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.

中文翻译：

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流形上的非线性广义函数


在这项工作中，我们采用一种基于平滑算子概念的新方法来构建流形上广义函数代数的全局理论。这以适合微分几何应用的形式产生了先前理论的概括。引入并证明了广义李导数是对 Schwartz 分布的李导数的扩展。该理论的一个新特征是能够定义广义标量场的协变导数，该导数在关联级别上扩展了分布的协变导数。最后我们概述了该理论的一些应用。这项工作还为分布几何的非线性理论奠定了基础，该理论在随后的论文中基于流形上张量分布的哥伦博代数而发展。