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 $d$ and then project the result on the horizontal subspace. This leads to a new operator $d_H$.
I think that it has to do with the typical situation of a function with parameters (like in high school). Consider $f(x)=ax^3+x^2-3x+1$. We have variables $x$ and $a$, but they are not of the same "status". Students ask why we derive $x$ but not $a$...
Even in function of several variables
$$ f(x,y)=ax^2+y^2 $$they don't have the same "status": $x$ and $y$ are more "variables" and $a$ is more fixed.
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...
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Author of the notes: Antonio J. Pan-Collantes
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