I recently came across the Darboux Derivative $\omega _f\equiv f^* \omega_G$, where $f:M \to G$ maps from a manifold to a Lie group, $f^*$ is its pullback, and $\omega _G$ is the Maurer-Cartan form of the Lie group $\omega _G(X)\equiv (T_g L_g)^{-1} X$ (Also, the Wikipedia article said that $T_g L_g$ is the derivative of the left multiplication function, but what does that mean and why is it written with a T?).
The Wikipedia article gives a local generalization of the fundamental theorem of calculus, but I’m not able to find the global generalization it talked about.
Seeing as there’s almost nothing I could find on the subject, anything would be appreciated.
Best Answer
Although I have not before seen Darboux’s name on this, this is a standard construction in differential geometry and applications of this together with the Frobenius integrability theorem go back certainly to E. Cartan. For a beautiful discussion, I recommend "On Cartan's Method of Lie Groups and Moving Frames as Applied to Uniqueness and Existence Questions in Differential Geometry," by Phillip Griffiths (Duke Math J. 41(4)(1974), pp. 775-814.
You can also find discussions in Spivak's Differential Geometry, Chern/Chen/Lam Lectures on Differentail Geometry, and numerous other places.
A typical example is the Fundamental Theorem of Surface Theory. One can prove existence/uniqueness (up to rigid motion) of a surface in $\Bbb R^3$ with prescribed first and second fundamental forms exactly by observing that the integrability conditions are given by pulling back the Maurer-Cartan form on $SO(3)$ to the frame bundle of the surface.