Torsion-free tetrad equivalence to Levi-Civita

For smooth vector fields $X,Y$ on $U$, evaluating exterior derivative and the wedge product gives $T^I(X,Y)=X(e^I(Y))-Y(e^I(X))-e^I([X,Y])+{\omega }^I{}_J(X)e^J(Y)-{\omega }^I{}_J(Y)e^J(X)$. Since $Y=e^J(Y)E_J$, the connection definition gives $e^I({\mathrm{\nabla }}_{X}Y)=X(e^I(Y))+{\omega }^I{}_J(X)e^J(Y)$; the corresponding formula with $X,Y$ exchanged shows $T^I(X,Y)=e^I({\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X-[X,Y])$. Thus all coframe torsion forms vanish exactly when the induced affine-connection torsion vanishes.

Write $\alpha :=d\mathrm{\Lambda }{\mathrm{\Lambda }}^{-1}$ and $\bar{\omega }:=\mathrm{\Lambda }\omega {\mathrm{\Lambda }}^{-1}$, so ${\omega }^{\mathrm{\prime }}=\bar{\omega }-\alpha$. By [F3] applied componentwise and [F4], differentiating $\mathrm{\Lambda }{\mathrm{\Lambda }}^{-1}=I$ gives $d{\mathrm{\Lambda }}^{-1}=-{\mathrm{\Lambda }}^{-1}d\mathrm{\Lambda }{\mathrm{\Lambda }}^{-1}$ and then $d\alpha =d^2\mathrm{\Lambda }{\mathrm{\Lambda }}^{-1}-d\mathrm{\Lambda }\wedge d{\mathrm{\Lambda }}^{-1}=\alpha \wedge \alpha \mathrm{.}$ The same graded product rule gives $d\bar{\omega }=d\mathrm{\Lambda }\wedge \omega {\mathrm{\Lambda }}^{-1}+\mathrm{\Lambda }d\omega {\mathrm{\Lambda }}^{-1}-\mathrm{\Lambda }\omega \wedge d{\mathrm{\Lambda }}^{-1}=\alpha \wedge \bar{\omega }+\mathrm{\Lambda }(d\omega ){\mathrm{\Lambda }}^{-1}+\bar{\omega }\wedge \alpha \mathrm{.}$ Expanding $d{\omega }^{\mathrm{\prime }}+{\omega }^{\mathrm{\prime }}\wedge {\omega }^{\mathrm{\prime }}$ and substituting these identities cancels the $\alpha$ terms and leaves $\mathrm{\Lambda }(d\omega +\omega \wedge \omega ){\mathrm{\Lambda }}^{-1}$, hence $F^{\mathrm{\prime }}=\mathrm{\Lambda }F{\mathrm{\Lambda }}^{-1}$. Also [F3] gives $d{e}^{\mathrm{\prime }}=d\mathrm{\Lambda }\wedge e+\mathrm{\Lambda }de$ and ${\omega }^{\mathrm{\prime }}\wedge e^{\mathrm{\prime }}=\mathrm{\Lambda }\omega \wedge e-d\mathrm{\Lambda }\wedge e$, hence $T^{\mathrm{\prime }}=\mathrm{\Lambda }T$.

Restrict the unique Levi-Civita connection supplied by [F2] to $U$. Its connection coefficients in $E$ are smooth; metric compatibility and the constant frame metric ${\eta }_{IJ}$ give ${\omega }_{IJ}=-{\omega }_{JI}$. Its tangent torsion is zero, so step 1.1 makes every $T^I$ zero. This proves existence of an $\eta$-skew solution.

Conversely, an $\eta$-skew $\omega$ induces a metric-compatible connection. For $V=V^I{E}_I$ and $W=W^J{E}_J$, the connection terms in $g({\mathrm{\nabla }}_{X}V,W)+g(V,{\mathrm{\nabla }}_{X}W)$ are ${\omega }_{KJ}{V}^J{W}^K+{\omega }_{IJ}{V}^I{W}^J=0$ after interchanging the summed indices in the first term and using ${\omega }_{IJ}=-{\omega }_{JI}$; the remaining terms equal $X(g(V,W))$. If also $T^I=0$, step 1.1 gives zero tangent torsion. By uniqueness in [F2], this connection is the Levi-Civita connection. Hence the solution from step 2.1 is unique, and the displayed torsion equation is equivalent to torsion-freeness.

Steps 1.1–3.1 establish the torsion equivalence and unique Levi-Civita identification; step 1.2 establishes both transformation laws.