Bactra Review `
` ` ` Geometrical Methods of Mathematical Physics

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?)