TITLE: What exactly is the difference between a Ricci-flat manifold and Minkowski space?
QUESTION [1 upvotes]: Minkowski space describes a flat spacetime and a Ricci-flat manifold is a solution of GRT without curvature. Shouldn't they be the same? So, what's the difference?

REPLY [9 votes]: A flat spacetime has a zero Riemann tensor, while a Ricci flat spacetime has a zero Ricci tensor.
The Ricci tensor is a contraction of the Riemann tensor, and it is possible for the Ricci tensor to be zero when the Riemann tensor is not. An obvious example of this is the Schwarzschild geometry that describes a static black hole. In this geometry the Ricci tensor is zero everywhere but the spacetime is most certainly not flat.

REPLY [4 votes]: Ricci flat is not necessarily "truly" flat. Minkowski space has the entire Riemann curvature tensor vanishing, ie, $R_{\mu\nu\rho\sigma}=0$. However, being just Ricci flat is a weaker condition that only requires that the Ricci tensor $R_{\mu\nu}$ vanish.
For example, Calabi-Yau manifolds are Ricci flat, but they are certainly not the same as Minkowski space.
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