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Question
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.

Answer
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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Clyde Oliver

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I can answer all questions up to, and including, graduate level mathematics. I am more likely to prefer questions beyond the level of calculus. I can answer any questions, from basic elementary number theory like how to prove the first three digits of powers of 2 repeat (they do, with period 100, starting at 8), all the way to advanced mathematics like proving Egorov's theorem or finding phase transitions in random networks.

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I am a PhD educated mathematician working in research at a major university.

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