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jbergman

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So just so I understand, you want to show the following, given an arbitrary connection, does there exist a metric compatible with it and what does it look like? You don't care about the signature of the resulting metric, whether it is lorentzian or Riemannian?Jianbing_Shao said:Thanks!

From metric compatibility equation, we can use parallel propagator(defined in Sean Carroll' book) to express the metric with connection.

Then the difference between a metric compatible with a torsion free connection and the metric compatible with the same torsion free connection plus a torsion. in other words if the torsion part have effects on the metric compatible with the connection.

My original problem is just to find out that if an arbitrary connection have a metric compatible with it then we put them into the equation can make the equation work and if such connections do exist then why.