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VB-structures and generalizations

Janusz Grabowski, Katarzyna Grabowska, Zohreh Ravanpak

Abstract

Motivated by properties of higher tangent lifts of geometric structures, we introduce concepts of weighted structures for various geometric objects on a manifold F equipped with a homogeneity structure. The latter is a smooth action on F of the monoid of multiplicative reals. Vector bundles are particular cases of homogeneity structures with the action being the multiplication of vectors by reals, and weighted structures on them we call VB-structures. In the case of Lie algebroids and Lie groupoids, the weighted structures include the concepts of VB-algebroids and VB-groupoids, intensively studied recently in the literature. Investigating various weighted structures, we prove some interesting results about their properties.

VB-structures and generalizations

Abstract

Motivated by properties of higher tangent lifts of geometric structures, we introduce concepts of weighted structures for various geometric objects on a manifold F equipped with a homogeneity structure. The latter is a smooth action on F of the monoid of multiplicative reals. Vector bundles are particular cases of homogeneity structures with the action being the multiplication of vectors by reals, and weighted structures on them we call VB-structures. In the case of Lie algebroids and Lie groupoids, the weighted structures include the concepts of VB-algebroids and VB-groupoids, intensively studied recently in the literature. Investigating various weighted structures, we prove some interesting results about their properties.

Paper Structure

This paper contains 23 sections, 49 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Graded bundles and homogeneity structures
  3. $\mathbb{Z}$-graded bundles
  4. Graded bundles
  5. Homogeneity structures
  6. Double graded bundles
  7. Tangent and phase lifts of homogeneity structures
  8. Duality, sections and tensor products of $\operatorname{ZGL}$-bundles
  9. Duality
  10. The degree of sections
  11. Tensor products
  12. Higher lifts of tensor fields and distributions
  13. Weighted structures
  14. Weighted tensor fields
  15. Weighted vector bundles and distributions
  16. ...and 8 more sections

Key Result

Proposition 2.4

Every $\mathbb{Z}$-graded bundle $焜:F\to M$ is canonically equipped with a globally defined weight vector field$\nabla_F$ which locally, in $\mathbb{Z}$-charts, looks like (lwvf). The weight vector field induces also a smooth action $h^F_t$, $t\ne 0$, of the multiplicative group $\mathbb{R}^\times$ If we use the convention that $0^w=0$ for $w\ne 0$, the above formula defines actually an action $h

Theorems & Definitions (134)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Example 2.9
  • Remark 2.10
  • ...and 124 more