  This article studies the smoothness of conformal mappings between two
Riemannian manifolds whose metric tensors have limited regularity. We show that
any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two
manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal (local)
diffeomorphism. This result was proved in [12, 27, 33], but we give a new proof
of this fact. The proof is based on $n$-harmonic coordinates, a generalization
of the standard harmonic coordinates that is particularly suited to studying
conformal mappings. We establish the existence of a $p$-harmonic coordinate
system for $1 < p < \infty$ on any Riemannian manifold.
