I've heard assertions of the sort:


1. Let there be a Riemann metric (not very smooth, say of class $C^1$ or $C^2$ or maybe $C$?) in a neighbourhood of a point on a manifold. Then it is possible to choose coordinates so that the metric is $C^\infty$ or even analytic in them.
2. In case of 3-dimensional manifolds it is possible to choose such coordinates globally, so the manifold becomes a smooth one. In the case of higher dimensions $n\ge4$ it is not true.


Are those assertions true? I've heard them some time ago and not sure I remember all the details. Is it a well-known thing? Are there some detailed references?



Questions on smoothness of Riemann metrics
1. NO. Given a Riemannian manifold, it might be possible to improve smoothness by changing atlas. The atlas with harmonic functions as coordinates is the best [proved by Samuil Shefel (1979) and rediscovered by Dennis DeTurck and Jerry Kazdan (1981)]. But, the obtained metric might be worse than $C^\infty$.
2. There is no local-global issue here, harmonic atlas is defined locally and it is the best one globally. So you get problems starting with dimension 2.


