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How can a person chart a manifold if he does not know how the manifold looks like ?

E.g. The 2-sphere manifold can have 2 charts and symmetric charts with the chart goes like this ( theta from zero to pi , psy from minus pi to pi ) but the problem for unknown manifold , e.g. In general relativity (4-manifold) when solving GR equations , (consider the vacuum solutions by Schwarzschild ) how to take a chart here to do analysis ? And further more he considers [ spherically symmetric spacetime manifold ] , does this mean that the spatial manifold is 3-sphere , so he can chart by considering ( r larger than zero , theta from 0 to pi , psy from minus pi to pi ) , but who says that the manifold is spherical S^3 ?

Thanks