In general relativity we introduce local inertial frames to be such frames where the laws of special relativity holds. Let $\xi^{\alpha}$ the coordinates in the local inertial frame, so we get $$ds^2=\eta_{\alpha \beta}d \xi^{\alpha} d \xi^{\beta}.$$ If we switch the frame of reference to coordinates $x^{\mu}$ : $\xi^{\alpha}= \xi^{\alpha}(x^0, x^1, x^2, x^3)$ and with $$g_{\mu \nu} (x)= \eta_{\alpha \beta} \frac{\partial \xi^{\alpha}}{\partial x^{\mu}} \frac{\partial \xi^{\beta}}{\partial x^{\nu}}$$ we get:
$$ds^2=g_{\mu \nu}d x^{\mu}(x) d x^{\nu}.$$
I don't understand why it isn't possible to find a transformation to get $$ds^2=\eta_{\alpha \beta}d \xi^{\alpha} d \xi^{\beta}$$ on the whole or almost the whole manifold? Because $g_{\mu \nu}(x)$ is still the same on the whole manifold?