In this lecture by Fredric Schuller it is said that the wave-function is not a wave-function. He attempts to find an appropriate coordinate independent derivative using "chart calculations" on the level of base manifold.
Application: Quantum mechanics on curved spaces - Lec 26 - Frederic Schuller
We are shown that what we call a wave function is indeed a pull back of a section of the associated bundle to the frame bundle or a vector fiber valued function on the frame bundle which is the principal bundle.
At the end of the lecture the co-variant derivative is found on the base manifold on a local chart. However, he explains that using this "chart calculation", we should go up in the frame bundle to find the connection one form living on there and also to find the exterior co-variant derivative on the frame bundle. But the lecture ends here and the rest of derivation is not shown.
Now the question is that where can I find the rest of the derivation and see how using chart calculation on the base manifold we can find the connection, the vector fiber valued function, and the exterior co-variant derivative up on the frame bundle?