mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a natural bundle is any

fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...

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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

M

together with their partial derivatives up to order at most

s

. 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. An example of natural bundle (of first order) is the

tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...

TM

of a manifold

M

Notes

References

* * * {{citation, last1 = Saunders, first1 = D.J., title = The geometry of jet bundles, year = 1989, publisher = Cambridge University Press, isbn = 0-521-36948-7, url-access = registration, url = https://archive.org/details/geometryofjetbun0000saun Differential geometry Manifolds Fiber bundles