Bactra Review    Geometrical Methods of Mathematical Physics

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More technically, the manifold looks like R^n locally. Yet more technically, for every point on the manifold there is a continuous 1-1 map from an open neighborhood of the point onto an open set in R^n. In general there isn't a single mapping, a single coordinate system, which can be extended consistently over the entire manifold: even spheres are like this. (What is the longitude of the poles?)