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Is there a simple proof for the theorem that a 3-dimensional spherically symmetric Riemannian space is conformally flat? I found the statement on web but no proof or reference.

Thanks for the interesting question!

A bit of research first leads us to the fact that being conformally flat (for a 3-dimensional space) is equivalent to the Cotton tensor vanishing:

https://en.wikipedia.org/wiki/Cotton_tensor

I believe it is easy to see that, based on the definition, the spherical symmetry should cause the terms ∇kRij and ∇jRik to cancel, and likewise with the other two such terms, giving you zero identically.

Differential geometry and physics are not my areas of greatest expertise, so please let me know if you have follow up questions or comments.

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Comment | Thank you for your answer. I'll try with the Cotton tensor. It is not as simple as I hoped but looks simple enough. |

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