TITLE: Can anyone give an example of Ricci flat Riemannian or Lorentzian Manifold that is not flat?
QUESTION [6 upvotes]: Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat?  And is there any theorm about Ricci-flat but not flat?  
I am especially interset in the case of Lorentzian Manifold whose sign signature is (- ,+ ,+ , + ). Of course, the example is not constricted in Lorentzian case.
I know there are many Ricci flat case in General Relativiy which is the vacuum solution to Einstein's equation. But what I know, such as Kerr solution, are all geodesic incomplete. So I want a geodesic complete example and reference. Thanks!

REPLY [5 votes]: Here's another explicit reference, on top of Ben Crowell's more general comment. The following paper discusses explicit examples of "pp-wave" spacetimes (Lorentzian, solving vacuum Einstein equations) that are geodesically complete: Causal structures of pp-waves by Veronika E. Hubeny and Mukund Rangamani (arXiv:hep-th/0211195).