Two dimensional manifold are conformally flat

[paweld]

Does anyone know why every 2D manifold is conformally flat.

 

[Trifis]

If you have access to d' Inverno's textbook, have a look at exercise 6.30 .

 

[bcrowell]

For insight, consider the example of latitude and longitude. The fact that the Earth is curved doesn't prevent you from mapping a neighborhood of the Earth's surface to Cartesian graph paper using lines of latitude and longitude. This mapping is conformal, because all the right angles remain right angles. [Oops, this isn't quite right. Only the Mercator mapping is conformal.]

Also consider that in two dimensions, the Riemann tensor only has one independent element, which is the Gaussian curvature. This means that you can't have a distinction between Ricci curvature and sectional curvature.