I'm looking for a good literature with a proof of Riemann theorem.
"A Riemannian manifold is locally flat iff R=0"
In particular I would like to see how $R=0$ implies that there exist a coorinate system, where $\Gamma^i_{jk}=0$
Edit: I know that this theorem works for Riemannian manifold $(M,g)$, i.e. "$(M,g)$ is flat $\iff \exists$ a local isometry $\phi$ to $(\mathbb{R}^n, \delta)$ "
I wonder if we can do similar for a differential manifold with an symmetric affine conection (not a Levi-Civita), i.e. without metric. In other words, is the following true?
"A manifold with an affine connection $(M,\nabla)$ is flat $\iff \exists$ a local diffeomorphsim $\phi$ to $(\mathbb{R}^n, \bar{\nabla})$"
$\bar{\nabla}$ - is an affine connection, whose Christoffel symbols are zero
Thanks,