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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2010

    polished and expanded Ehresmann connection

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeNov 30th 2011
    • (edited Nov 30th 2011)

    I have improved (hopefully) the section on the definition via horizontal subspaces at Ehresmann connection. On the other hand, I think (and wikipedia agrees) that the statement about the terminology is wrong at two places. One is the statement in the entry that the Ehresmann connection must be on a principal bundle (but must be on a fiber bundle) to be called such and another is suspicious phrase “Cartan-Ehresmann connection”, in my opinion Cartan connection is by the definition in a smaller generality then Ehresmann.

    Finally the Ehresmann connections on a principal and its associated bundles are in 1-1 correspondence: If T HPTPT^H P\subset T P is the smooth horizontal distrubution of subspaces defining the principal connection on a principal GG-bundle PP over XX, where GG is a Lie group and FF a smooth left GG-space, then consider the total space P× GFP\times_G F of the associated bundle with typical fiber FF. Then, for a fixed fFf\in F one defines a map ρ f:PP× GF\rho_f : P\to P\times_G F assigning the class [p,f][p,f] to pPp\in P. If (T pρ f)(T p HP)=:T [p,f] HP× GF(T_p \rho_f)(T^H_p P) =: T_{[p,f]}^H P\times_G F defines the horizontal subspace T [p,f] HP× GFT [p,f]P× GFT_{[p,f]}^H P\times_G F\subset T_{[p,f]} P\times_G F, the collection of such subspaces does not depend on the choice of (p,f)(p,f) in the class [p,f][p,f], and the correspondence pT [p,f] HP× GFp\mapsto T_{[p,f]}^H P\times_G F is a connection on the associated bundle P× GFXP\times_G F\to X. I added now this reasoning to the entry as well.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2012
    • (edited Jun 28th 2012)

    I have added to Ehresmann connection a pointer to the formalization of flat Ehresmann connections in cohesive homotopy type theory.

    I have just posted a little more chat about this here to the nnCafé.

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 14th 2021

    Added a link to the PDF file of the original paper.

    diff, v19, current

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeFeb 16th 2021

    Urs, 3: link to nnCafé seem not to work.

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 16th 2021
    • (edited Feb 16th 2021)

    I wrote:

    Even if you go to the café and find that day’s entry, the follow up gives the same error. The problem is thus in the Café.

    but when I checked back the problem did not seem to exist any more. Strange.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeFeb 17th 2021

    It does again. Though the main page works.

    • CommentRowNumber8.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 17th 2021

    There is no problem for me at the moment.