A chart with constant metric components has vanishing metric-selected Christoffel symbols and curvature

(The zero-coefficient connection.) On the open submanifold $U$, consider the coordinate connection ${\mathrm{\nabla }}^{\mathrm{c}}$ whose Christoffel symbols in $(U,x)$ are all zero; this is an affine connection on $U$ by [L1]. Its coefficients satisfy ${\mathrm{\Gamma }}^a{}_{bc}=0={\mathrm{\Gamma }}^a{}_{cb}$, so it is torsion-free; and the compatibility identity of [L2] holds because both sides vanish: the left side is ${\mathrm{\partial }}_a{g}_{bc}=0$ since the metric components are constant, and the right side is a sum of terms each containing a Christoffel symbol. Hence ${\mathrm{\nabla }}^{\mathrm{c}}$ is a torsion-free metric-compatible connection for the restricted metric $g{\mathrm{\mid }}_U$; by uniqueness in [L2] it equals the metric-selected connection of $(U,g{\mathrm{\mid }}_U)$, which is the restriction $\mathrm{\nabla }{\mathrm{\mid }}_U$ of the metric-selected connection of $M$ (its restriction is torsion-free and metric-compatible on $U$). Therefore every Christoffel symbol of $\mathrm{\nabla }$ in $(U,x)$ vanishes on $U$, which is (a).

(The curvature.) Substituting the vanishing coefficients of step 1.1 into the coordinate formula [L3] makes every term vanish identically on $U$, so $R^a{}_{bcd}=0$ on $U$ for all indices; since the curvature is a tensor field, $R=0$ on $U$. This is (b).

(Coordinate curves.) Let $y(s)=(s,c^1,c^2,c^3)$ be the coordinate path of a coordinate curve; then $\dot{y}=(1,0,0,0)$ and $\ddot{y}=0$. The right side of the geodesic equation of [L4] vanishes because every ${\mathrm{\Gamma }}^k{}_{ij}$ vanishes by step 1.1, so the equation holds and the coordinate curve is a geodesic in the affine parameter $s$. This proves (c) and completes the proof.