Connection, soldering, curvature and torsion forms

If we introduce, as in Section 1.4, a local parametrization of $\mathcal{S}_n$, starting from the explicit formulas (1.9) and (1.31) for the vector fields $A_{\alpha}$ and from the definition (1.53), we can compute the following explicit expressions of the forms $\omega^{\beta}$:

$$\omega^i = e_{\lambda}^{i} d x^{\lambda}.$$ (1.56)

$$\omega^{[ik]} = g^{kj} e^i_{\mu} (d e^{\mu}_j + e^{\nu}_j \Gamma_{\nu \lambda}^{\mu} d x^{\lambda}),$$ (1.57)

$$\omega^a = \chi^a - D^a{}_b(g^{-1}) a_{\lambda}^b d x^{\lambda},$$ (1.58)

where $\chi^a$ is the Maurer-Cartan form of $\mathcal{G}$ defined in Section 1.4. One can show by means of eq. (1.1) that the expression (1.57) is antisymmetric in the indices $i, k$. It is a useful exercise to show that eq. (1.54) is satisfied.

The one-forms $\omega^{[ik]}$ and $\omega^a$ are called the components of the connection form, which takes its values in $\mathcal{T}_V$, namely in the Lie algebra of the structural group. The one-forms $\omega^i$ are called the components of the soldering form, also called the canonical form, that takes its values in $\mathcal{T}_H$.

Other useful quantities defined in the literature on fibre bundles [25,26,27,28] are the curvature form, a two-form taking values in $\mathcal{T}_V$, with components

$$\Omega^{[ik]} = - 2^{-1} F_{jl}^{[ik]} \omega^j \wedge \omeg... ... \hat F_{[jl] [mn]}^{[ik]} \omega^{[jl]} \wedge \omega^{[mn]},$$ (1.59)

$$\Omega^a = - 2^{-1} F_{jl}^a \omega^j \wedge \omega^l = d \omega^a + 2^{-1} \hat F_{bc}^a \omega^b \wedge \omega^c,$$ (1.60)

and the torsion form, a two-form taking values in $\mathcal{T}_H$ with components

$$\Omega^{i} = - 2^{-1} F_{jl}^{i} \omega^j \wedge \omega^l =... ...ga^i + 2^{-1} \hat F_{[kl] j}^i \omega^{[kl]} \wedge \omega^j.$$ (1.61)

Note that these formulas, called structure equations agree with eq. (1.54).