I am confused about conformal transformations on Riemannian manifolds. Here's what I have so far. 1. Under a conformal transformation the metric changes by: g' -> Ω2g 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)(du2 + dv2 ) 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?