Natural bundle

In mathematics, a natural bundle is any fiber bundle associated to the s-frame bundle for some . It turns out that its transition functions depend functionally on local changes of coordinates in the base manifold together with their partial derivatives up to order at most .[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 of a manifold .

Notes

  1. ↑ Palais, Richard; Terng, Chuu-Lian (1977), "Natural bundles have finite order", Topology, 16: 271–277
  2. ↑ 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
  • 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
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