Geometric structures on Lie groups with flat bi - invariant metric Vicente Cortés

Let L ⊂ V = R 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 η ∈ Λ3L ⊂ Λ3V . If η is of type (3, 0)+(0, 3) with respect to a skew-symmetric endomorphism J with J2 = ǫId, then the Lie group L(η) is endowed with a left-invariant nearly Kähler structure if ǫ = −1 and with a left-invariant nearly para-Kähler structure if ǫ = +1. This construction exhausts all complete simply connected flat nearly (para-)Kähler manifolds. If η 6= 0 has rational coefficients with respect to some basis, then L(η) admits a lattice Γ, and the quotient Γ \ L(η) is a compact inhomogeneous nearly (para-)Kähler manifold. The first non-trivial example occurs in six dimensions. MSC(2000): 53C50, 53C15.