In differential geometry, a field in mathematics, a natural bundle is any fiber bundle associated to the higher order frame bundle , for some . In other words, its transition functions depend functionally on local changes of coordinates in the base manifold together with their partial derivatives up to order at most .
The concept of a natural bundle was introduced in 1972 by Albert Nijenhuis as a modern reformulation of the classical concept of an arbitrary bundle of geometric objects.
Let denote the category of smooth manifolds and smooth maps and the category of smooth -dimensional manifolds and local diffeomorphisms. Consider also the category 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 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 and every point , one hasNatural bundles of order coincide with the associated fibre bundles to the -th order frame bundles .
After various intermediate cases, it was proved by Epstein and Thurston that all natural bundles have finite order.
The notion of natural -bundle arises from that of natural bundle by restricting to the suitable categories of -manifolds and of -fibred manifolds, where is a pseudogroup. The case when is the pseudogroup of all diffeomorphisms between open subsets of recovers the ordinary notion of natural bundle.
Under suitable assumptions, natural -bundles have finite order as well.
An example of natural bundle (of first order) is the tangent bundle of a manifold .
Other examples include the cotangent bundles, the bundles of metrics of signature and the bundle of linear connections.