In differential geometry, a field in mathematics, a natural bundle is any fiber bundle associated to the s-frame bundle
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.
{\displaystyle Mf}
denote the category of smooth manifolds and smooth maps and
the category of smooth
-dimensional manifolds and local diffeomorphisms.
of fibred manifolds and bundle morphisms, and the functor
associating to any fibred manifold its base manifold.
A natural bundle (or bundle functor) is a functor
satisfying the following three properties: As a consequence of the first condition, one has a natural transformation
A natural bundle
{\displaystyle F:Mf_{n}\to Mf}
is called of finite order
if, for every local diffeomorphism
and every point
, the map
depends only on the jet
Equivalently, for every local diffeomorphisms
Natural bundles of order
coincide with the associated fibre bundles to the
-th order frame bundles
A classical result by Epstein and Thurston shows that all natural bundles have finite order.
[3] An example of natural bundle (of first order) is the tangent bundle
Other examples include the cotangent bundles, the bundles of metrics of signature
and the bundle of linear connections.