Abstract We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between (n + 1) points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of defining the scalar curvature on a non-smooth metric space. In the second part we will discuss this issue, focusing in particular on Alexandrov spaces and surfaces with bounded integral curvature.

中文翻译：

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通过局部范围的标量曲率

摘要 我们给出了光滑黎曼流形的标量曲率的度量表征，分析了一个点的无穷小邻域中 (n + 1) 个点之间的最大距离。由于这种表征纯粹是根据距离函数，它可以用来解决在非光滑度量空间上定义标量曲率的问题。在第二部分中，我们将讨论这个问题，特别关注 Alexandrov 空间和具有有界积分曲率的曲面。