Natural bundle


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.

Definition

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 is a functor satisfying the following three properties:
  1. , i.e. is a fibred manifold over, with projection denoted by ;
  2. if is an open submanifold, with inclusion map, then coincides with, and is the inclusion ;
  3. for any smooth map such that is a local diffeomorphism for every, then the function is smooth.
As a consequence of the first condition, one has a natural transformation.

Finite order natural bundles

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.

Natural \Gamma-bundles

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.

Examples

An example of natural bundle 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.