Natural bundle

In mathematics, a natural bundle is any fiber bundle associated to the s-frame bundle $F^{s}(M)$ for some $s\geq 1$ . It turns out that its transition functions depend functionally on local changes of coordinates in the base manifold $M$ together with their partial derivatives up to order at most $s$ .^[1]

The concept of a natural bundle was introduced by Albert Nijenhuis as a modern reformulation of the classical concept of an arbitrary bundle of geometric objects.^[2]

An example of natural bundle (of first order) is the tangent bundle $TM$ of a manifold $M$ .

Notes

^ Palais, Richard; Terng, Chuu-Lian (1977), "Natural bundles have finite order", Topology, 16: 271–277, doi:10.1016/0040-9383(77)90008-8, hdl:10338.dmlcz/102222
^ A. Nijenhuis (1972), Natural bundles and their general properties, Tokyo: Diff. Geom. in Honour of K. Yano, pp. 317–334

References

Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2017-08-15
Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, ISBN 80-210-0165-8
Saunders, D.J. (1989), The geometry of jet bundles, Cambridge University Press, ISBN 0-521-36948-7

[1] Palais, Richard; Terng, Chuu-Lian (1977), "Natural bundles have finite order", Topology, 16: 271–277, doi:10.1016/0040-9383(77)90008-8, hdl:10338.dmlcz/102222

[2] A. Nijenhuis (1972), Natural bundles and their general properties, Tokyo: Diff. Geom. in Honour of K. Yano, pp. 317–334

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