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There is a exterior covariant derivative $$ D \alpha_a = d \alpha_a -\omega^b{}_a \wedge \alpha_b\;, \tag 1 $$ and another $D$ in the vierbein postulate $$ D_\mu e_{I\alpha}=\partial_\mu e_{I\alpha}-\omega_\mu{}^J{}_Ie_{J\alpha}-\Gamma_\mu{}^{\lambda}{}_\alpha e_{I\lambda}=0\;.\tag 2 $$ The question is: Do they the same thing?

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  • $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$ – Qmechanic Feb 22 '17 at 16:33
  • $\begingroup$ No. The clue is in the term "exterior differential" - which is a skew differential. Regarding the orthogonal moving frame, $e_{I\alpha}$ - which I presume is the frame basis vector and not a $2$-form - which typically should be written as $e_{a}(x)=e^{i}_{a}(x)\frac{\partial}{\partial x_{i}}$ where $a$ is the frame index and $i$ is the tangent space index. $\endgroup$ – Cinaed Simson Jan 3 at 20:37
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As far as I known the anti-symmetric part of (2) equals to (1).

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