What is the least number of charts needed to specify a given smooth manifold, for simplicity of dimension 2? For instance the minimum number of charts to cover a torus, or a 2-sphere or a 2D 1-sheet hyperboloid? I would think it goes with the definition of manifold that in any case you need at least two charts are needed to cover the whole manifold, I think in the case of the 2-sphere is enough with two, but the torus needs 3, but another poster seems to think that in general it is enough with one chart of coordinates but I know that it is only possible trivially with the spaces like E^n in the case of Riemannian manifolds or M^n in case of pseudoriemannian, because by definition the Euclidean space R^n is a smooth manifold with a single chart, and the same in the case of a (-,+,+,+) signature with Minkowski space would need just one chart. But other than this I'd say you need more than one chart otherwise it defeats the manifold concept. Anyone agrees? Thanks in advance.