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 \((\mathbb {R},\cdot )\) of multiplicative reals. Vector bundles are particular cases of homogeneity structures and weighted structures on them we call \(\mathrm{V\!B}\)-structures. In the case of Lie algebroids and Lie groupoids, the weighted structures include the concepts of \(\mathrm{V\!B}\)-algebroids and \(\mathrm{V\!B}\)-groupoids, intensively studied recently in the literature. Investigating various weighted structures, we prove some interesting results about their properties.