Are All Two-Dimensional Manifolds Conformally Flat?

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SUMMARY

All two-dimensional manifolds are conformally flat, as established by the ability to find coordinates where the metric is a scalar multiple of the standard flat metric. This fundamental result was first proven by Carl Friedrich Gauss. Additionally, manifolds with constant sectional curvature are also conformally flat, although the proof for this statement requires further exploration and consideration of partial differential equations (PDEs).

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Why all two dimensional manifolds are conformally flat?
Why all manifolds with constant sectional curvature are conformally flat?
Does anyone know proofs of above statements.
Thanks in advance.
 
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paweld said:
Why all two dimensional manifolds are conformally flat?
Why all manifolds with constant sectional curvature are conformally flat?
Does anyone know proofs of above statements.
Thanks in advance.

On a surface one can always find coordinates where the metric is a scalar times the standard flat metric on the coordinate tangent vectors. You should try to prove this yourself. It was first proved by Gauss. You will get a PDE.

For constant sectional curvature I will have to think about it.
 

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