I wish to study curved spacetime with torsion, however, the trouble is how do I go about with the variational principle? Should I assume the connection $\Gamma^{\alpha}_{\beta\gamma}$ and the metric $g_{\mu\nu}$ as independent variables? Maybe this is fine when metricity is relaxed but what if metricity is imposed i.e. $\nabla_{\mu}g_{\alpha\beta} = 0$ then $$\partial_{\mu}g_{\alpha\beta}-\Gamma^{\lambda}_{~~~\alpha\mu}g_{\lambda\beta}-\Gamma^{\lambda}_{~~~\beta\mu}g_{\alpha\lambda} = 0$$ then clearly the connection is not independent of the metric. Please advice.
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