奇维单纯流形的角亏损
The angle defect for odd-dimensional simplicial manifolds
摘要：
在1967年的一篇论文中，班乔夫指出，某种类型的多面体曲率，适用于所有有限多面体，是零在所有顶点的奇数维多面体流形;然后得到一个基本的证明，奇数维流形有零欧拉示性数。在以前的论文中，作者定义了一个不同的方法，曲率为任意单纯复形，基于一个直接推广的角度亏损。在每个奇维流形的单形处，广义角亏损不为零。本文利用Bernoulli数序列定义了有限单纯复形的角亏损的一个变体，它仍然满足Gauss-Bonnet型定理，但在奇数维单纯复形K的任一单纯形处也为零（尺寸至少为3），使得每个i-单形的链路的欧拉示性数等于2，其中i是小于n的非负偶数整数。作为推论，给出了一个初等证明，证明了任何这样的单纯复形的欧拉示性数为零。
英文摘要：
In a 1967 paper, Banchoff stated that a certain type of polyhedral curvature, that applies to all finite polyhedra, was zero at all vertices of an odd-dimensional polyhedral manifold; one then obtains an elementary proof that odd-dimensional manifolds have zero Euler characteristic. In a previous paper, the author defined a different approach to curvature for arbitrary simplicial complexes, based upon a direct generalization of the angle defect. The generalized angle defect is not zero at the simplices of every odd-dimensional manifold. In this paper we use a sequence based upon the Bernoulli numbers to define a variant of the angle defect for finite simplicial complexes that still satisfies a Gauss-Bonnet type theorem, but is also zero at any simplex of an odd-dimensional simplicial complex K (of dimension at least 3), such that the Euler characteristic of the link of each i-simplex equals 2, where i is a non-negative even integer that is less than n. As a corollary, an elementary proof is given that any such simplicial complex has Euler characteristic zero.
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