具有平坦实射影结构的流形到（n-1）-凸流形和凹仿射流形的典范分解
Canonical decomposition of manifolds with flat real projective structure into (n-1)-convex manifolds and concave affine manifolds
摘要：
我们试图理解$n$ -流形（$n\geq 2$）的几何性质，其几何结构以$(\bR P^n, \PGL(n+1, \bR))$为模型，即，$n$ -具有射影平坦无挠仿射联络的流形。我们定义的概念$i$ -凸的流形由于Carriére为整数$i$，$1 \leq i \leq n-1$，这是推广的凸性。给定一个真正的投影$n$ -流形$M$，我们表明，$(n-1)$ -凸$M$的失败意味着存在一定的几何对象，$n$ -新月，在完成$\che M$的通用覆盖$\tilde M$的$M$。我们表明，这进一步意味着存在一个特定类型的仿射子流形$M$，并给出一个自然的分解$M$到更简单的真正的投影流形，其中一些是$(n-1)$ -凸和其他仿射，更具体地凹仿射。我们认为进行这种分解是有用的，特别是在三维空间。我们的结果将帮助我们以后研究辐射仿射三维流形的几何和拓扑性质，导致他们的分类。我们得到了仿射李群的一个结果。
英文摘要：
We try to understand the geometric properties of $n$-manifolds ($n\geq 2$) with geometric structures modeled on $(\bR P^n, \PGL(n+1, \bR))$, i.e., $n$-manifolds with projectively flat torsion free affine connections. We define the notion of $i$-convexity of such manifolds due to Carriére for integers $i$, $1 \leq i \leq n-1$, which are generalization of convexity. Given a real projective $n$-manifold $M$, we show that the failure of an $(n-1)$-convexity of $M$ implies an existence of a certain geometric object, $n$-crescent, in the completion $\che M$ of the universal cover $\tilde M$ of $M$. We show that this further implies the existence of a particular type of affine submanifold in $M$ and give a natural decomposition of $M$ into simpler real projective manifolds, some of which are $(n-1)$-convex and others are affine, more specifically concave affine. We feel that it is useful to have such decomposition particularly in dimension three. Our result will later aid us to study the geometric and topological properties of radiant affine 3-manifolds leading to their classification. We get a consequence for affine Lie groups.
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