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Journal of Lie Theory 19 (2009), No. 2, 423--437
Copyright Heldermann Verlag 2009

Geometric Structures on Lie Groups with Flat Bi-Invariant Metric

Vicente Cortés
Department Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany

Lars Schäfer
Institut für Differentialgeometrie, Leibniz Universität, Welfengarten 1, 30167 Hannover, Germany


\def\R{{\Bbb R}} \def\e{{\varepsilon}} \def\Id{\mathop{\rm Id}\nolimits} Let $L\subset V=\R^{k,l}$ 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 $\eta \in \Lambda^3L\subset \Lambda^3V$. 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.

Keywords: Flat Lie-groups, bi-invariant metrics, nearly para-Kaehler manifolds, flat almost para-Hermitian manifolds, almost para-complex structures.

MSC: 53C50, 53C15

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