Graded bundles and homogeneity structures

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 n Q 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 ( R ≥ 0 , ⋅ ) 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.