Write Torsion Tensor: Definition, Metric Tensor & Equation

[Physicist97]

Would it be possible to write the torsion tensor in terms of the metric? I know that a symmetric Christoffel Symbol can be written in terms of the partial derivatives of the metric. This definition of the christoffel symbols does not apply if they are not symmetric. Is it possible to write a definition of the non-symmetric christoffel symbols in terms of a metric, and from that find an equation for the torsion tensor, or is it independent of the metric tensor?

 

[Orodruin]

The torsion free condition is required to uniquely define a metric compatible connection. If you let it go, there are several possible metrics (one of which is the torsion free one).