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EDIT, 16 July: I was looking for an example of said topological obstructions, and with an assist fom Willie Wong it has worked out: the ordinary sphere $$\mathbb S^2 \subseteq \mathbb R^3$$ does not possess a signature $(+,-)$ metric which I suppose ought to be called Lorentzian for this dimension. The topological obstruction is that $\mathbb S^2$ cannot have a smooth (tangent) "line field," just as it cannot have a smooth nonzero tangent vector field by Brouwer. Now, a Lorentzian metric would give (pointwise) null cones, in this case a pair of distinct but intersecting lines in each tangent plane. As we are using $\mathbb R^3,$ we can cheat and define a line field from the angle bisector of the $+$ part of the cone.