Exterior covariant derivative
Schuller_2013 page 197
Olver_2014 page 31
Olver_1995 page 123-124
Idea: when we have a decomposition of the cotangent bundle in horizontal and vertical subspaces, we can use the usual exterior derivative operator
I think that it has to do with the typical situation of a function with parameters (like in high school). Consider
Even in function of several variables
they don't have the same "status":
I think that the exterior covariant derivative is an "external device" that we introduce in order to specify this distinction. Indeed I guess that the connection itself has this goal...