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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 16th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexwrote a definition and short discussion of [[covariant derivative]] in the spirit of oo-Chern-Weil theory

   wrote a definition and short discussion of covariant derivative in the spirit of oo-Chern-Weil theory

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 16th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 16th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexokay, I had a chance to go through the entry again and harmonize the index-notation.

   okay, I had a chance to go through the entry again and harmonize the index-notation.

4. • CommentRowNumber4.
   • CommentAuthorEric
   • CommentTimeSep 17th 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexParallel 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorDmitri Pavlov
   • CommentTimeSep 12th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexReplaced broken link. <a href="https://ncatlab.org/nlab/revision/diff/covariant+derivative/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/covariant+derivative/18">v18</a>, <a href="https://ncatlab.org/nlab/show/covariant+derivative">current</a>

   Replaced broken link.

   diff, v18, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMar 12th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Jan Schouten]], *On the Conditions of Integrability of Covariant Differential Equations*, Transactions of the American Mathematical Society, **27** 4 (1925) 441-473 &lbrack;[doi:10.2307/1989236](https://doi.org/10.2307/1989236), [jstor:1989236](https://www.jstor.org/stable/1989236)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/covariant+derivative/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/covariant+derivative/20">v20</a>, <a href="https://ncatlab.org/nlab/show/covariant+derivative">current</a>

   added pointer to:

   • Jan Schouten, On the Conditions of Integrability of Covariant Differential Equations, Transactions of the American Mathematical Society, 27 4 (1925) 441-473 [doi:10.2307/1989236, jstor:1989236]

   diff, v20, current