Reference request-(Criterion for a manifold to be real analytic)

Question

Let $(M , g)$ be a Riemannian manifold. If there exists a subatlas of normal coordinate systems such that the $g_{ij}$ are real analytic functions with respect to each normal coordinate system in the subatlas, then $(M , g)$ is a real analytic Riemannian manifold with respect to this subatlas.

Answer 1

If you are asking for a necessary and sufficient conditions for existence of such a subatlas, then it is completely hopeless (except for tautologically correct answers). Try first to answer the same question in the one-dimensional case:

Find necessary and sufficient conditions for a positive function $f: R\to R$ to admit a smooth change of variables such that the result is real-analytic.

However, if you want some interesting sufficient conditions, then say, harmonic metrics or metrics given by the Ricci flow provide good examples.