vectors in the tangential vector space of different space time points. They are defined by the condition that the metric shall be left invariant by the covariant derivative

$$D_\mu (g_{\rho\sigma})=0
\label{38}$$

and have the following shape

$$\Gamma^\rho_{\mu\nu}=\frac{1}{2}g^{\rho\sigma}(\partial_\nu g_{\mu\sigma}+\partial_\mu g_{\nu\sigma}-\partial_\sigma g_{\mu\nu}).
\label{41}$$

(The explicit derivation of this relation can be found in any book on general relativity, e.g. [@[book4]].)

We can express the covariant derivative of the vierbein by applying the Christoffel symbol to each index leading to

$$D_\mu e^m_\nu=\partial_\mu e^m_\nu+\Gamma^m_{\mu n}e^n_\nu-\Gamma^\rho_{\mu\nu}e^m_{\rho}.
\label{42}$$

Inserting Eq. $(\ref{28})$ into Eq. $(\ref{38})$ and considering the case $\nu=\rho$ one obtains

$$\begin{aligned}
D_\mu(g_{\nu\nu})&=D_\mu (e_\nu^m e_\nu^n \eta_{mn})=D_\mu ((e_\nu^m)^2 \eta_{mm})\nonumber\\
&=\eta_{mm} D_\mu (e_\nu^m)^2=2\eta_{mm}e_\nu^m D_\mu e_\nu^m=0.
\label{43}\end{aligned}$$

which means that the covariant derivative applied to the vierbein gives zero

$$D_\mu e^m_\nu=0.
\label{44}$$

Here we have used that $D_\mu\eta_{mn}=0$ and $\eta_{mn}$ is a diagonal matrix. Combining Eq. $(\ref{42})$ and $(\ref{44})$ results in

$$\partial_\mu e^m_\nu+\Gamma^m_{\mu n}e^n_\nu-\Gamma^\rho_{\mu\nu}e^m_{\rho}=0.
\label{45}$$

By moving the second term of Eq. $(\ref{45})$ to the r.h.s. side of the equation, multiplying with $e^\nu_n$ and raising the flat coordinate indices with $\eta^{\alpha\beta}$ we arrive at a relation between the above Dirac Spinor coefficients and the Christoffel symbols in arbitrary coordinates.

$$\omega_\mu^{mn}=\Gamma_{\mu}^{mn}=-\eta^{n p}e^{\nu}_{p}(\partial_\mu e_\nu^m-\Gamma^\rho_{\mu\nu} e_\rho^m).
\label{46}$$

Here, we can identify the 