Flat manifolds and reducibility
Flat manifolds and reducibility
复制标题
扁平流形和还原性
DOI：
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
P. Piccione
中科院分区：
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作者：
A. Derdzinski;P. Piccione
希斯和Szczepanski在1991年证明了，任何紧致平坦黎曼流形的holonomy群，维数至少为2，通过第一Bieberbach定理，作用于与流形相关的欧几里得格的有理跨度上。几何上，他们的结果指出，这样的流形必须承认一个非零的正确的平行分布与紧凑的叶子。我们研究了由于上述定理而存在的子格跨距完整不变子空间的代数和几何性质，以及由此产生的紧致平坦流形的紧致叶叶理。由前一个子空间组成的类，除了在跨距和交集下是闭的之外，还允许（通常是非正交的）补集。至于后者的叶理，我们提供的描述，首先-和最重要的-的内在几何形状的通用叶在原来的平面流形，其次-作为一个基本上明显的事后-的叶空间orbifold。一般的结论，然后说明了广义克莱因瓶的形式的例子。
Hiss and Szczepanski proved in 1991 that the holonomy group of any compact flat Riemannian manifold, of dimension at least two, acts reducibly on the rational span of the Euclidean lattice associated with the manifold via the first Bieberbach theorem. Geometrically, their result states that such a manifold must admit a nonzero proper parallel distribution with compact leaves. We study algebraic and geometric properties of the sublattice-spanned holonomy-invariant subspaces that exist due to the above theorem, and of the resulting compact-leaf foliations of compact flat manifolds. The class consisting of the former subspaces, in addition to being closed under spans and intersections, also turns out to admit (usually nonorthogonal) complements. As for the latter foliations, we provide descriptions, first -- and foremost -- of the intrinsic geometry of their generic leaves in terms of that of the original flat manifold and, secondly -- as an essentially obvious afterthought -- of the leaf-space orbifold. The general conclusions are then illustrated by examples in the form of generalized Klein bottles.