VB-structures and generalizations

TL;DR

This paper introduces weighted geometric structures on manifolds with homogeneity actions, generalizing vector bundle concepts and exploring properties of VB-structures, VB-algebroids, and VB-groupoids.

Contribution

It defines and studies weighted structures on manifolds with homogeneity actions, extending VB-structures to new geometric contexts and analyzing their properties.

Findings

Properties of weighted structures are established.

Weighted structures include VB-algebroids and VB-groupoids.

Results generalize existing concepts in geometric structures.

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.