Gauss theorem egregium says that it is possible for the inhabitants of a 2d surface to calculate the surface curvature without knowing that it is embedded in a 3d euclidean space, simply calculating distances and angles.
But, in order to calculate distances and angles they need a metric (related to their 2 curvilinear coordinates). How can they discover the proper metric if they cannot refer their curvilinear coordinates to the embedding euclidean space?
If possible, please make some example