This paper builds on the theory of nonlinear generalized functions begun in Nigsch & Vickers (Nigsch, Vickers 2021 Proc. R. Soc. A 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 C2 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.

中文翻译：

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分布几何的非线性理论


本文以 Nigsch 和 Vickers 开始的非线性广义函数理论为基础 (Nigsch, Vickers 2021 Proc. R. Soc. A 20200640 (doi:10.1098/rspa.2020.0640))，并将其扩展到广义微分同胚不变非线性理论具有轴属性的张量场。引入广义李导数并证明其与分布张量场的嵌入可交换，而广义协变导数可与关联级别的嵌入交换。引入广义度量的概念并用于发展微分几何的非光滑理论。结果表明，连续度量的嵌入会产生具有明确定义的连接和曲率的广义度量，并且对于 C2 度量，嵌入会保留关联级别的曲率。最后，我们考虑 Geroch-Traschen 类之外的圆锥度量的示例，并表明曲率与 delta 函数相关。