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\begin{document}

\( T^{p}_{uv}=\Gamma ^{p}_{uv}-\Gamma ^{p}_{vu} \)

So you can define \( T^{p}=\frac{1}{2}T^{p}_{uv}dx^{u}\wedge dx^{v} \)

So in the vielbein basis, \( T^{a}=\frac{1}{2}T^{a}_{bg}\theta ^{b}\wedge \theta ^{g}=d\theta ^{a}+w^{a}_{b}\theta ^{b} \)

Where the last equality is the first cartan structure equation.

Proof:

\( T^{a}(e_{b},e_{g})=T^{a}_{bg} \)

\( (d\theta ^{a}+w^{a}_{b}\theta ^{b})(e_{b},e_{g}) \)

\( d\theta ^{a}=d(e^{a}_{u}dx^{u})=\partial _{v}e^{a}_{u}dx^{v}\wedge dx^{u} \)

so \( d\theta ^{a}(e_{b},e_{g})=\partial _{v}e^{a}_{u}(e^{v}_{b}e^{u}_{g}-e^{u}_{b}e^{v}_{g}) \)

and \( w^{a}_{d}\wedge \theta ^{d}=w_{ed}^{a}\theta ^{e}\wedge \theta ^{g} \)

The second cartan structure equation is: \( R^{a}_{b}=dw^{a}_{b}+w^{a}_{g}\wedge w^{g}_{b} \)

\end{document}
