Hello, I've been struggling with the so often spoken idea that a metric tensor gives you all necessary information about the geometry of a given space. I accept that from the mathematical point of view as every important calculation (speaking as a physicist with respect to GTR rather than differential geometry itself) comes down to the metric. However, that is not enough for me. I would like to be able to come up with an aswer that doesn't rely on pure mathematics. I guess my question boils down to this. How can I know from the validity of Pythagoras' theorem on a given 2D plane that the said plane is flat? To me, metric tensor is essentially local measuring. The problem is I don't see the connection between this local measurment and saying that, for example, a surface is either flat or curved. I would appreciate your advice in this. Thank you.