Hi,

I see that the tetrad postulate:

$\nabla_{\mu}e_{\nu}^{I}=\partial_{\mu}e_{\nu}^{I}-\Gamma_{\mu\nu}^{\rho}e_{\rho}^{I}+\omega_{\mu J}^{I}e_{\nu}^{J}=0$ Can be merely derived from writing a tensor in two different basis (pure natural-coordinates $\{\partial_\mu\}$ and mixed $\{\partial_\mu\} + \{e_a\}$), my questions are:

1. Does this imply the metricity of the connexion or the inverse?
2. If it is the inverse, how?
3. In writing $g_{\mu\nu}=\eta_{IJ}e_{\mu}^{I}e_{\nu}^{J}$ do we need to impose a metricity on $\eta$?

ps: metricity = metric compatible