Minkowski metric on a surface

Question

Do closed surfaces admit a metric with lorentzian signature?

Any reference?

Answer 1

Not in general. A closed connected orientable manifold admits a Lorentzian metric if and only if its Euler characteristic vanishes. So the torus $S^1 \times S^1$ admits a Lorentzian metric (it can be described as $\mathbb{R}^{1, 1} / \mathbb{Z}^2$), and that's it for the orientable surfaces. In the nonorientable case it's necessary that the Euler characteristic vanishes but I'm not sure if it's sufficient. For surfaces it's clear that the Klein bottle also admits a Lorentzian metric.