Graded bundles and homogeneity structures

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 TnQTnQ 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,…,n0,1,…,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,⋅)(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   (rr-tuple, in general) graded bundle  –a broad generalization of the concept of a double (rr-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.

► The concept of a graded bundle is proposed. ► This generalizes the concept of vector bundles. ► Canonical examples are higher tangent bundles. ► An equivalent description is given via homogeneity structures. ► Applications are also given for double structures.