Journal of Lie Theory

no. 2
Geometric Structures on Lie Groups with Flat Bi-Invariant Metric
Vicente Cortés1; Lars Schäfer2
1Department Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
2Institut für Differentialgeometrie, Leibniz Universität, Welfengarten 1, 30167 Hannover, Germany
Journal of Lie Theory, Volume 19 (2009) no. 2, pp. 423-437
\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.
DOI: 10.5802/jolt.561
Classification: 53C50, 53C15
Keywords: Flat Lie-groups, bi-invariant metrics, nearly para-Kaehler manifolds, flat almost para-Hermitian manifolds, almost para-complex structures

Vicente Cortés  1 ; Lars Schäfer  2

1 Department Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
2 Institut für Differentialgeometrie, Leibniz Universität, Welfengarten 1, 30167 Hannover, Germany 
Vicente Cortés; Lars Schäfer. Geometric Structures on Lie Groups with Flat Bi-Invariant Metric. Journal of Lie Theory, Volume 19 (2009) no. 2, pp. 423-437. doi: 10.5802/jolt.561
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     title = {Geometric {Structures} on {Lie} {Groups} with {Flat} {Bi-Invariant} {Metric}},
     journal = {Journal of Lie Theory},
     pages = {423--437},
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