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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

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  • $\begingroup$ A well-made piece of string will go a long way, no matter how small. $\endgroup$ – TotallyRhombus Aug 30 at 20:13
  • $\begingroup$ @totallyrhombus sorry I don't get your point. Can you explain in mathematical terms? $\endgroup$ – Federico Toso Aug 30 at 20:18
  • $\begingroup$ By constructing a frame using the tangent vectors to the surface and the dot product - and their dual frames. From the dual frames you can calculate the connection without knowing the covariant derivative from the structual equations. Once you have the connection, you can probe the intrinsic geometry. I recommend you move this to math.stackexchange.com/questions/tagged/differential-geometry. $\endgroup$ – Cinaed Simson Aug 30 at 20:23
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Coordinates are just arbitrary labels to keep track of different points in spacetime. They don't come down from a Euclidean embedding space as if from Descartes himself. The inhabitants of the 2D surface are free to assign arbitrary coordinates to each point, taking care only that nearby points have nearby coordinates. (If you have Misner, Thorne, and Wheeler’s Gravitation, read section 1.2, “Spacetime with and without coordinates.”) Then they can measure distances between various pairs of points, and figure out what metric produces those distances from the coordinates they assigned.

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