闭平坦仿射3-流形是素的
Closed flat affine 3-manifolds are prime
摘要：
一个（平坦的）仿射$3$ -流形是一个$3$ -流形，它具有一个到仿射空间${\mathbf R}^3$的图表集，该仿射空间${\mathbf R}^3$具有仿射变换群$Aff({\mathbf R}^3)$中的过渡映射。等价仿射$3$ -流形是具有平坦无挠仿射联络的$3$ -流形。证明了闭仿射$3$ -流形是不可约的或被仿射Hopf流形复盖.一个实射影$3$ -流形是一个流形，它有一个到一个实射影空间${\mathbf R} P^3$的图表集，该空间的转移映射在射影变换群$PGL(4, {\mathbf R})$中。利用实射影流形的凸凹分解，我们证明了闭实射影$3$ -流形可以分解为凹仿射子流形、凸$π$ -子流形和$2$ -凸实射影流形.
英文摘要：
An (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an affine space ${\mathbf R}^3$ with transition maps in the affine transformation group $Aff({\mathbf R}^3)$. Equivalently an affine $3$-manifold is a $3$-manifold with a flat torsion-free affine connection. We show that a closed affine $3$-manifold is either irreducible or is finitely covered by an affine Hopf manifold. A real projective $3$-manifold is a manifold with an atlas of charts to a real projective space ${\mathbf R} P^3$ with transition maps in the projective transformation group $PGL(4, {\mathbf R})$. Using the convex concave decomposition of real projective manifolds, we will show that a closed real projective $3$-manifold decomposes into concave affine submanifolds, toral $π$-submanifolds and $2$-convex real projective manifolds.
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