An explicit Lorentzian metric on the Klein bottle

Question

I want to construct an explicit Lorentzian metric on the (abstract) Klein bottle but have no idea where to start. Could someone please give me a hint and/or guide me in the right direction? Thanks.

Answer 1

Start with the standard Lorentzian metric $dx^2 - dy^2$ on the Euclidean plane. Restrict that metric to the unit square $[-1,1] \times [-1,1]$. Notice that the two gluing maps which are used to construct the Klein bottle, namely the maps $(x,y) \to (x+2,y)$ and $(x,y) \to (x,y+2)$, are isometries of the standard Lorentz metric. Therefore, the metric restricted to the square descends to a Lorentz metric on the Klein bottle.

I'll throw out that this exact same line of reasoning is what one uses to prove that the torus or Klein bottle has a Euclidean metric, and many other similar metric constructions.