The Cartan structure equations for a connection and various associated 1-forms can be checked in a straightforward algebraic manner. But is there a geometric or global significance to the equations- can one visualize the proof?



What is the geometric significance of Cartan's structure equations?
There is a nice grand story behind all this. I don't know if you like thinking that way, but things do clarify when one looks at it from a more general perspective of [oo-Lie algebroid valued differential forms](http://ncatlab.org/schreiber/show/%E2%88%9E-Lie+algebroid+valued+differential+forms) with [curvature](http://ncatlab.org/schreiber/show/curvature+of+%E2%88%9E-Lie+algebroid+valued+differential+forms).


I am still working on these entries, trying to expose some of my work with Jim Stasheff and Hisham Sati. If you bug me with questions, there is a good chance that I'll improve the exposition taylor-made for your needs.


