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I'm currently taking a course in Semi-Euclidean (Semi-Riemannian) Geometry and am given the task to give an example of a non-orientable surface $S$ in $\mathbb{R}^3_1$ which is time-type.

With the Lorentzian dot product: $$ \langle u,v\rangle = u_1v_1 + u_2v_2 - u_3v_3 $$ We say that a vector is time-type whenever $\langle v,v\rangle < 0$. And we say that a surface is time-type iff the tangent plane to at any point in $S$ contains a time-type vector.

I cannot think of an example that is non-orientable and time-type. We know the classic example of the mobius band which is non-orientable but there is at least one point where the tangent plane does not contain one such vector.

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