special relativity - Can Minkowski spacetime be redefined as a non-flat riemannian manifold?

Minkowski space time is defined in terms of a flat pseudo-Riemannian manifold. I have wondered if it can be redefined as Riamannian manifold and in the case what type of curvature would there appear.

Formally:

Let M be a semi-Riemannian manifold of dimension 4, corresponding to the Minkowski space, and let g be the metric tensor (non positive definite), T be the Riemann curvature tensor and P a generic point of M.

Question 1

Which (if any) of the following is true?

a. at every P there exists one system of coordinates for which the metric g becomes Riemannian (positive definite) in a ball of radius R non infinitesimal centred in P

b. there exists one system of coordinates for which g is Riemannian (positive definite) at all P of M

Comment: in words, can we, with a change of coordinates, get rid of semi-Riemannianity – either in finite region or globally?

If this is the case, how do we pay it in terms of curvature? This the next question:

Question 2

c. if previous a) is true, is it true that T cannot be null in the entire ball? And what type of curvature T "displays"?

d. if previous b) is true, is it true that T cannot be null in the entire ball? And what type of curvature T "displays"?

Thanks a lot