Proving That a Lorentzian Structure Cannot be Put on S^2

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SUMMARY

The discussion centers on the impossibility of endowing the two-sphere (S^2) with a Lorentzian structure, specifically a metric with signature +1, -1. While every manifold can be given a Riemannian structure using partitions of unity, this approach fails for signed metrics. It is established that the existence of a Lorentz metric necessitates a non-vanishing vector field, which the two-sphere lacks, thereby proving that a Lorentzian structure cannot be applied to S^2.

PREREQUISITES
  • Understanding of Riemannian geometry
  • Familiarity with Lorentzian metrics
  • Knowledge of vector fields and their properties
  • Basic concepts of manifold theory
NEXT STEPS
  • Study the properties of Riemannian and Lorentzian metrics
  • Explore the concept of partitions of unity in manifold theory
  • Investigate the implications of non-vanishing vector fields
  • Examine examples of manifolds that can and cannot support Lorentzian structures
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Mathematicians, theoretical physicists, and students of differential geometry interested in the properties of manifolds and the implications of metric signatures.

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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