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wrote a definition and short discussion of covariant derivative in the spirit of oo-Chern-Weil theory
I have now considerably expanded the entry covariant derivative providing now a full derivation of the equivalence of the definition I give with the standard one, in particular a derivation of the standard $\nabla \nabla \sigma = F_\nabla \sigma$.
I also added some guiding comments on what’s going on. Should be readable now.
But I am in a huge rush now and might have introduced some symbol mix up when I decided to switch the names of my inices for the local formulas. Will fix that later, if need be.
okay, I had a chance to go through the entry again and harmonize the index-notation.
Parallel transport is one of the things I’ve spent years meditating about and think I have a decent intuitive feel for. I was never very happy with the traditional presentations you’d find in standard texts. Of course, none of those were from the nPOV. I like the general presentation at covariant derivative but will need to meditate on it before it sinks in. I hope the ideas presented there become standard in introductory texts. It “feels” better.
How much “higher” category theory would be needed to cover most of the standard material and rewrite some introductory differential geometry texts? For example, I learned formally from Boothby, but spent most of my time with Nakahara and Frankel.
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