Postulate of Special Relativity leads to the conclusion that the metric in flat Minkowski space $$ds^2=c^2t^2-\textbf{r}^2=\eta_{\mu\nu}dx^\mu dx^\nu\tag{1}$$ is Lorentz invariant. This follows as a consequence of Lorentz transformations. Using this, we can derive that the metric tensor transforms as $$\eta_{\rho\sigma}=\eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma}.\tag{2}$$ Now consider the metric of a curved space: $$ds^2=g_{\mu\nu}(x)dx^\mu dx^\nu\tag{3}$$

$\bullet$ Do we say that $(3)$ is also Lorentz invariant? Do we assume it or does it follow?

$\bullet$ If so, does $g_{\mu\nu}$ transform in the same way under Lorentz transformation as $\eta_{\mu\nu}$ in $(2)$?