Abstract Differential Geometry, Differential Algebras of Generalized Functions, and de Rham Cohomology
Abstract Differential Geometry, Differential Algebras of Generalized Functions, and de Rham Cohomology
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抽象微分几何、广义函数的微分代数和德拉姆上同调
DOI：
10.1023/a:1006106718337
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发表时间：
1999
期刊：
影响因子：
--
通讯作者：
E. Rosinger
中科院分区：
文献类型：
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作者：
A. Mallios;E. Rosinger
抽象微分几何是经典微分几何在光滑流形上的最新推广，但它不再使用任何微积分的概念。与光滑函数不同，我们从一个代数层开始，即在任意拓扑空间上考虑的结构层，它是随后涉及的所有层的基空间。此外，还讨论了与适当的‘微分’，即适当的‘Leibniz’层态射相关的模层序列，它们将构成‘微分复形’。这种抽象的方法抓住了许多经典微分几何的本质，因为它为我们提供了一个强大的工具，可以复制并因此推广基本的经典结果。这篇论文的目的是通过包括迄今为止所处理的最大类奇点，给出这个装置可以在多大程度上超越经典框架的指示。因此，我们可以从一个大得多的非光滑的所谓无处稠密的广义函数微分代数开始，而不是从经典的光滑函数代数的结构束开始。这些包含Schwartz分布的后一类代数也提供了任意解析非线性偏微分方程组的整体解。此外，与分布不同的是，作为一个物理问题，这些代数可以处理更大类别的奇点，这些奇点集中在任意封闭的、无处稠密的子集上，因此可以有任意大的正勒贝格测度。在抽象的微分几何背景下，证明了从这些无处稠密的微分代数作为结构束出发，可以重新获得相应的De Rham复形的正确性，也可以得到短的指数序列。这些结果是沿着经典线条和抽象线条发展微分几何的两个基本成分。虽然这里使用的是对易框架，但人们可以很容易地处理一类奇点，这类奇点比迄今为止所处理的任何其他奇点都要大得多，包括在非对易理论中。
Abstract differential geometry is a recent extension of classical differential geometry on smooth manifolds which, however, does no longer use any notion of Calculus. Instead of smooth functions, one starts with a sheaf of algebras, i.e., the structure sheaf, considered on an arbitrary topological space, which is the base space of all the sheaves subsequently involved. Further, one deals with a sequence of sheaves of modules, interrelated with appropriate ‘differentials’, i.e., suitable ‘Leibniz’ sheaf morphisms, which will constitute the ‘differential complex’. This abstract approach captures much of the essence of classical differential geometry, since it places a powerful apparatus at our disposal which can reproduce and, therefore, extend fundamental classical results. The aim of this paper is to give an indication of the extent to which this apparatus can go beyond the classical framework by including the largest class of singularities dealt with so far. Thus, it is shown that, instead of the classical structure sheaf of algebras of smooth functions, one can start with a significantly larger, and nonsmooth, sheaf of so-called nowhere dense differential algebras of generalized functions. These latter algebras, which contain the Schwartz distributions, also provide global solutions for arbitrary analytic nonlinear PDEs. Moreover, unlike the distributions, and as a matter of physical interest, these algebras can deal with the vastly larger class of singularities which are concentrated on arbitrary closed, nowhere dense subsets and, hence, can have an arbitrary large positive Lebesgue measure. Within the abstract differential geometric context, it is shown that, starting with these nowhere dense differential algebras as a structure sheaf, one can recapture the exactness of the corresponding de Rham complex, and also obtain the short exponential sequence. These results are the two fundamental ingredients in developing differential geometry along classical, as well as abstract lines. Although the commutative framework is used here, one can easily deal with a class of singularities which is far larger than any other one dealt with so far, including in noncommutative theories.
DOI：
10.1007/978-3-642-66243-0
发表时间：
1976
期刊：
Energy Sources, Part B: Economics, Planning, and Policy
影响因子：
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作者：
A. Kirillov
通讯作者：
A. Kirillov