Gauss remarkable theorem proves that the inhabitants of a 2D curved surface can discover that their surface is curved without being aware of the embedding dimension, simply calculating distances and angles.
However: in order to calculate distsnces and angles they must use the proper metric, expressed in their curvilinear coordinates. And in order to find this expression, they have to relate their curvilinear coordinates to the embedding euclidean space. So they need to "look outside".
Isn't it a circular reasoning? They must "look outside" to find the metric, which they then use to find the curvature intrinsically (???) Where am I going wrong?
This Is misleading in mathematics I agree. what they are saying that the curvature is invariant with respect to the embedding. this is like saying a tensor is invariant when you change coordinates. but this doesn't tell you what it is irrespective if coordinates, just that it doesn't matter what coordinates you use.
Measure Circle circumferences And radiuses intrinsically on a surface in the neighborhood of a point, and if circumferences turn out to be 2 pi *r, than the surface is not curved at that point.