My (physics) lecture notes for differentiable manifolds say that the charts will necessarily overlap and therefore we need smooth transitions functions. Is this strictly true, can't we always define charts such that they don't overlap?
2026-03-30 02:05:44.1774836344
Why must charts on a manifold necessarily overlap?
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Since the charts are defined to be open, this means that, if we are to cover the entire manifold, we will necessarily have to have charts with a non-zero intersection.