added pointer to:

- Demeter Krupka, Josef Janyška, Part 2 of:
*Lectures on differential invariants*, Univerzita J. E. Purkyně, Brno (1990) [ISBN:80-210-165-8, researchgate]

added pointer to:

- Chuu-Lian Terng,
*Natural vector bundles and natural differential operators*, Amer. J. Math. 100 (1978) 775-828 [doi:10.2307/2373910, jstor:2373910]

Added links to PDF files.

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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

]]>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…

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

]]>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 ?

]]>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).

]]>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…)

]]>Wrote some minimum at *natural bundle*.