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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

    bcrowell

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    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
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