On the concept of a filtered bundle

TL;DR

This paper introduces filtered bundles as a generalization of graded bundles, relaxing transformation laws, and explores their structures, including double filtered bundles, with applications in geometric mechanics and field theory.

Contribution

It defines filtered bundles with polynomial transformation laws and introduces double filtered bundles, expanding the framework of geometric structures used in mechanics and field theory.

Findings

Filtered bundles generalize graded bundles with more flexible coordinate transformations.

Double filtered bundles extend double vector and affine bundles.

Linearisation of filtered bundles is well-defined and useful.

Abstract

We present the notion of a filtered bundle as a generalisation of a graded bundle. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of the coordinates by allowing more general polynomial transformation laws. The key examples of such bundles include affine bundles and various jet bundles, both of which play fundamental roles in geometric mechanics and classical field theory. We also present the notion of double filtered bundles which provide natural generalisations of double vector bundles and double affine bundles. Furthermore, we show that the linearisation of a filtered bundle - which can be seen as a partial polarisation of the admissible changes of local coordinates - is well defined.