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I am studying general relativity and i have a question as follow. We have the 2- sphere can be scanned totally by a coordinate system (theta, phi) with the metric tensor written in terms of theta and phi. Now i want to divide the 2-sphere into charts 4 charts then each will have its own coordinate system. Then what will happen for the metric? initially we were having the same formula for the metric for the whole sphere but now shall i say that each chart will have its own metric formula?

Thank you