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nLab > Latest Changes: connection on a bundle

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2010

    added to connection on a bundle

    • a Definition (nPOV-flavor, of course)

    • a Properties-section with statement and proof of the fact that every bundle does admit a connection.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJul 29th 2010

    Good. There should be some remark stating that the existence of connection is in a smooth category but not in analytic category nor in algebraic category. In this vain, I have checked a bit about Atiyah Lie algebroid and corrected few things around. For example Lie algebroid had an incomplete definition in explicit terms: the anchor is not only a vector bundle map, but its differential respects the brackets (please correct my notation if you have better). I added few references including the Pradines 1967 with explicit definition of Lie algebroid. Of course, Atiyah’s work is 1967, and the first references on Lie pseudoalgebras aka Lie-Rinehart algebras/pairs are 1953 and 1955 and the Rinehart’s paper itself is 1963 and many more coming in those years. In any case the Courant late 1980s paper is far not among the first references on Lie algebroids, but popularity and width of research sparked more after it, especially in non-French literature. I hope you agree with the corrections and literature enhancements.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2010
    • (edited Jul 29th 2010)

    but not in analytic category

    Yes, I was thinking about the need to re-write the entry with more general assumptions on everything. We can eventually do that.

    As the existence proof shows, it relies crucially on the existence of a partition of unity subordinate to a good cover, and works precisely if such exists.

    Generally speaking, I can give a definition of connection so general that it will work in cases even undreamed of, but for the moment I wanted to record some standard facts, in order to be able to point to them in my examples-section, for comparison.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJul 29th 2010
    • (edited Jul 29th 2010)

    Wow, this is a good insight, that the proof still goes when one can do the splitting on a partition. This local splitting is always about the Galois type-theory with connection or without it. I wish the current effort of John about separable algebras and Igor’s about bicategorical Glaois theory ends up with some nlab use :) I mean somebody competent should write some content into my stub categorical Galois theory which has so far only references.

    Generally speaking, I can give a definition of connection so general that it will work in cases even undreamed of

    You are giving us a dangerous appetizer!

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 31st 2010

    Igor’s about bicategorical Galois theory

    is this secret, or contained in some public work of his?

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeAug 2nd 2010

    Igor was just writing these things in last 3-4 weeks, email him if he has some draft version or will have soon. He is hoping of having a final version by the end of the summer, but he is doing 3-4 projects in parallel.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeMar 9th 2011
    • (edited Mar 9th 2011)
    S. Morita (Geometry of characteristic classes, page 78) says that not every connection in the sense of Ehresmann i.e. in the sense of a distribution of horizontal subspaces gives a parallel transport if the base manifold is not compact; i.e. a parallel transport is defined along some but not all smooth curves, unlike in the compact case. If the parallel translation is defined for all curves, he says that the Ehresmann connection is strict.
    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 9th 2011

    He is hoping of having a final version by the end of the summer

    @Zoran - I did email Igor about this, but got no response. I don’t suppose you know if this was finished or not?

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeMar 9th 2011
    I do not think he was recently working quite on that subject. I hope he will let us know soon.
    • CommentRowNumber10.
    • CommentAuthorjim_stasheff
    • CommentTimeMar 9th 2011
    S. Morita (Geometry of characteristic classes, page 78) - I can't access now
    Could you give us a counterexample
    Apparently the smooth case is even more different from the topologicial
    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeOct 18th 2011
    • (edited Oct 18th 2011)

    I changed the “principal connection on XX” to “affine connection on XX” in Cartan connection for formal reason. Principal connection on XX does not make sense to me as XX is not a bundle. Principal connection may be on the frame bundle of XX, and this is called or equivalent to what is usually called an affine connection.

    Jim: did I send you the book ? If not let me know, I will send it.

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