construction of nearly (para-)Ke4hler structures on these groups.

Existence of compact inhomogeneous nearly (para-)Ke4hler manifolds via lattices.

Abstract

Let L\subset V=\bR^{k,l} be a maximally isotropic subspace. It is shown that any simply connected Lie group with a bi-invariant flat pseudo-Riemannian metric of signature (k,l) is 2-step nilpotent and is defined by an element \eta \in \Lambda^3L\subset \Lambda^3V. If \eta is of type (3,0)+(0,3) with respect to a skew-symmetric endomorphism J with J^2=\e Id, then the Lie group {\cal L}(\eta) is endowed with a left-invariant nearly K\"ahler structure if \e =-1 and with a left-invariant nearly para-K\"ahler structure if \e =+1. This construction exhausts all complete simply connected flat nearly (para-)K\"ahler manifolds. If \eta \neq 0 has rational coefficients with respect to some basis, then {\cal L}(\eta) admits a lattice \Gamma, and the quotient \Gamma\setminus {\cal L}(\eta) is a compact inhomogeneous nearly (para-)K\"ahler manifold. The first non-trivial example occurs in six dimensions.