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Ricci tensor equals zero implies flat splace?

  1. Apr 29, 2015 #1
    Hi, my question is the title, if Ricci tensor equals zero implies flat space? Thanks for your help
  2. jcsd
  3. Apr 29, 2015 #2


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    If by flat space you mean a flat Riemannian manifold, the answer is no. There are examples of both compact and non-compact Ricci flat manifolds that cannot be given flat Riemannian metrics.

    One thing to note is that for compact manifolds,a flat Riemannian metric implies that the Euler characteristic is zero - because the Euler class can be expressed as a polynomial in the curvature 2 form. If you look around you will find Calabi -Yau manifolds with non-zero Euler characteristic.
    Last edited: Apr 30, 2015
  4. May 2, 2015 #3


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  5. May 3, 2015 #4


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    As a physics application, in the semi-Riemannian case, a vacuum has a Ricci tensor equal to zero. A vacuum doesn't have to be flat. A vacuum be curved by gravity. An example would be the space above the earth's surface where you're sitting right now.
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