Geometric Structures on Lie Groups with Flat Bi-invariant Metric

Let L subset of V = R(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 is an element of Lambda(3)L subset of Lambda(3)V. If eta is of type (3, 0) + (0, 3) with respect to a skew-symmetric endomorphism J with J(2) = epsilon Id, then the Lie group L(eta) is endowed with a left-invariant nearly Kahler structure if epsilon = -1 and with a left-invariant nearly para-Kahler structure if epsilon = +1. This construction exhausts all complete simply connected flat nearly (para-) Kahler manifolds. If eta not equal 0 has rational coefficients with respect to some basis, then L(eta) admits a lattice Gamma, and the quotient Gamma/L(eta) is a compact inhomogeneous nearly (para-)Kahler manifold. The first non-trivial example occurs in six dimensions.