# Geometric structures on Lie groups with flat bi-invariant metric - Mathematics > Differential Geometry

Abstract: Let L\subset V=\bR^{k,l} be a maximally isotropic subspace. It is shown thatany simply connected Lie group with a bi-invariant flat pseudo-Riemannianmetric 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 respectto a skew-symmetric endomorphism J with J^2=\e Id, then the Lie group {\calL}\eta is endowed with a left-invariant nearly K\-ahler structure if \e =-1and with a left-invariant nearly para-K\-ahler structure if \e =+1. Thisconstruction exhausts all complete simply connected flat nearly para-K\-ahlermanifolds. If \eta eq 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. Thefirst non-trivial example occurs in six dimensions.

Author: Vicente Cortés, Lars Schäfer

Source: https://arxiv.org/