The definition of flat metric has two definitions: 1. given a metric norm $F$ on manifold $M$, there exists coordinate charts s.t. for every point $p$, all differentials of the norm is zero, i.e. $\partial^{(n)} F(v) = 0$. 2. For Riemannian metric, Christopher symbols of Levi-Civita connection are all zero. $\nabla_{\frac{\partial}{\partial x_i}} \frac{\partial}{\partial x_j} = \Gamma_{ij}^k {\frac{\partial}{\partial x_k}}$, here $\Gamma_{ij}^k = 0$.
My question is that: Is flat metric related to flat modules in homological algebra and flat vector bundles? Can you show it using coordinates symbols?