In this paper, we study affine manifolds endowed with linear foliations. These are foliations defined by vector subspaces invariant by the linear holonomy. We show that an n-dimensional compact, complete, and oriented affine manifold endowed with a codimension 1 linear foliation $\mathcal{F}$ is homeomorphic to the n-dimensional torus if the leaves of $\mathcal{F}$ are simply connected. Let $(M,{\mathrm{\nabla }}_M)$ be a 3-dimensional compact affine manifold endowed with a codimension 1 linear foliation. We prove that $(M,{\mathrm{\nabla }}_M)$ has a finite cover which is homeomorphic to the total space of a bundle over the circle if its developing map is injective, and has a convex image.

中文翻译：

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仿射流形上的线性叶理

在本文中，我们研究了具有线性叶理的仿射流形。这些是由线性完整法不变的向量子空间定义的叶面。我们展示了一个n维紧凑、完整和定向的仿射流形，其具有辅维 1 线性叶理$\mathcal{F。}$同胚于n维环面，如果$\mathcal{F。}$只是连接。让$(\mathrm{M。},{\mathrm{\nabla }}_{\mathrm{M。}})$是一个 3 维紧凑仿射流形，赋予了一个 codimension 1 线性叶理。我们证明$(\mathrm{M。},{\mathrm{\nabla }}_{\mathrm{M。}})$ 有一个有限覆盖，如果它的展开图是单射的，它同胚于圆上的丛的总空间，并且有一个凸像。