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**FRW from spaces of constant curvature**

I'm reading how the Friedmann Robertson Walker metric (with its k=-1,0,+1) is derived by considering spaces of constant curvature in order to satisfy the cosmological principle that space is the same everywhere. But I'm not able to find any derivation of these metrics of constant curvature. And that's where we get k=-1,0,+1. Any help out there? Thanks.

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