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I am coming from this question, which has not being completely answered but I think is very interesting. In several works ([Chern], [Griffiths] and [Clelland]) the Maurer-Cartan form for $E(n)$ is ...

I'm basically wondering how to make "curved" the first column of the diagram $\require{AMScd}$ \begin{CD} P_1 @>\textrm{inclusion} >> G\\ @V \omega_0 V P_1\cap P_2 V @V\omega V P_2 V\\...

The question is about the family of tensors that are naturally associated to any nice Lie group. Take the Mauer-Cartan form, $\omega=g^{-1} dg$ and I would like to make the covariant index of this one-...