Euclidean rather than Riemannian

I think that you mean Euclidean rather than Minkowskian. Euclidean vs Riemannian has to do with whether spacetime is curved (Euclidean no, Riemannian yes), while Euclidean vs Minkowskian has to do with whether the metric treats the time coordinate differently (Euclidean no, Minkowskian yes). (And then the spacetime of classical general relativity, which answers both questions yes, is Lorentzian.)