Let $M$ be complete flat pseudo-Riemannian homogeneous manifold and$\Gamma\subset\Iso(\RR^n_s)$ its fundamental group. We show that $M$ is atrivial fiber bundle $G/\Gamma\to M\to\RR^{n-k}$, where $G$ is the Zariskiclosure of $\Gamma$ in $\Iso(\RR^n_s)$. Moreover, we show that the $G$-orbitsin $\RR^n_s$ are affinely diffeomorphic to $G$ endowed with the (0)-connection.If the induced metric on the $G$-orbits is non-degenerate, then $G$ (and hence$\Gamma$) has linear abelian holonomy. If additionally $G$ is not abelian, then$G$ contains a certain subgroup of dimension 6. In particular, for non-abelian$G$ orbits with non-degenerate metric can appear only if $\dim G\geq 6$.
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机译：令$ M $是完全平坦的伪黎曼齐次流形，$ \ Gamma \ subset \ Iso（\ RR ^ n_s）$是其基群。我们显示$ M $是从光纤束$ G / \ Gamma \到M \ to \ RR ^ {nk} $，其中$ G $是$ \ Iso（\ RR ^ n_s）中$ \ Gamma $的Zariskiclosure $。此外，我们证明$ \ RR ^ n_s $中的$ G $轨道与赋予（0）连接的$ G $仿射微分，如果在$ G $轨道上的诱导度量是非简并的，则$ G $（因此$ \ Gamma $）具有线性阿贝尔完整性。如果另外$ G $不是abelian，则$ G $包含维6的某个子组。特别是，对于非abelian $ G $，具有非简并度度量的轨道只有在$ \ dim G \ geq 6 $时才能出现。
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