# Two dimensional manifold are conformally flat

1. May 9, 2010

### paweld

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

2. Sep 19, 2012

### Trifis

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

3. Sep 19, 2012

### bcrowell

Staff Emeritus
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.

Last edited: Jun 20, 2015