Nigsch E A, Vickers J A
Institute of Analysis and Scientific Computing, TU Wien, 1040 Vienna, Austria.
School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK.
Proc Math Phys Eng Sci. 2020 Dec;476(2244):20200640. doi: 10.1098/rspa.2020.0640. Epub 2020 Dec 16.
In this work, we adopt a new approach to the construction of a global theory of algebras of generalized functions on manifolds based on the concept of smoothing operators. This produces a generalization of previous theories in a form which is suitable for applications to differential geometry. The generalized Lie derivative is introduced and shown to extend the Lie derivative of Schwartz distributions. A new feature of this theory is the ability to define a covariant derivative of generalized scalar fields which extends the covariant derivative of distributions at the level of association. We end by sketching some applications of the theory. This work also lays the foundations for a nonlinear theory of distributional geometry that is developed in a subsequent paper that is based on Colombeau algebras of tensor distributions on manifolds.
在这项工作中，我们采用一种新方法来构建基于平滑算子概念的流形上广义函数代数的全局理论。这以一种适用于微分几何应用的形式对先前的理论进行了推广。引入了广义李导数，并证明它扩展了施瓦茨分布的李导数。该理论的一个新特点是能够定义广义标量场的协变导数，它在关联层面扩展了分布的协变导数。我们通过概述该理论的一些应用来结束本文。这项工作还为后续一篇基于流形上张量分布的科洛姆博代数所发展的分布几何非线性理论奠定了基础。
