Undergrad Metric tensor derived from a geodesic

[VladZH]

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?

 

[zwierz]

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.

 

[VladZH]

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.

Thanks for reply! Could you tell me what is the complexity? As I see I need to solve partial differential equations. That's it?

 

[zwierz]

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

 

[VladZH]

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?

 

[Orodruin]

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?

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).

 

[VladZH]

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).

Can my chosen curve/curves family give me information about torsion? Should it have second derivative to have a torsion?

 

[VladZH]

Is there really no answer?

 

[robphy]

possibly interesting reading:
https://mathoverflow.net/questions/132244/can-one-recover-a-metric-from-geodesics

 

[VladZH]

possibly interesting reading:
https://mathoverflow.net/questions/132244/can-one-recover-a-metric-from-geodesics

Thanks for reply. I have seen this question. For now my question is: if we define geodesics over a subset of a manifold, can we just say that there is no torsion, put geodesic curves into geodesic equation, solve PDE and get a metric or bunch of metrics for the subset?