论文标题
封闭的仿射歧管和不变线
Closed Affine Manifolds with an Invariant Line
论文作者
论文摘要
封闭的仿射歧管是一个封闭的歧管，其坐标贴在仿射空间中，其过渡图是仿射自动形态的限制。这样的结构引起了从歧管的普遍覆盖到仿射空间的局部差异性，而仿射空间则相对于从基本群体到仿射自动形态组的同态性相对于同态性。局部的差异和同态分别称为发展的图和全体图。如果线性载体保留了一个共同的向量，则构建了某些“大”开放子集，而开发图是其图像上的差异性。辐射歧管不能在发育图像中具有其固定点的事实的修改证明。结合了这些结果，本文解决了某些封闭仿射歧管的不存在，这些封闭仿射歧管的载体使仿射线不变。具体而言，如果仿射单位纯粹由不变线上的翻译起作用，则开发图像将无法满足这一线。
A closed affine manifold is a closed manifold with coordinate patches into affine space whose transition maps are restrictions of affine automorphisms. Such a structure gives rise to a local diffeomorphism from the universal cover of the manifold to affine space that is equivariant with respect to a homomorphism from the fundamental group to the group of affine automorphisms. The local diffeomorphism and homomorphism are referred to as the developing map and holonomy respectively. In the case where the linear holonomy preserves a common vector, certain `large' open subsets upon which the developing map is a diffeomorphism onto its image are constructed. A modified proof of the fact that a radiant manifold cannot have its fixed point in the developing image is presented. Combining these results, this paper addresses the non-existence of certain closed affine manifolds whose holonomy leaves invariant an affine line. Specifically, if the affine holonomy acts purely by translations on the invariant line, then the developing image cannot meet this line.
