Einstein manifold | Wikipedia audio article
Channel: Subhajit Sahu
Date: 2018-12-24

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Einstein_manifold


00:00:56 1 The Einstein condition and Einstein's equation
00:01:01 2 Examples
00:02:42 3 Applications
00:03:40 4 See also
00:05:28 5 References
00:06:55 See also



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SUMMARY
=======
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to the four-dimensional Lorentzian manifolds usually studied in general relativity.
If M is the underlying n-dimensional manifold and g is its metric tensor the Einstein condition means that

  
    
      
        
          R
          i
          c
        
        =
        k
        g
      
    
    {\displaystyle \mathrm {Ric} =kg}
  for some constant k, where Ric denotes the Ricci tensor of g. Einstein manifolds with k = 0 are called Ricci-flat manifolds.