Given a manifold $\mathcal{M}$ with fixed "shape" (say a hemisphere), we may define two sets of Riemannian metrics and connections for $\mathcal{M}$, say $g_{ij},\Gamma_{i,j}^k$ and $g'_{ij}, \Gamma_{i,j}^{'k}$. My questions are as follows:
Does different choices of metrics+connections simply imply different choices of coordinate systems for $\mathcal{M}$? If so, how to find a map between the two coordinate systems based on $g_{ij},\Gamma_{i,j}^k$ and $g'_{ij}, \Gamma_{i,j}^{'k}$?
Is it sufficient to say that a $\mathcal{M}$ is flat if we can find a map $f:R\rightarrow R$ so that $[f(x_1),\ldots,f(x_d)]=[y_1,\ldots,y_d]$, where $[x_1,\ldots,x_d]$ are the coordinates of an arbitrary coordinate system for $\mathcal{M}$ and $[y_1,\ldots,y_d]$ are Euclidean coordinates?
Thanks!