Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Two dimensional manifold are conformally flat

  1. May 9, 2010 #1
    Does anyone know why every 2D manifold is conformally flat.
  2. jcsd
  3. Sep 19, 2012 #2
    If you have access to d' Inverno's textbook, have a look at excercise 6.30 .
  4. Sep 19, 2012 #3


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - dimensional manifold conformally Date
A Two Dimensional Ricci curvature Mar 5, 2018
5D Space-time (and higher ) Mar 4, 2015
Null geodesic in 2 dimensional manifold Mar 30, 2010
Weyl tensor on 3-dimensional manifold Aug 7, 2006