Riemannian manifold
Regularity of the Riemannian metric

The Riemannian metric g {\displaystyle g} is continuous if its components g i j: U → R {\displaystyle g_{ij}:U\to \mathbb {R} } are continuous in any smooth coordinate chart (U, x). {\displaystyle (U,x).} The Riemannian metric g {\displaystyle g} is smooth if its components g i j {\displaystyle g_{ij}} are smooth in any smooth coordinate chart. One can consider many other types of Riemannian metrics in this spirit, such as Lipschitz Riemannian metrics or measurable Riemannian metrics.

There are situations in geometric analysis in which one wants to consider non-smooth Riemannian metrics. See for instance (Gromov 1999) and (Shi and Tam 2002). However, in this article, g {\displaystyle g} is assumed to be smooth unless stated otherwise.
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