I'd like a hint for the following exercise: Find a conformally flat transformation for the 2D metric of the sphere $$ds^{2}=d\theta^{2}+sin^{2}(\theta)d\phi^{2}$$ Or at least some bibliography. Any help is greatly appreciated.
Conformally flat transformation of a metric
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differential-geometry
coordinate-systems
1 Answers
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You know about stereographic projection from (most of) the sphere to the plane, and stereographic projection is conformal. So it should not come as a surprise that the sphere is conformally flat.
Here's a hint: Try multiplying through by something to remove the $\sin^2(\theta)$ from $d\phi^2$. Why is the resulting metric flat?