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On the concept of a filtered bundle

Andrew James Bruce, Katarzyna Grabowska, Janusz Grabowski

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.

On the concept of a filtered bundle

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.

Paper Structure

This paper contains 15 sections, 21 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polynomial bundles
  4. Graded manifolds and graded bundles
  5. Filtered polynomial algebras
  6. Filtered Bundles
  7. The definition of a filtered bundle
  8. The category of filtered bundles
  9. Duality
  10. Bachelor--Gawȩdzki like-theorems
  11. Towers of affine bundles vs filtered bundles
  12. Multiple structures and linearisation
  13. Filtered-linear bundles
  14. Linearisation
  15. Double and multiple filtered structures

Key Result

Theorem 2.5

(Grabowski-Rotkiewicz Grabowski:2012) Any graded bundle $(F,\textnormal{h})$ is a locally trivial fibration $\tau:F\to M$ with a typical fiber ${\mathbb R}^\mathbf{d}$, for some $\mathbf{d}=(d_1,\dots,d_k)$, and the homogeneity structure locally equivalent to the one in Example e1, so that the trans

Theorems & Definitions (80)

  • Definition 2.1: Adapted from Bertram:2014 and Voronov:2010
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Example 2.7
  • Remark 2.8
  • Definition 2.9
  • Example 2.10
  • ...and 70 more