Graded bundles and homogeneity structures

TL;DR

This paper introduces graded bundles as a generalization of vector bundles, characterizes them via homogeneity structures, and explores their local triviality and higher-order generalizations.

Contribution

It formalizes graded bundles and homogeneity structures, establishing their equivalence and introducing double graded bundles as a broad generalization.

Findings

Graded bundles generalize higher tangent bundles.

Homogeneity structures provide an equivalent description of graded bundles.

Double graded bundles are locally trivial with compatible homogeneous coordinates.

Abstract

We introduce the concept of a graded bundle which is a natural generalization of the concept of a vector bundle and whose standard examples are higher tangent bundles T^nQ playing a fundamental role in higher order Lagrangian formalisms. Graded bundles are graded manifolds in the sense that we can choose an atlas whose local coordinates are homogeneous functions of degrees 0,1,...,n. We prove that graded bundles have a convenient equivalent description as homogeneity structures, i.e. manifolds with a smooth action of the multiplicative monoid of non-negative reals. The main result states that each homogeneity structure admits an atlas whose local coordinates are homogeneous. Considering a natural compatibility condition of homogeneity structures we formulate, in turn, the concept of a double (r-tuple, in general) graded bundle - a broad generalization of the concept of a double (r-tuple) vector bundle. Double graded bundles are proven to be locally trivial in the sense that we can find local coordinates which are simultaneously homogeneous with respect to both homogeneity structures.