I am confused about conformal transformations on Riemannian manifolds. Here's what I have so far.(adsbygoogle = window.adsbygoogle || []).push({});

1. Under a conformal transformation the metric changes by:

g' -> Ω^{2}g

2. Under a Weyl transformation the metric changes by:

g' -> exp(-2f)g

3. Any 2D Riemann manifold is locally conformally flat and the metric can be defined in terms of isothermal coordinates.

g = exp(f)(du^{2}+ dv^{2})

Where u and v are Euclidean.

How are these all tied together? There appears to be a commonality in form but the multiplying functions are different. Where does the exponential come from?

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# A Confusion regarding conformal transformations

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