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.