Natural bundle
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In mathematics, a natural bundle is any fiber bundle associated to the s-frame bundle [math]\displaystyle{ F^s(M) }[/math] for some [math]\displaystyle{ s \geq 1 }[/math]. It turns out that its transition functions depend functionally on local changes of coordinates in the base manifold [math]\displaystyle{ M }[/math] together with their partial derivatives up to order at most [math]\displaystyle{ s }[/math].[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 [math]\displaystyle{ TM }[/math] of a manifold [math]\displaystyle{ M }[/math].
Notes
- ↑ Palais, Richard; Terng, Chuu-Lian (1977), "Natural bundles have finite order", Topology 16: 271–277, doi:10.1016/0040-9383(77)90008-8
- ↑ 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) (PDF), Natural operators in differential geometry, Springer-Verlag, http://www.emis.de/monographs/KSM/kmsbookh.pdf, 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, https://archive.org/details/geometryofjetbun0000saun
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