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nLab > Latest Changes: natural bundle

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2014

    Wrote some minimum at natural bundle.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeSep 15th 2014
    • (edited Sep 15th 2014)

    I slightly object to calling it a “type of bundle”, as it is not a single bundle but a functorial assignement of a bundle to a manifold, for suitable category of manifolds. It is like saying that a characteristic function is a number. No, it is an assignement which assigns a number to an element which indicates if the element is in a given subset. (Of course, some bundles can be said not to be in image of any natural functor, as their structure group is such, but still…)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 15th 2014

    But a characteristic function is a type of function. Here the tangent bundle is a type of bunde (or kind of bundle or the like).

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeSep 15th 2014

    Remark 3 makes no sense to me. I agree that natural bundle is a type of functor to bundles. Parallelly char fcn is a fcs, but it is not a number. Its value on an element is a number. But you do not say that characteristic function s a type of a number, It is a function from a set to numberS. So it is function. Not number. Natural bundle s a functoR from manifolds to bundleS. So it s a functor, Not bundle.
    Otherwise it makes no sense to apply the definition.

    “The tangent bundle” ? Which the ? For which concrete manifold ?

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeSep 16th 2014

    It's a manifold-dependent bundle, a bundle in the context of a free variable for a manifold.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeSep 16th 2014

    Yes, Toby, but qualification that it is a type of a bundle would mean that the type/kind/subsort is specified in the domain. But, no – there are no restriction on kind of a bundle in domain, the restriction is on a kind of depending. It is not about classification of bundles but classification of free variable behaviour…

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 2nd 2023

    Added:

    Originally introduced in

    • Albert Nijenhuis, Geometric aspects of formal differential operations on tensor fields, Proceedings of the International Congress of Mathematicians 1958, 463–469.

    • Albert Nijenhuis, Natural bundles and their general properties, in: Differential Geometry (in honor of Kentaro Yano), Kinokuniya, Tokyo, 1972, pp. 317–334.

    A comprehensive reference is available in

    diff, v8, current

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 4th 2023

    Added links to PDF files.

    diff, v11, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2023

    added pointer to:

    diff, v12, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2023

    added pointer to:

    diff, v12, current

1 to 10 of 10

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