Lorentzian structure

  • Thread starter MilesReid
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It is fairly easy to prove that each manifold can be given a Riemannian structure. The argument is standard: locally you give the riemannian structure and then you use partions of unity. This proof breaks down for signed metrics. Even for a manifold requiring only two charts. For example, I've been told that you cannot put on S^2 a metric with signature +1, -1. This is quite remarkable, since on S^2-{p} diffeo to R^2 you can! Any ideas on proving it?
 

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The existence of a Lorentz metric implies the existence of a non-vanishing vector field (in physical terms, time's arrow is always defined). The two-sphere doesn't have one.
 

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