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Is there any relation between symmetries of spacetime and the curvature invariants? For example is spherical symmetric spacetimes, necessarily have positive curvature? Could we define any spherical symmetric spacetime with negative curvature?

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    $\begingroup$ All of the spacetimes described by the Friedmann equations are isotropic (and homogeneous) so they are all spherically symmetric. $\endgroup$ – John Rennie Nov 15 '19 at 16:56
  • $\begingroup$ Are you talking about the curvature of space or spacetime? For example, the Milne model is based on the flat Minkowski spacetime, but can be described by the Friedmann metric with a negative spatial curvature. $\endgroup$ – safesphere Nov 15 '19 at 17:14
  • $\begingroup$ @JohnRennie Do we have any static solution with these properties? $\endgroup$ – Astrolabe Nov 15 '19 at 17:33
  • $\begingroup$ @safesphere I mean the curvature of spacetime which is derived from Ricci tensor contraction $\endgroup$ – Astrolabe Nov 15 '19 at 17:46
  • $\begingroup$ Do we have any static solution with these properties? If you want that to be the question, please edit the question. I mean the curvature of spacetime which is derived from Ricci tensor contraction Please edit the question to clarify that you mean the Ricci scalar. When people ask for clarification of a question, don't provide it in comments, just edit the question. $\endgroup$ – user4552 Nov 15 '19 at 22:10

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