Ricci tensor equals zero implies flat splace?

[Abrahamsk8]

Hi, my question is the title, if Ricci tensor equals zero implies flat space? Thanks for your help

 

[lavinia]

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.

 

[robphy]

This might be helpful:
http://en.wikipedia.org/wiki/Ricci_decomposition
http://en.wikipedia.org/wiki/Curvature_of_Riemannian_manifolds#Weyl_curvature_tensor

 

[bcrowell]

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.