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I Metric tensor derived from a geodesic

  1. Apr 17, 2017 #1
    Let we have a 2D manifold. We choose a coordinate system where we can construct all geodesics through any point. Is it enough to derive a metric from geodesic equation? Or do we need to define something else for the manifold?
  2. jcsd
  3. Apr 17, 2017 #2
    Geodesic curves are generated by connection; you can have a connection but do not have metric. Restoring metric compatible with symmetric connection is not a simple problem.
  4. Apr 17, 2017 #3
    Thanks for reply! Could you tell me what is the complexity? As I see I need to solve partial differential equations. That's it?
  5. Apr 17, 2017 #4
    Yes, you must solve the PDE to restore metric by known connection; these PDE are not obliged to have a suitable solution as well as they are not obliged to have a unique solution
  6. Apr 19, 2017 #5
    There is a question, though. Let we have a curve equation not w.r.t. the affine parameter but a 2D chart. Can I use the geodesic equation with this curve equation to find connection coefficients? If so, will solving further PDE give me a correct metric in this coordinate system?
  7. Apr 19, 2017 #6


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    In general, no. The geodesic equation does not depend on the anti-symmetric part of the connection coefficients. You can only find the connection coefficients if you also specify the anti-symmetric part (i.e., you need to specify the torsion).
  8. Apr 20, 2017 #7
    Can my chosen curve/curves family give me information about torsion? Should it have second derivative to have a torsion?
  9. May 8, 2017 #8
    Is there really no answer?
  10. May 8, 2017 #9


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  11. May 12, 2017 #10
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