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So my understanding of Riemann geometry is that whether a surface has curvature or not depends entirely on the metric the manifold is equipped with. Furthermore there are many other metrics a manifold can have and the identify function should act as a pullback between them - since those are Riemann metrics i don't see how they could have incompatible topology amongst each other to to break the smoothness of ##id##.

So given a problem (e.g. an equation) on a specific Riemann manifold there should be no issue to pull it back to any other geometry (same manifold, different Riemann metric) and have an equivalent description of the problem there. My question is now if there is anything that distinguishes a specific and unique metric as in "the real depiction of the surface e.g. how it's really curved". Until now my understanding was that one metric is as good as another and if a distinction is made it's just for pragmatic reasons.

So far my naive view was that curvature was merely something resulting from how we chose to measure distances and therefore by itself not inherently a physical entity, which is dictated by nature. On the other hand i am very well aware that any information contained in the curvature of a physical problem is crucial, but i wonder if that's the only way to depict it.