# Metric tensor derived from a geodesic

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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?

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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.

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?

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

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
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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).

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?