Direct Connections and Chern Character
Direct Connections and Chern Character
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直接联系和陈省身性格
DOI：
10.1142/9789812707499_0039
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
Nicolae Teleman
中科院分区：
文献类型：
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作者：
Nicolae Teleman
我们证明了如何利用循环同调从测地距离函数中提取光滑流形的切丛的Chern特征。这样的考虑导致我们在第5节中定义了向量丛中的线性直接连接的概念，该概念推广了线性连接的概念。线性直接连接和线性连接之间的基本区别包括这样的事实，尽管线性连接提供了纤维沿曲线的传输，而线性直接连接提供了光纤从点到点的直接传输。因此，线性直接连接可以在不可微性的上下文中定义。接下来，我们展示了构造Chern特征符的代数过程，本文对A.Connes[1，2]的非对易陈氏特征标给出了几何解释。我们证明了这种解释与N.Teleman[11]和C.Teleman[10]的解释是一致的。这里讨论的论点可以推广到群胚的语言。在接下来的一篇文章中，我们将改进这里提出的一些考虑，并将它们的应用范围扩展到更奇异的情况。备注0.1本文引入的线性直接联系的概念取代了作者在[12]中引入的线性拟联系的概念。
We show how the Chern character of the tangent bundle of a smooth manifold may be extracted from the geodesic distance function by means of cyclic homology. Such considerations lead us to define in Sect. 5 the notion oflinear direct connectionin vector bundles, notion which generalizes the notion of linear connection.The basic difference between a linear direct connection and a linear connection consists of the fact that while a linear connection provides a transport of fibers alongcurves, a linear direct connection provides adirecttransport of fibresfrom point to point.For this reason, linear direct connections could be defined in contexts where differentiability is not available.We show next that the algebraic procedure for constructing the Chern character, discussed in Sect. 4 applies also in the case of linear direct connections.This paper provides a geometric interpretation of the non commutative Chern character due to A. Connes [1,2]. We show that this interpretation goes along the same lines as those presented by N. Teleman [11] and C. Teleman [10].The arguments discussed here may be extended to the language of groupoids. In a subsequent paper we are going to improve some of the considerations presented here and extend their field of application to more singular situations.Remark 0.1.The notion of linear direct connection, introduced in this paper, replaces the notion oflinear quasi connection, introduced by the author in [12].