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While Minkowski space and Euclidean space both have identically zero curvature tensors it seems that a flat Lorentz manifold in general, may not admit a flat Riemannian metric. Such a manifold is the quotient of Minkowski space by the action of a properly discontinuous group of Lorentz isometries. This group in general is not a subgroup of the group of rigid motions of Euclidean space and so can not have a flat Riemannian metric.
What can one say about the curvature of Riemannian metrics on flat Lorentz manifolds?
What can one say about the curvature of Riemannian metrics on flat Lorentz manifolds?