Can a manifold be reconstructed from its charts? I'm learning special relativity and I am having a confusion on this mathematical point. Whenever any sort of motion or non motion happens in the world, it can only be perceived by the scientist in a chart (inertial frame). But, we say there exists a common manifold which is being charted by different observers when doing observation.
How can one from just seeing the charts with transition maps and metric (spacetime - metric) know the existence of a manifold?
In a more simplistic sense, could I reconstruct how the earth sits in space just from seeing pages the world map (with marked distances)?

To my mind there are two ways to answer this question. From a mathematical point of view, a variety is a set with a set of maps which are bijections with open sets of $R^n$, subject to some conditions. So in principle, from the definition, we know what is the variety. From another point of view, we can also start from a set of charts and try to put two by two pieces together, as we do with road maps, or with an atlas. It is more intuitive, but it is not the usual definition.
