User:Amben/sandbox

$$ \begin{align} \text{d}\pi &= \theta^i \varepsilon_i\\ \text{d}\varepsilon_i &= {\omega^k}_i \varepsilon_k \end{align} $$

And then

$$ \begin{align} 0 = \text{d}^2 \pi &= (\text{d} \theta^i)\varepsilon_i + (\text{d}\varepsilon_i) \wedge \theta^i\\ &= (\text{d} \theta^i)\varepsilon_i + ({\omega^k}_i \varepsilon_k) \wedge \theta^i\\ &= (\text{d} \theta^i)\varepsilon_i + ({\omega^i}_j \varepsilon_i) \wedge \theta^j\\ &= (\text{d} \theta^i+ {\omega^i}_j \wedge \theta^j) \varepsilon_i \end{align} $$

as well as

$$ \begin{align} 0 = \text{d}^2 \varepsilon_i &= (\text{d}{\omega^k}_i) \varepsilon_k + (\text{d}\varepsilon_k) \wedge {\omega^k}_i\\ &= (\text{d}{\omega^k}_i) \varepsilon_k + ({\omega^l}_k \varepsilon_l) \wedge {\omega^k}_i\\ &= (\text{d}{\omega^k}_i) \varepsilon_k + ({\omega^k}_j \varepsilon_k) \wedge {\omega^j}_i\\ &= (\text{d}{\omega^k}_i + {\omega^k}_j \wedge {\omega^j}_i ) \varepsilon_k. \end{align} $$