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

Notes

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