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):20200642. doi: 10.1098/rspa.2020.0642. Epub 2020 Dec 16.
This paper builds on the theory of nonlinear generalized functions begun in Nigsch & Vickers (Nigsch, Vickers 2021 20200640 (doi:10.1098/rspa.2020.0640)) and extends this to a diffeomorphism-invariant nonlinear theory of generalized tensor fields with the sheaf property. The generalized Lie derivative is introduced and shown to commute with the embedding of distributional tensor fields and the generalized covariant derivative commutes with the embedding at the level of association. The concept of a generalized metric is introduced and used to develop a non-smooth theory of differential geometry. It is shown that the embedding of a continuous metric results in a generalized metric with well-defined connection and curvature and that for metrics the embedding preserves the curvature at the level of association. Finally, we consider an example of a conical metric outside the Geroch-Traschen class and show that the curvature is associated to a delta function.
本文基于Nigsch和Vickers（Nigsch，Vickers 2021 20200640（doi：10.1098/rspa.2020.0640））中开始的非线性广义函数理论，并将其扩展为具有层性质的广义张量场的微分同胚不变非线性理论。引入了广义李导数，并证明其与分布张量场的嵌入可交换，广义协变导数在关联层面与嵌入可交换。引入了广义度量的概念，并用于发展非光滑微分几何理论。结果表明，连续度量的嵌入会产生具有明确定义的联络和曲率的广义度量，并且对于度量而言，嵌入在关联层面保持曲率。最后，我们考虑一个不在Geroch-Traschen类中的锥形度量示例，并表明曲率与一个狄拉克函数相关。
