## A Nijenhuis on the Formal Definition of Natural Bundles

This appendix contains excerpts from Nijenhuis (1994, pp. 109–110).

Natural bundles …are defined through functors that, for each type of geometric object, associate a fiber bundle with each manifold. To formalize this, we need two categories.

First, consider the category of -dimensional (smooth) manifolds. Its morphisms are diffeomorphisms (into). Every open set of an -manifold belongs to : the theory is fundamentally a local one.

Second, consider the category of fibered manifolds where are manifolds and is a surjective submersion. The inverse images , for , are the fibers, is the total space, and the base space. The morphisms of are the fiber-preserving (smooth) maps. The base functor assigns to each fibered manifold its base manifold and to each morphism in the induced map on the base spaces.

With these definitions, a bundle functor on , or a natural bundle over -manifolds, is a covariant functor with these simple properties:

(1) (Prolongation) The base space of the fibered manifold is itself.

(2) (Locality) If is an open subset of , then the total space of is , the part of above .

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The concept of natural bundle was first formalized by the reviewer. It was little more than a definition, however, until some real theorems were proved. Work by Epstein and Thurston shows that natural bundles are of finite order (i.e., the structure group is a homomorphic image of finite-order jets of diffeomorphisms of fixing the origin). In addition, their work shows that natural bundles have a natural smooth structure that automatically satisfies a regularity condition (not stated here) in the original definition of natural bundle. Basic to all of this is Peetre’s Theorem, with a number of refinements. Other fundamental work, initiated by Palais and Terng, deals with the classification of natural vector bundles.

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The pursuit of D[ifferntial]G[eometry] consists to a large extent of performing operations on sections of natural bundles. Connections are constructed from Riemann metrics, covariant derivatives are taken, Lie brackets of vector fields are formed, etc. These operations are natural; they commute with point transformations, yield smooth sections in natural bundles from the same, and have non-increasing supports. All such natural operations are, as implied by Peetre-like theorems, of finite order and so induce natural transformations between corresponding jet bundles.